Article BCIALG_1, MML version 4.99.1005
:: BCIALG_1:structnot 1 => BCIALG_1:struct 1
definition
struct(1-sorted) BCIStr(#
carrier -> set,
InternalDiff -> Function-like quasi_total Relation of [:the carrier of it,the carrier of it:],the carrier of it
#);
end;
:: BCIALG_1:attrnot 1 => BCIALG_1:attr 1
definition
let a1 be BCIStr;
attr a1 is strict;
end;
:: BCIALG_1:exreg 1
registration
cluster strict BCIStr;
end;
:: BCIALG_1:aggrnot 1 => BCIALG_1:aggr 1
definition
let a1 be set;
let a2 be Function-like quasi_total Relation of [:a1,a1:],a1;
aggr BCIStr(#a1,a2#) -> strict BCIStr;
end;
:: BCIALG_1:selnot 1 => BCIALG_1:sel 1
definition
let a1 be BCIStr;
sel the InternalDiff of a1 -> Function-like quasi_total Relation of [:the carrier of a1,the carrier of a1:],the carrier of a1;
end;
:: BCIALG_1:exreg 2
registration
cluster non empty strict BCIStr;
end;
:: BCIALG_1:funcnot 1 => BCIALG_1:func 1
definition
let a1 be BCIStr;
let a2, a3 be Element of the carrier of a1;
func A2 \ A3 -> Element of the carrier of a1 equals
(the InternalDiff of a1) .(a2,a3);
end;
:: BCIALG_1:def 1
theorem
for b1 being BCIStr
for b2, b3 being Element of the carrier of b1 holds
b2 \ b3 = (the InternalDiff of b1) .(b2,b3);
:: BCIALG_1:structnot 2 => BCIALG_1:struct 2
definition
struct(BCIStr, ZeroStr) BCIStr_0(#
carrier -> set,
InternalDiff -> Function-like quasi_total Relation of [:the carrier of it,the carrier of it:],the carrier of it,
ZeroF -> Element of the carrier of it
#);
end;
:: BCIALG_1:attrnot 2 => BCIALG_1:attr 2
definition
let a1 be BCIStr_0;
attr a1 is strict;
end;
:: BCIALG_1:exreg 3
registration
cluster strict BCIStr_0;
end;
:: BCIALG_1:aggrnot 2 => BCIALG_1:aggr 2
definition
let a1 be set;
let a2 be Function-like quasi_total Relation of [:a1,a1:],a1;
let a3 be Element of a1;
aggr BCIStr_0(#a1,a2,a3#) -> strict BCIStr_0;
end;
:: BCIALG_1:exreg 4
registration
cluster non empty strict BCIStr_0;
end;
:: BCIALG_1:funcnot 2 => BCIALG_1:func 2
definition
let a1 be non empty BCIStr_0;
let a2 be Element of the carrier of a1;
func A2 ` -> Element of the carrier of a1 equals
(0. a1) \ a2;
end;
:: BCIALG_1:def 2
theorem
for b1 being non empty BCIStr_0
for b2 being Element of the carrier of b1 holds
b2 ` = (0. b1) \ b2;
:: BCIALG_1:attrnot 3 => BCIALG_1:attr 3
definition
let a1 be non empty BCIStr_0;
attr a1 is being_B means
for b1, b2, b3 being Element of the carrier of a1 holds
((b1 \ b2) \ (b3 \ b2)) \ (b1 \ b3) = 0. a1;
end;
:: BCIALG_1:dfs 3
definiens
let a1 be non empty BCIStr_0;
To prove
a1 is being_B
it is sufficient to prove
thus for b1, b2, b3 being Element of the carrier of a1 holds
((b1 \ b2) \ (b3 \ b2)) \ (b1 \ b3) = 0. a1;
:: BCIALG_1:def 3
theorem
for b1 being non empty BCIStr_0 holds
b1 is being_B
iff
for b2, b3, b4 being Element of the carrier of b1 holds
((b2 \ b3) \ (b4 \ b3)) \ (b2 \ b4) = 0. b1;
:: BCIALG_1:attrnot 4 => BCIALG_1:attr 4
definition
let a1 be non empty BCIStr_0;
attr a1 is being_C means
for b1, b2, b3 being Element of the carrier of a1 holds
((b1 \ b2) \ b3) \ ((b1 \ b3) \ b2) = 0. a1;
end;
:: BCIALG_1:dfs 4
definiens
let a1 be non empty BCIStr_0;
To prove
a1 is being_C
it is sufficient to prove
thus for b1, b2, b3 being Element of the carrier of a1 holds
((b1 \ b2) \ b3) \ ((b1 \ b3) \ b2) = 0. a1;
:: BCIALG_1:def 4
theorem
for b1 being non empty BCIStr_0 holds
b1 is being_C
iff
for b2, b3, b4 being Element of the carrier of b1 holds
((b2 \ b3) \ b4) \ ((b2 \ b4) \ b3) = 0. b1;
:: BCIALG_1:attrnot 5 => BCIALG_1:attr 5
definition
let a1 be non empty BCIStr_0;
attr a1 is being_I means
for b1 being Element of the carrier of a1 holds
b1 \ b1 = 0. a1;
end;
:: BCIALG_1:dfs 5
definiens
let a1 be non empty BCIStr_0;
To prove
a1 is being_I
it is sufficient to prove
thus for b1 being Element of the carrier of a1 holds
b1 \ b1 = 0. a1;
:: BCIALG_1:def 5
theorem
for b1 being non empty BCIStr_0 holds
b1 is being_I
iff
for b2 being Element of the carrier of b1 holds
b2 \ b2 = 0. b1;
:: BCIALG_1:attrnot 6 => BCIALG_1:attr 6
definition
let a1 be non empty BCIStr_0;
attr a1 is being_K means
for b1, b2 being Element of the carrier of a1 holds
(b1 \ b2) \ b1 = 0. a1;
end;
:: BCIALG_1:dfs 6
definiens
let a1 be non empty BCIStr_0;
To prove
a1 is being_K
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1 holds
(b1 \ b2) \ b1 = 0. a1;
:: BCIALG_1:def 6
theorem
for b1 being non empty BCIStr_0 holds
b1 is being_K
iff
for b2, b3 being Element of the carrier of b1 holds
(b2 \ b3) \ b2 = 0. b1;
:: BCIALG_1:attrnot 7 => BCIALG_1:attr 7
definition
let a1 be non empty BCIStr_0;
attr a1 is being_BCI-4 means
for b1, b2 being Element of the carrier of a1
st b1 \ b2 = 0. a1 & b2 \ b1 = 0. a1
holds b1 = b2;
end;
:: BCIALG_1:dfs 7
definiens
let a1 be non empty BCIStr_0;
To prove
a1 is being_BCI-4
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1
st b1 \ b2 = 0. a1 & b2 \ b1 = 0. a1
holds b1 = b2;
:: BCIALG_1:def 7
theorem
for b1 being non empty BCIStr_0 holds
b1 is being_BCI-4
iff
for b2, b3 being Element of the carrier of b1
st b2 \ b3 = 0. b1 & b3 \ b2 = 0. b1
holds b2 = b3;
:: BCIALG_1:attrnot 8 => BCIALG_1:attr 8
definition
let a1 be non empty BCIStr_0;
attr a1 is being_BCK-5 means
for b1 being Element of the carrier of a1 holds
b1 ` = 0. a1;
end;
:: BCIALG_1:dfs 8
definiens
let a1 be non empty BCIStr_0;
To prove
a1 is being_BCK-5
it is sufficient to prove
thus for b1 being Element of the carrier of a1 holds
b1 ` = 0. a1;
:: BCIALG_1:def 8
theorem
for b1 being non empty BCIStr_0 holds
b1 is being_BCK-5
iff
for b2 being Element of the carrier of b1 holds
b2 ` = 0. b1;
:: BCIALG_1:prednot 1 => BCIALG_1:attr 3
notation
let a1 be non empty BCIStr_0;
synonym a1 is_B for being_B;
end;
:: BCIALG_1:prednot 2 => BCIALG_1:attr 4
notation
let a1 be non empty BCIStr_0;
synonym a1 is_C for being_C;
end;
:: BCIALG_1:prednot 3 => BCIALG_1:attr 5
notation
let a1 be non empty BCIStr_0;
synonym a1 is_I for being_I;
end;
:: BCIALG_1:prednot 4 => BCIALG_1:attr 6
notation
let a1 be non empty BCIStr_0;
synonym a1 is_K for being_K;
end;
:: BCIALG_1:prednot 5 => BCIALG_1:attr 7
notation
let a1 be non empty BCIStr_0;
synonym a1 is_BCI-4 for being_BCI-4;
end;
:: BCIALG_1:prednot 6 => BCIALG_1:attr 8
notation
let a1 be non empty BCIStr_0;
synonym a1 is_BCK-5 for being_BCK-5;
end;
:: BCIALG_1:funcnot 3 => BCIALG_1:func 3
definition
func BCI-EXAMPLE -> BCIStr_0 equals
BCIStr_0(#1,op2,op0#);
end;
:: BCIALG_1:def 9
theorem
BCI-EXAMPLE = BCIStr_0(#1,op2,op0#);
:: BCIALG_1:funcreg 1
registration
cluster BCI-EXAMPLE -> non empty trivial strict;
end;
:: BCIALG_1:funcreg 2
registration
cluster BCI-EXAMPLE -> being_B being_C being_I being_BCI-4 being_BCK-5;
end;
:: BCIALG_1:exreg 5
registration
cluster non empty strict being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0;
end;
:: BCIALG_1:modenot 1
definition
mode BCI-algebra is non empty being_B being_C being_I being_BCI-4 BCIStr_0;
end;
:: BCIALG_1:modenot 2
definition
mode BCK-algebra is non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0;
end;
:: BCIALG_1:modenot 3 => BCIALG_1:mode 1
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
mode SubAlgebra of A1 -> non empty being_B being_C being_I being_BCI-4 BCIStr_0 means
0. it = 0. a1 &
the carrier of it c= the carrier of a1 &
the InternalDiff of it = (the InternalDiff of a1) || the carrier of it;
end;
:: BCIALG_1:dfs 10
definiens
let a1, a2 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
To prove
a2 is SubAlgebra of a1
it is sufficient to prove
thus 0. a2 = 0. a1 &
the carrier of a2 c= the carrier of a1 &
the InternalDiff of a2 = (the InternalDiff of a1) || the carrier of a2;
:: BCIALG_1:def 10
theorem
for b1, b2 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b2 is SubAlgebra of b1
iff
0. b2 = 0. b1 &
the carrier of b2 c= the carrier of b1 &
the InternalDiff of b2 = (the InternalDiff of b1) || the carrier of b2;
:: BCIALG_1:th 1
theorem
for b1 being non empty BCIStr_0 holds
b1 is non empty being_B being_C being_I being_BCI-4 BCIStr_0
iff
b1 is being_I &
b1 is being_BCI-4 &
(for b2, b3, b4 being Element of the carrier of b1 holds
((b2 \ b3) \ (b2 \ b4)) \ (b4 \ b3) = 0. b1 &
(b2 \ (b2 \ b3)) \ b3 = 0. b1);
:: BCIALG_1:prednot 7 => BCIALG_1:pred 1
definition
let a1 be non empty BCIStr_0;
let a2, a3 be Element of the carrier of a1;
pred A2 <= A3 means
a2 \ a3 = 0. a1;
end;
:: BCIALG_1:dfs 11
definiens
let a1 be non empty BCIStr_0;
let a2, a3 be Element of the carrier of a1;
To prove
a2 <= a3
it is sufficient to prove
thus a2 \ a3 = 0. a1;
:: BCIALG_1:def 11
theorem
for b1 being non empty BCIStr_0
for b2, b3 being Element of the carrier of b1 holds
b2 <= b3
iff
b2 \ b3 = 0. b1;
:: BCIALG_1:th 2
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Element of the carrier of b1 holds
b2 \ 0. b1 = b2;
:: BCIALG_1:th 3
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2, b3, b4 being Element of the carrier of b1
st b2 \ b3 = 0. b1 & b3 \ b4 = 0. b1
holds b2 \ b4 = 0. b1;
:: BCIALG_1:th 4
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2, b3, b4 being Element of the carrier of b1
st b2 \ b3 = 0. b1
holds (b2 \ b4) \ (b3 \ b4) = 0. b1 & (b4 \ b3) \ (b4 \ b2) = 0. b1;
:: BCIALG_1:th 5
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2, b3, b4 being Element of the carrier of b1
st b2 <= b3
holds b2 \ b4 <= b3 \ b4 & b4 \ b3 <= b4 \ b2;
:: BCIALG_1:th 6
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2, b3 being Element of the carrier of b1
st b2 \ b3 = 0. b1
holds (b3 \ b2) ` = 0. b1;
:: BCIALG_1:th 7
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 \ b3) \ b4 = (b2 \ b4) \ b3;
:: BCIALG_1:th 8
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2, b3 being Element of the carrier of b1 holds
b2 \ (b2 \ (b2 \ b3)) = b2 \ b3;
:: BCIALG_1:th 9
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2, b3 being Element of the carrier of b1 holds
(b2 \ b3) ` = b2 ` \ (b3 `);
:: BCIALG_1:th 10
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2, b3 being Element of the carrier of b1 holds
((b2 \ (b2 \ b3)) \ (b3 \ b2)) \ (b2 \ (b2 \ (b3 \ (b3 \ b2)))) = 0. b1;
:: BCIALG_1:th 11
theorem
for b1 being non empty BCIStr_0 holds
b1 is non empty being_B being_C being_I being_BCI-4 BCIStr_0
iff
b1 is being_BCI-4 &
(for b2, b3, b4 being Element of the carrier of b1 holds
((b2 \ b3) \ (b2 \ b4)) \ (b4 \ b3) = 0. b1 &
b2 \ 0. b1 = b2);
:: BCIALG_1:th 12
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
st for b2 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b3, b4 being Element of the carrier of b2 holds
b3 \ (b3 \ b4) = b4 \ (b4 \ b3)
holds b1 is non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0;
:: BCIALG_1:th 13
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
st for b2 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b3, b4 being Element of the carrier of b2 holds
(b3 \ b4) \ b4 = b3 \ b4
holds b1 is non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0;
:: BCIALG_1:th 14
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
st for b2 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b3, b4 being Element of the carrier of b2 holds
b3 \ (b4 \ b3) = b3
holds b1 is non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0;
:: BCIALG_1:th 15
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
st for b2 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b3, b4, b5 being Element of the carrier of b2 holds
(b3 \ b4) \ b4 = (b3 \ b5) \ (b4 \ b5)
holds b1 is non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0;
:: BCIALG_1:th 16
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
st for b2 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b3, b4 being Element of the carrier of b2 holds
(b3 \ b4) \ (b4 \ b3) = b3 \ b4
holds b1 is non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0;
:: BCIALG_1:th 17
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
st for b2 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b3, b4 being Element of the carrier of b2 holds
(b3 \ b4) \ ((b3 \ b4) \ (b4 \ b3)) = 0. b2
holds b1 is non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0;
:: BCIALG_1:th 18
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is being_K
iff
b1 is non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0;
:: BCIALG_1:funcnot 4 => BCIALG_1:func 4
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
func BCK-part A1 -> non empty Element of bool the carrier of a1 equals
{b1 where b1 is Element of the carrier of a1: 0. a1 <= b1};
end;
:: BCIALG_1:def 12
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
BCK-part b1 = {b2 where b2 is Element of the carrier of b1: 0. b1 <= b2};
:: BCIALG_1:th 19
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
0. b1 in BCK-part b1;
:: BCIALG_1:th 20
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2, b3 being Element of BCK-part b1 holds
b2 \ b3 in BCK-part b1;
:: BCIALG_1:th 21
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Element of the carrier of b1
for b3 being Element of BCK-part b1 holds
b2 \ b3 <= b2;
:: BCIALG_1:th 22
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is SubAlgebra of b1;
:: BCIALG_1:attrnot 9 => BCIALG_1:attr 9
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
let a2 be SubAlgebra of a1;
attr a2 is proper means
a2 <> a1;
end;
:: BCIALG_1:dfs 13
definiens
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
let a2 be SubAlgebra of a1;
To prove
a2 is proper
it is sufficient to prove
thus a2 <> a1;
:: BCIALG_1:def 13
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being SubAlgebra of b1 holds
b2 is proper(b1)
iff
b2 <> b1;
:: BCIALG_1:exreg 6
registration
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
cluster non empty being_B being_C being_I being_BCI-4 non proper SubAlgebra of a1;
end;
:: BCIALG_1:attrnot 10 => BCIALG_1:attr 10
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
let a2 be Element of the carrier of a1;
attr a2 is atom means
for b1 being Element of the carrier of a1
st b1 \ a2 = 0. a1
holds b1 = a2;
end;
:: BCIALG_1:dfs 14
definiens
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
let a2 be Element of the carrier of a1;
To prove
a2 is atom
it is sufficient to prove
thus for b1 being Element of the carrier of a1
st b1 \ a2 = 0. a1
holds b1 = a2;
:: BCIALG_1:def 14
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Element of the carrier of b1 holds
b2 is atom(b1)
iff
for b3 being Element of the carrier of b1
st b3 \ b2 = 0. b1
holds b3 = b2;
:: BCIALG_1:funcnot 5 => BCIALG_1:func 5
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
func AtomSet A1 -> non empty Element of bool the carrier of a1 equals
{b1 where b1 is Element of the carrier of a1: b1 is atom(a1)};
end;
:: BCIALG_1:def 15
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
AtomSet b1 = {b2 where b2 is Element of the carrier of b1: b2 is atom(b1)};
:: BCIALG_1:th 23
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
0. b1 in AtomSet b1;
:: BCIALG_1:th 24
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Element of the carrier of b1 holds
b2 in AtomSet b1
iff
for b3 being Element of the carrier of b1 holds
b3 \ (b3 \ b2) = b2;
:: BCIALG_1:th 25
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Element of the carrier of b1 holds
b2 in AtomSet b1
iff
for b3, b4 being Element of the carrier of b1 holds
(b4 \ b3) \ (b4 \ b2) = b2 \ b3;
:: BCIALG_1:th 26
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Element of the carrier of b1 holds
b2 in AtomSet b1
iff
for b3, b4 being Element of the carrier of b1 holds
b2 \ (b4 \ b3) <= b3 \ (b4 \ b2);
:: BCIALG_1:th 27
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Element of the carrier of b1 holds
b2 in AtomSet b1
iff
for b3, b4, b5 being Element of the carrier of b1 holds
(b2 \ b5) \ (b4 \ b3) <= (b3 \ b5) \ (b4 \ b2);
:: BCIALG_1:th 28
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Element of the carrier of b1 holds
b2 in AtomSet b1
iff
for b3 being Element of the carrier of b1 holds
b3 ` \ (b2 `) = b2 \ b3;
:: BCIALG_1:th 29
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Element of the carrier of b1 holds
b2 in AtomSet b1
iff
b2 ` ` = b2;
:: BCIALG_1:th 30
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Element of the carrier of b1 holds
b2 in AtomSet b1
iff
for b3 being Element of the carrier of b1 holds
(b3 \ b2) ` = b2 \ b3;
:: BCIALG_1:th 31
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Element of the carrier of b1 holds
b2 in AtomSet b1
iff
for b3 being Element of the carrier of b1 holds
(b2 \ b3) ` ` = b2 \ b3;
:: BCIALG_1:th 32
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Element of the carrier of b1 holds
b2 in AtomSet b1
iff
for b3, b4 being Element of the carrier of b1 holds
b3 \ (b3 \ (b2 \ b4)) = b2 \ b4;
:: BCIALG_1:th 33
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Element of AtomSet b1
for b3 being Element of the carrier of b1 holds
b2 \ b3 in AtomSet b1;
:: BCIALG_1:funcnot 6 => BCIALG_1:func 6
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
let a2, a3 be Element of AtomSet a1;
redefine func a2 \ a3 -> Element of AtomSet a1;
end;
:: BCIALG_1:th 34
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Element of the carrier of b1 holds
b2 ` in AtomSet b1;
:: BCIALG_1:th 35
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Element of the carrier of b1 holds
ex b3 being Element of AtomSet b1 st
b3 <= b2;
:: BCIALG_1:attrnot 11 => BCIALG_1:attr 11
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
attr a1 is generated_by_atom means
for b1 being Element of the carrier of a1 holds
ex b2 being Element of AtomSet a1 st
b2 <= b1;
end;
:: BCIALG_1:dfs 16
definiens
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
To prove
a1 is generated_by_atom
it is sufficient to prove
thus for b1 being Element of the carrier of a1 holds
ex b2 being Element of AtomSet a1 st
b2 <= b1;
:: BCIALG_1:def 16
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is generated_by_atom
iff
for b2 being Element of the carrier of b1 holds
ex b3 being Element of AtomSet b1 st
b3 <= b2;
:: BCIALG_1:funcnot 7 => BCIALG_1:func 7
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
let a2 be Element of AtomSet a1;
func BranchV A2 -> non empty Element of bool the carrier of a1 equals
{b1 where b1 is Element of the carrier of a1: a2 <= b1};
end;
:: BCIALG_1:def 17
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Element of AtomSet b1 holds
BranchV b2 = {b3 where b3 is Element of the carrier of b1: b2 <= b3};
:: BCIALG_1:th 36
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is generated_by_atom;
:: BCIALG_1:th 37
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2, b3 being Element of AtomSet b1
for b4 being Element of BranchV b3 holds
b2 \ b4 = b2 \ b3;
:: BCIALG_1:th 38
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Element of AtomSet b1
for b3 being Element of BCK-part b1 holds
b2 \ b3 = b2;
:: BCIALG_1:th 39
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2, b3 being Element of AtomSet b1
for b4 being Element of BranchV b2
for b5 being Element of BranchV b3 holds
b4 \ b5 in BranchV (b2 \ b3);
:: BCIALG_1:th 40
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Element of AtomSet b1
for b3, b4 being Element of BranchV b2 holds
b3 \ b4 in BCK-part b1;
:: BCIALG_1:th 41
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2, b3 being Element of AtomSet b1
for b4 being Element of BranchV b2
for b5 being Element of BranchV b3
st b2 <> b3
holds not b4 \ b5 in BCK-part b1;
:: BCIALG_1:th 42
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2, b3 being Element of AtomSet b1
st b2 <> b3
holds (BranchV b2) /\ BranchV b3 = {};
:: BCIALG_1:modenot 4 => BCIALG_1:mode 2
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
mode Ideal of A1 -> non empty Element of bool the carrier of a1 means
0. a1 in it &
(for b1, b2 being Element of the carrier of a1
st b1 \ b2 in it & b2 in it
holds b1 in it);
end;
:: BCIALG_1:dfs 18
definiens
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
let a2 be non empty Element of bool the carrier of a1;
To prove
a2 is Ideal of a1
it is sufficient to prove
thus 0. a1 in a2 &
(for b1, b2 being Element of the carrier of a1
st b1 \ b2 in a2 & b2 in a2
holds b1 in a2);
:: BCIALG_1:def 18
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being non empty Element of bool the carrier of b1 holds
b2 is Ideal of b1
iff
0. b1 in b2 &
(for b3, b4 being Element of the carrier of b1
st b3 \ b4 in b2 & b4 in b2
holds b3 in b2);
:: BCIALG_1:attrnot 12 => BCIALG_1:attr 12
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
let a2 be Ideal of a1;
attr a2 is closed means
for b1 being Element of a2 holds
b1 ` in a2;
end;
:: BCIALG_1:dfs 19
definiens
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
let a2 be Ideal of a1;
To prove
a2 is closed
it is sufficient to prove
thus for b1 being Element of a2 holds
b1 ` in a2;
:: BCIALG_1:def 19
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Ideal of b1 holds
b2 is closed(b1)
iff
for b3 being Element of b2 holds
b3 ` in b2;
:: BCIALG_1:exreg 7
registration
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
cluster non empty closed Ideal of a1;
end;
:: BCIALG_1:th 43
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
{0. b1} is closed Ideal of b1;
:: BCIALG_1:th 44
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
the carrier of b1 is closed Ideal of b1;
:: BCIALG_1:th 45
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
BCK-part b1 is closed Ideal of b1;
:: BCIALG_1:th 46
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being non empty Element of bool the carrier of b1
st b2 is Ideal of b1
for b3, b4 being Element of the carrier of b1
st b3 in b2 & b4 <= b3
holds b4 in b2;
:: BCIALG_1:attrnot 13 => BCIALG_1:attr 13
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
attr a1 is associative means
for b1, b2, b3 being Element of the carrier of a1 holds
(b1 \ b2) \ b3 = b1 \ (b2 \ b3);
end;
:: BCIALG_1:dfs 20
definiens
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
To prove
a1 is associative
it is sufficient to prove
thus for b1, b2, b3 being Element of the carrier of a1 holds
(b1 \ b2) \ b3 = b1 \ (b2 \ b3);
:: BCIALG_1:def 20
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is associative
iff
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 \ b3) \ b4 = b2 \ (b3 \ b4);
:: BCIALG_1:attrnot 14 => BCIALG_1:attr 14
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
attr a1 is quasi-associative means
for b1 being Element of the carrier of a1 holds
b1 ` ` = b1 `;
end;
:: BCIALG_1:dfs 21
definiens
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
To prove
a1 is quasi-associative
it is sufficient to prove
thus for b1 being Element of the carrier of a1 holds
b1 ` ` = b1 `;
:: BCIALG_1:def 21
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is quasi-associative
iff
for b2 being Element of the carrier of b1 holds
b2 ` ` = b2 `;
:: BCIALG_1:attrnot 15 => BCIALG_1:attr 15
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
attr a1 is positive-implicative means
for b1, b2 being Element of the carrier of a1 holds
(b1 \ (b1 \ b2)) \ (b2 \ b1) = b1 \ (b1 \ (b2 \ (b2 \ b1)));
end;
:: BCIALG_1:dfs 22
definiens
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
To prove
a1 is positive-implicative
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1 holds
(b1 \ (b1 \ b2)) \ (b2 \ b1) = b1 \ (b1 \ (b2 \ (b2 \ b1)));
:: BCIALG_1:def 22
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is positive-implicative
iff
for b2, b3 being Element of the carrier of b1 holds
(b2 \ (b2 \ b3)) \ (b3 \ b2) = b2 \ (b2 \ (b3 \ (b3 \ b2)));
:: BCIALG_1:attrnot 16 => BCIALG_1:attr 16
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
attr a1 is weakly-positive-implicative means
for b1, b2, b3 being Element of the carrier of a1 holds
(b1 \ b2) \ b3 = ((b1 \ b3) \ b3) \ (b2 \ b3);
end;
:: BCIALG_1:dfs 23
definiens
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
To prove
a1 is weakly-positive-implicative
it is sufficient to prove
thus for b1, b2, b3 being Element of the carrier of a1 holds
(b1 \ b2) \ b3 = ((b1 \ b3) \ b3) \ (b2 \ b3);
:: BCIALG_1:def 23
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is weakly-positive-implicative
iff
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 \ b3) \ b4 = ((b2 \ b4) \ b4) \ (b3 \ b4);
:: BCIALG_1:attrnot 17 => BCIALG_1:attr 17
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
attr a1 is implicative means
for b1, b2 being Element of the carrier of a1 holds
(b1 \ (b1 \ b2)) \ (b2 \ b1) = b2 \ (b2 \ b1);
end;
:: BCIALG_1:dfs 24
definiens
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
To prove
a1 is implicative
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1 holds
(b1 \ (b1 \ b2)) \ (b2 \ b1) = b2 \ (b2 \ b1);
:: BCIALG_1:def 24
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is implicative
iff
for b2, b3 being Element of the carrier of b1 holds
(b2 \ (b2 \ b3)) \ (b3 \ b2) = b3 \ (b3 \ b2);
:: BCIALG_1:attrnot 18 => BCIALG_1:attr 18
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
attr a1 is weakly-implicative means
for b1, b2 being Element of the carrier of a1 holds
(b1 \ (b2 \ b1)) \ ((b2 \ b1) `) = b1;
end;
:: BCIALG_1:dfs 25
definiens
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
To prove
a1 is weakly-implicative
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1 holds
(b1 \ (b2 \ b1)) \ ((b2 \ b1) `) = b1;
:: BCIALG_1:def 25
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is weakly-implicative
iff
for b2, b3 being Element of the carrier of b1 holds
(b2 \ (b3 \ b2)) \ ((b3 \ b2) `) = b2;
:: BCIALG_1:attrnot 19 => BCIALG_1:attr 19
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
attr a1 is p-Semisimple means
for b1, b2 being Element of the carrier of a1 holds
b1 \ (b1 \ b2) = b2;
end;
:: BCIALG_1:dfs 26
definiens
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
To prove
a1 is p-Semisimple
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1 holds
b1 \ (b1 \ b2) = b2;
:: BCIALG_1:def 26
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is p-Semisimple
iff
for b2, b3 being Element of the carrier of b1 holds
b2 \ (b2 \ b3) = b3;
:: BCIALG_1:attrnot 20 => BCIALG_1:attr 20
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
attr a1 is alternative means
for b1, b2 being Element of the carrier of a1 holds
b1 \ (b1 \ b2) = (b1 \ b1) \ b2 &
(b1 \ b2) \ b2 = b1 \ (b2 \ b2);
end;
:: BCIALG_1:dfs 27
definiens
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
To prove
a1 is alternative
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1 holds
b1 \ (b1 \ b2) = (b1 \ b1) \ b2 &
(b1 \ b2) \ b2 = b1 \ (b2 \ b2);
:: BCIALG_1:def 27
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is alternative
iff
for b2, b3 being Element of the carrier of b1 holds
b2 \ (b2 \ b3) = (b2 \ b2) \ b3 &
(b2 \ b3) \ b3 = b2 \ (b3 \ b3);
:: BCIALG_1:funcreg 3
registration
cluster BCI-EXAMPLE -> associative positive-implicative weakly-positive-implicative implicative weakly-implicative p-Semisimple;
end;
:: BCIALG_1:exreg 8
registration
cluster non empty being_B being_C being_I being_BCI-4 associative positive-implicative weakly-positive-implicative implicative weakly-implicative p-Semisimple BCIStr_0;
end;
:: BCIALG_1:th 47
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is associative
iff
for b2 being Element of the carrier of b1 holds
b2 ` = b2;
:: BCIALG_1:th 48
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
for b2, b3 being Element of the carrier of b1 holds
b3 \ b2 = b2 \ b3
iff
b1 is associative;
:: BCIALG_1:th 49
theorem
for b1 being non empty BCIStr_0 holds
b1 is non empty being_B being_C being_I being_BCI-4 associative BCIStr_0
iff
for b2, b3, b4 being Element of the carrier of b1 holds
(b3 \ b2) \ (b4 \ b2) = b4 \ b3 &
b2 \ 0. b1 = b2;
:: BCIALG_1:th 50
theorem
for b1 being non empty BCIStr_0 holds
b1 is non empty being_B being_C being_I being_BCI-4 associative BCIStr_0
iff
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 \ b3) \ (b2 \ b4) = b4 \ b3 &
b2 ` = b2;
:: BCIALG_1:th 51
theorem
for b1 being non empty BCIStr_0 holds
b1 is non empty being_B being_C being_I being_BCI-4 associative BCIStr_0
iff
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 \ b3) \ (b2 \ b4) = b3 \ b4 &
b2 \ 0. b1 = b2;
:: BCIALG_1:th 52
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is p-Semisimple
iff
for b2 being Element of the carrier of b1 holds
b2 is atom(b1);
:: BCIALG_1:th 53
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
st b1 is p-Semisimple
holds BCK-part b1 = {0. b1};
:: BCIALG_1:th 54
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is p-Semisimple
iff
for b2 being Element of the carrier of b1 holds
b2 ` ` = b2;
:: BCIALG_1:th 55
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is p-Semisimple
iff
for b2, b3 being Element of the carrier of b1 holds
b3 \ (b3 \ b2) = b2;
:: BCIALG_1:th 56
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is p-Semisimple
iff
for b2, b3, b4 being Element of the carrier of b1 holds
(b4 \ b3) \ (b4 \ b2) = b2 \ b3;
:: BCIALG_1:th 57
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is p-Semisimple
iff
for b2, b3, b4 being Element of the carrier of b1 holds
b2 \ (b4 \ b3) = b3 \ (b4 \ b2);
:: BCIALG_1:th 58
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is p-Semisimple
iff
for b2, b3, b4, b5 being Element of the carrier of b1 holds
(b2 \ b5) \ (b4 \ b3) = (b3 \ b5) \ (b4 \ b2);
:: BCIALG_1:th 59
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is p-Semisimple
iff
for b2, b3 being Element of the carrier of b1 holds
b3 ` \ (b2 `) = b2 \ b3;
:: BCIALG_1:th 60
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is p-Semisimple
iff
for b2, b3 being Element of the carrier of b1 holds
(b2 \ b3) ` ` = b2 \ b3;
:: BCIALG_1:th 61
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is p-Semisimple
iff
for b2, b3, b4 being Element of the carrier of b1 holds
b4 \ (b4 \ (b2 \ b3)) = b2 \ b3;
:: BCIALG_1:th 62
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is p-Semisimple
iff
for b2 being Element of the carrier of b1
st b2 ` = 0. b1
holds b2 = 0. b1;
:: BCIALG_1:th 63
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is p-Semisimple
iff
for b2, b3 being Element of the carrier of b1 holds
b2 \ (b3 `) = b3 \ (b2 `);
:: BCIALG_1:th 64
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is p-Semisimple
iff
for b2, b3, b4, b5 being Element of the carrier of b1 holds
(b2 \ b3) \ (b4 \ b5) = (b2 \ b4) \ (b3 \ b5);
:: BCIALG_1:th 65
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is p-Semisimple
iff
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 \ b3) \ (b4 \ b3) = b2 \ b4;
:: BCIALG_1:th 66
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is p-Semisimple
iff
for b2, b3, b4 being Element of the carrier of b1 holds
b2 \ (b3 \ b4) = (b4 \ b3) \ (b2 `);
:: BCIALG_1:th 67
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is p-Semisimple
iff
for b2, b3, b4 being Element of the carrier of b1
st b3 \ b2 = b4 \ b2
holds b3 = b4;
:: BCIALG_1:th 68
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is p-Semisimple
iff
for b2, b3, b4 being Element of the carrier of b1
st b2 \ b3 = b2 \ b4
holds b3 = b4;
:: BCIALG_1:th 69
theorem
for b1 being non empty BCIStr_0 holds
b1 is non empty being_B being_C being_I being_BCI-4 p-Semisimple BCIStr_0
iff
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 \ b3) \ (b2 \ b4) = b4 \ b3 &
b2 \ 0. b1 = b2;
:: BCIALG_1:th 70
theorem
for b1 being non empty BCIStr_0 holds
b1 is non empty being_B being_C being_I being_BCI-4 p-Semisimple BCIStr_0
iff
b1 is being_I &
(for b2, b3, b4 being Element of the carrier of b1 holds
b2 \ (b3 \ b4) = b4 \ (b3 \ b2) &
b2 \ 0. b1 = b2);
:: BCIALG_1:th 71
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is quasi-associative
iff
for b2 being Element of the carrier of b1 holds
b2 ` <= b2;
:: BCIALG_1:th 72
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is quasi-associative
iff
for b2, b3 being Element of the carrier of b1 holds
(b2 \ b3) ` = (b3 \ b2) `;
:: BCIALG_1:th 73
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is quasi-associative
iff
for b2, b3 being Element of the carrier of b1 holds
b2 ` \ b3 = (b2 \ b3) `;
:: BCIALG_1:th 74
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is quasi-associative
iff
for b2, b3 being Element of the carrier of b1 holds
(b2 \ b3) \ (b3 \ b2) in BCK-part b1;
:: BCIALG_1:th 75
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is quasi-associative
iff
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 \ b3) \ b4 <= b2 \ (b3 \ b4);
:: BCIALG_1:th 76
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2, b3 being Element of the carrier of b1
st b1 is alternative
holds b2 ` = b2 & b2 \ (b2 \ b3) = b3 & (b2 \ b3) \ b3 = b2;
:: BCIALG_1:th 77
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2, b3, b4 being Element of the carrier of b1
st b1 is alternative & b2 \ b3 = b2 \ b4
holds b3 = b4;
:: BCIALG_1:th 78
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2, b3, b4 being Element of the carrier of b1
st b1 is alternative & b2 \ b3 = b4 \ b3
holds b2 = b4;
:: BCIALG_1:th 79
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2, b3 being Element of the carrier of b1
st b1 is alternative & b2 \ b3 = 0. b1
holds b2 = b3;
:: BCIALG_1:th 80
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2, b3, b4 being Element of the carrier of b1
st b1 is alternative & (b2 \ b3) \ b4 = 0. b1
holds b3 = b2 \ b4 & b4 = b2 \ b3;
:: BCIALG_1:condreg 1
registration
cluster non empty being_B being_C being_I being_BCI-4 alternative -> associative (BCIStr_0);
end;
:: BCIALG_1:condreg 2
registration
cluster non empty being_B being_C being_I being_BCI-4 associative -> alternative (BCIStr_0);
end;
:: BCIALG_1:condreg 3
registration
cluster non empty being_B being_C being_I being_BCI-4 alternative -> implicative (BCIStr_0);
end;
:: BCIALG_1:th 81
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2, b3 being Element of the carrier of b1
st b1 is alternative
holds (b2 \ (b2 \ b3)) \ (b3 \ b2) = b2;
:: BCIALG_1:th 82
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2, b3 being Element of the carrier of b1
st b1 is alternative
holds b2 \ (b2 \ (b3 \ (b3 \ b2))) = b2;
:: BCIALG_1:condreg 4
registration
cluster non empty being_B being_C being_I being_BCI-4 associative -> weakly-positive-implicative (BCIStr_0);
end;
:: BCIALG_1:condreg 5
registration
cluster non empty being_B being_C being_I being_BCI-4 p-Semisimple -> weakly-positive-implicative (BCIStr_0);
end;
:: BCIALG_1:th 83
theorem
for b1 being non empty BCIStr_0 holds
b1 is non empty being_B being_C being_I being_BCI-4 implicative BCIStr_0
iff
for b2, b3, b4 being Element of the carrier of b1 holds
((b2 \ b3) \ (b2 \ b4)) \ (b4 \ b3) = 0. b1 &
b2 \ 0. b1 = b2 &
(b2 \ (b2 \ b3)) \ (b3 \ b2) = b3 \ (b3 \ b2);
:: BCIALG_1:th 84
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is weakly-positive-implicative
iff
for b2, b3 being Element of the carrier of b1 holds
b2 \ b3 = ((b2 \ b3) \ b3) \ (b3 `);
:: BCIALG_1:condreg 6
registration
cluster non empty being_B being_C being_I being_BCI-4 positive-implicative -> weakly-positive-implicative (BCIStr_0);
end;
:: BCIALG_1:condreg 7
registration
cluster non empty being_B being_C being_I being_BCI-4 alternative -> weakly-positive-implicative (BCIStr_0);
end;
:: BCIALG_1:th 85
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
st b1 is non empty being_B being_C being_I being_BCI-4 weakly-positive-implicative BCIStr_0
for b2, b3 being Element of the carrier of b1 holds
(b2 \ (b2 \ b3)) \ (b3 \ b2) = ((b3 \ (b3 \ b2)) \ (b3 \ b2)) \ (b2 \ b3);
:: BCIALG_1:th 86
theorem
for b1 being non empty BCIStr_0 holds
b1 is non empty being_B being_C being_I being_BCI-4 positive-implicative BCIStr_0
iff
for b2, b3, b4 being Element of the carrier of b1 holds
((b2 \ b3) \ (b2 \ b4)) \ (b4 \ b3) = 0. b1 &
b2 \ 0. b1 = b2 &
b2 \ b3 = ((b2 \ b3) \ b3) \ (b3 `) &
(b2 \ (b2 \ b3)) \ (b3 \ b2) = ((b3 \ (b3 \ b2)) \ (b3 \ b2)) \ (b2 \ b3);