Article BCIALG_2, MML version 4.99.1005
:: BCIALG_2:funcnot 1 => BCIALG_2:func 1
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
let a2, a3 be Element of the carrier of a1;
let a4 be Element of NAT;
func (A2,A3)to_power A4 -> Element of the carrier of a1 means
ex b1 being Function-like quasi_total Relation of NAT,the carrier of a1 st
it = b1 . a4 &
b1 . 0 = a2 &
(for b2 being Element of NAT
st b2 < a4
holds b1 . (b2 + 1) = (b1 . b2) \ a3);
end;
:: BCIALG_2:def 1
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
for b4 being Element of NAT
for b5 being Element of the carrier of b1 holds
b5 = (b2,b3)to_power b4
iff
ex b6 being Function-like quasi_total Relation of NAT,the carrier of b1 st
b5 = b6 . b4 &
b6 . 0 = b2 &
(for b7 being Element of NAT
st b7 < b4
holds b6 . (b7 + 1) = (b6 . b7) \ b3);
:: BCIALG_2:th 1
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)to_power 0 = b2;
:: BCIALG_2:th 2
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)to_power 1 = b2 \ b3;
:: BCIALG_2:th 3
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)to_power 2 = (b2 \ b3) \ b3;
:: BCIALG_2:th 4
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
for b4 being Element of NAT holds
(b2,b3)to_power (b4 + 1) = ((b2,b3)to_power b4) \ b3;
:: BCIALG_2:th 5
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 NAT holds
(b2,0. b1)to_power (b3 + 1) = b2;
:: BCIALG_2:th 6
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Element of NAT holds
(0. b1,0. b1)to_power b2 = 0. b1;
:: BCIALG_2: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
for b5 being Element of NAT holds
((b2,b3)to_power b5) \ b4 = (b2 \ b4,b3)to_power b5;
:: BCIALG_2: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
for b4 being Element of NAT holds
(b2,b2 \ (b2 \ b3))to_power b4 = (b2,b3)to_power b4;
:: BCIALG_2:th 9
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 NAT holds
((0. b1,b2)to_power b3) ` = (0. b1,b2 `)to_power b3;
:: BCIALG_2: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
for b4, b5 being Element of NAT holds
((b2,b3)to_power b4,b3)to_power b5 = (b2,b3)to_power (b4 + b5);
:: BCIALG_2:th 11
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
for b5, b6 being Element of NAT holds
((b2,b3)to_power b5,b4)to_power b6 = ((b2,b4)to_power b6,b3)to_power b5;
:: BCIALG_2:th 12
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 NAT holds
((0. b1,b2)to_power b3) ` ` = (0. b1,b2)to_power b3;
:: BCIALG_2:th 13
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, b4 being Element of NAT holds
(0. b1,b2)to_power (b3 + b4) = ((0. b1,b2)to_power b3) \ (((0. b1,b2)to_power b4) `);
:: BCIALG_2:th 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
for b3, b4 being Element of NAT holds
((0. b1,b2)to_power (b3 + b4)) ` = ((0. b1,b2)to_power b3) ` \ ((0. b1,b2)to_power b4);
:: BCIALG_2:th 15
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, b4 being Element of NAT holds
((0. b1,(0. b1,b2)to_power b3)to_power b4) ` = (0. b1,b2)to_power (b3 * b4);
:: BCIALG_2:th 16
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, b4 being Element of NAT
st (0. b1,b2)to_power b3 = 0. b1
holds (0. b1,b2)to_power (b3 * b4) = 0. b1;
:: BCIALG_2:th 17
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
for b4 being Element of NAT
st b2 \ b3 = b2
holds (b2,b3)to_power b4 = b2;
:: BCIALG_2:th 18
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
for b4 being Element of NAT holds
(0. b1,b2 \ b3)to_power b4 = ((0. b1,b2)to_power b4) \ ((0. b1,b3)to_power b4);
:: BCIALG_2:th 19
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
for b5 being Element of NAT
st b2 <= b3
holds (b2,b4)to_power b5 <= (b3,b4)to_power b5;
:: BCIALG_2:th 20
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
for b5 being Element of NAT
st b2 <= b3
holds (b4,b3)to_power b5 <= (b4,b2)to_power b5;
:: BCIALG_2:th 21
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
for b5 being Element of NAT holds
((b2,b3)to_power b5) \ ((b4,b3)to_power b5) <= b2 \ b4;
:: BCIALG_2:th 22
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
for b4 being Element of NAT holds
((b2,b2 \ b3)to_power b4,b3 \ b2)to_power b4 <= b2;
:: BCIALG_2:attrnot 1 => BCIALG_1:attr 10
notation
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;
synonym minimal for atom;
end;
:: BCIALG_2:attrnot 2 => BCIALG_2:attr 1
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 positive means
0. a1 <= a2;
end;
:: BCIALG_2:dfs 2
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 positive
it is sufficient to prove
thus 0. a1 <= a2;
:: BCIALG_2:def 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 is positive(b1)
iff
0. b1 <= b2;
:: BCIALG_2:attrnot 3 => BCIALG_2:attr 2
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 least means
for b1 being Element of the carrier of a1 holds
a2 <= b1;
end;
:: BCIALG_2:dfs 3
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 least
it is sufficient to prove
thus for b1 being Element of the carrier of a1 holds
a2 <= b1;
:: BCIALG_2:def 3
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 least(b1)
iff
for b3 being Element of the carrier of b1 holds
b2 <= b3;
:: BCIALG_2:attrnot 4 => BCIALG_2:attr 3
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 maximal means
for b1 being Element of the carrier of a1
st a2 <= b1
holds b1 = a2;
end;
:: BCIALG_2:dfs 4
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 maximal
it is sufficient to prove
thus for b1 being Element of the carrier of a1
st a2 <= b1
holds b1 = a2;
:: BCIALG_2:def 4
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 maximal(b1)
iff
for b3 being Element of the carrier of b1
st b2 <= b3
holds b3 = b2;
:: BCIALG_2:attrnot 5 => BCIALG_2:attr 4
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 greatest means
for b1 being Element of the carrier of a1 holds
b1 <= a2;
end;
:: BCIALG_2:dfs 5
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 greatest
it is sufficient to prove
thus for b1 being Element of the carrier of a1 holds
b1 <= a2;
:: BCIALG_2:def 5
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 greatest(b1)
iff
for b3 being Element of the carrier of b1 holds
b3 <= b2;
:: BCIALG_2:exreg 1
registration
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
cluster positive Element of the carrier of a1;
end;
:: BCIALG_2:funcreg 1
registration
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
cluster 0. a1 -> atom positive;
end;
:: BCIALG_2:th 23
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 holds
b2 \ b3 = b3 ` \ (b2 `);
:: BCIALG_2: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 ` is atom(b1)
iff
for b3 being Element of the carrier of b1
st b3 <= b2
holds b2 ` = b3 `;
:: BCIALG_2: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 ` is atom(b1)
iff
for b3, b4 being Element of the carrier of b1 holds
((b2 \ b4) \ (b3 \ b4)) ` ` = b3 ` \ (b2 `);
:: BCIALG_2:th 26
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
st 0. b1 is maximal(b1)
for b2 being Element of the carrier of b1 holds
b2 is atom(b1);
:: BCIALG_2:th 27
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
st ex b2 being Element of the carrier of b1 st
b2 is greatest(b1)
for b2 being Element of the carrier of b1 holds
b2 is positive(b1);
:: BCIALG_2: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 \ (b2 ` `) is positive Element of the carrier of b1;
:: BCIALG_2: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 is atom(b1)
iff
b2 ` ` = b2;
:: BCIALG_2: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 is atom(b1)
iff
ex b3 being Element of the carrier of b1 st
b2 = b3 `;
:: BCIALG_2:attrnot 6 => BCIALG_2:attr 5
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 nilpotent means
ex b1 being non empty Element of NAT st
(0. a1,a2)to_power b1 = 0. a1;
end;
:: BCIALG_2:dfs 6
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 nilpotent
it is sufficient to prove
thus ex b1 being non empty Element of NAT st
(0. a1,a2)to_power b1 = 0. a1;
:: BCIALG_2:def 6
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 nilpotent(b1)
iff
ex b3 being non empty Element of NAT st
(0. b1,b2)to_power b3 = 0. b1;
:: BCIALG_2:attrnot 7 => BCIALG_2:attr 6
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
attr a1 is nilpotent means
for b1 being Element of the carrier of a1 holds
b1 is nilpotent(a1);
end;
:: BCIALG_2:dfs 7
definiens
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
To prove
a1 is nilpotent
it is sufficient to prove
thus for b1 being Element of the carrier of a1 holds
b1 is nilpotent(a1);
:: BCIALG_2:def 7
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is nilpotent
iff
for b2 being Element of the carrier of b1 holds
b2 is nilpotent(b1);
:: BCIALG_2:funcnot 2 => BCIALG_2:func 2
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;
assume a2 is nilpotent(a1);
func ord A2 -> non empty Element of NAT means
(0. a1,a2)to_power it = 0. a1 &
(for b1 being Element of NAT
st (0. a1,a2)to_power b1 = 0. a1 & b1 <> 0
holds it <= b1);
end;
:: BCIALG_2:def 8
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
st b2 is nilpotent(b1)
for b3 being non empty Element of NAT holds
b3 = ord b2
iff
(0. b1,b2)to_power b3 = 0. b1 &
(for b4 being Element of NAT
st (0. b1,b2)to_power b4 = 0. b1 & b4 <> 0
holds b3 <= b4);
:: BCIALG_2:funcreg 2
registration
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
cluster 0. a1 -> nilpotent;
end;
:: BCIALG_2: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 is positive Element of the carrier of b1
iff
b2 is nilpotent(b1) & ord b2 = 1;
:: BCIALG_2:th 32
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
b1 is non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0
iff
for b2 being Element of the carrier of b1 holds
ord b2 = 1 & b2 is nilpotent(b1);
:: BCIALG_2:th 33
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 NAT holds
(0. b1,b2 `)to_power b3 is atom(b1);
:: BCIALG_2: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
st b2 is nilpotent(b1)
holds ord b2 = ord (b2 `);
:: BCIALG_2:modenot 1 => BCIALG_2:mode 1
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
mode Congruence of A1 -> symmetric transitive total Relation of the carrier of a1,the carrier of a1 means
for b1, b2, b3, b4 being Element of the carrier of a1
st [b1,b2] in it & [b3,b4] in it
holds [b1 \ b3,b2 \ b4] in it;
end;
:: BCIALG_2:dfs 9
definiens
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
let a2 be symmetric transitive total Relation of the carrier of a1,the carrier of a1;
To prove
a2 is Congruence of a1
it is sufficient to prove
thus for b1, b2, b3, b4 being Element of the carrier of a1
st [b1,b2] in a2 & [b3,b4] in a2
holds [b1 \ b3,b2 \ b4] in a2;
:: BCIALG_2:def 9
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being symmetric transitive total Relation of the carrier of b1,the carrier of b1 holds
b2 is Congruence of b1
iff
for b3, b4, b5, b6 being Element of the carrier of b1
st [b3,b4] in b2 & [b5,b6] in b2
holds [b3 \ b5,b4 \ b6] in b2;
:: BCIALG_2:modenot 2 => BCIALG_2:mode 2
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
mode L-congruence of A1 -> symmetric transitive total Relation of the carrier of a1,the carrier of a1 means
for b1, b2 being Element of the carrier of a1
st [b1,b2] in it
for b3 being Element of the carrier of a1 holds
[b3 \ b1,b3 \ b2] in it;
end;
:: BCIALG_2:dfs 10
definiens
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
let a2 be symmetric transitive total Relation of the carrier of a1,the carrier of a1;
To prove
a2 is L-congruence of a1
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1
st [b1,b2] in a2
for b3 being Element of the carrier of a1 holds
[b3 \ b1,b3 \ b2] in a2;
:: BCIALG_2:def 10
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being symmetric transitive total Relation of the carrier of b1,the carrier of b1 holds
b2 is L-congruence of b1
iff
for b3, b4 being Element of the carrier of b1
st [b3,b4] in b2
for b5 being Element of the carrier of b1 holds
[b5 \ b3,b5 \ b4] in b2;
:: BCIALG_2:modenot 3 => BCIALG_2:mode 3
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
mode R-congruence of A1 -> symmetric transitive total Relation of the carrier of a1,the carrier of a1 means
for b1, b2 being Element of the carrier of a1
st [b1,b2] in it
for b3 being Element of the carrier of a1 holds
[b1 \ b3,b2 \ b3] in it;
end;
:: BCIALG_2:dfs 11
definiens
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
let a2 be symmetric transitive total Relation of the carrier of a1,the carrier of a1;
To prove
a2 is R-congruence of a1
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1
st [b1,b2] in a2
for b3 being Element of the carrier of a1 holds
[b1 \ b3,b2 \ b3] in a2;
:: BCIALG_2:def 11
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being symmetric transitive total Relation of the carrier of b1,the carrier of b1 holds
b2 is R-congruence of b1
iff
for b3, b4 being Element of the carrier of b1
st [b3,b4] in b2
for b5 being Element of the carrier of b1 holds
[b3 \ b5,b4 \ b5] in b2;
:: BCIALG_2:modenot 4 => BCIALG_2:mode 4
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
let a2 be Ideal of a1;
mode I-congruence of A1,A2 -> Relation of the carrier of a1,the carrier of a1 means
for b1, b2 being Element of the carrier of a1 holds
[b1,b2] in it
iff
b1 \ b2 in a2 & b2 \ b1 in a2;
end;
:: BCIALG_2:dfs 12
definiens
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
let a2 be Ideal of a1;
let a3 be Relation of the carrier of a1,the carrier of a1;
To prove
a3 is I-congruence of a1,a2
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1 holds
[b1,b2] in a3
iff
b1 \ b2 in a2 & b2 \ b1 in a2;
:: BCIALG_2:def 12
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Ideal of b1
for b3 being Relation of the carrier of b1,the carrier of b1 holds
b3 is I-congruence of b1,b2
iff
for b4, b5 being Element of the carrier of b1 holds
[b4,b5] in b3
iff
b4 \ b5 in b2 & b5 \ b4 in b2;
:: BCIALG_2:condreg 1
registration
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
let a2 be Ideal of a1;
cluster -> symmetric transitive total (I-congruence of a1,a2);
end;
:: BCIALG_2:funcnot 3 => BCIALG_2:func 3
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
func IConSet A1 -> set means
for b1 being set holds
b1 in it
iff
ex b2 being Ideal of a1 st
b1 is I-congruence of a1,b2;
end;
:: BCIALG_2:def 13
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being set holds
b2 = IConSet b1
iff
for b3 being set holds
b3 in b2
iff
ex b4 being Ideal of b1 st
b3 is I-congruence of b1,b4;
:: BCIALG_2:funcnot 4 => BCIALG_2:func 4
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
func ConSet A1 -> set equals
{b1 where b1 is Congruence of a1: TRUE};
end;
:: BCIALG_2:def 14
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
ConSet b1 = {b2 where b2 is Congruence of b1: TRUE};
:: BCIALG_2:funcnot 5 => BCIALG_2:func 5
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
func LConSet A1 -> set equals
{b1 where b1 is L-congruence of a1: TRUE};
end;
:: BCIALG_2:def 15
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
LConSet b1 = {b2 where b2 is L-congruence of b1: TRUE};
:: BCIALG_2:funcnot 6 => BCIALG_2:func 6
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
func RConSet A1 -> set equals
{b1 where b1 is R-congruence of a1: TRUE};
end;
:: BCIALG_2:def 16
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
RConSet b1 = {b2 where b2 is R-congruence of b1: TRUE};
:: BCIALG_2:th 35
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Congruence of b1 holds
Class(b2,0. b1) is closed Ideal of b1;
:: BCIALG_2:th 36
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being symmetric transitive total Relation of the carrier of b1,the carrier of b1 holds
b2 is Congruence of b1
iff
b2 is R-congruence of b1 & b2 is L-congruence of b1;
:: BCIALG_2:th 37
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Ideal of b1
for b3 being I-congruence of b1,b2 holds
b3 is Congruence of b1;
:: BCIALG_2:modenot 5 => BCIALG_2:mode 5
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
let a2 be Ideal of a1;
redefine mode I-congruence of a1,a2 -> Congruence of a1;
end;
:: BCIALG_2:th 38
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Ideal of b1
for b3 being I-congruence of b1,b2 holds
Class(b3,0. b1) c= b2;
:: BCIALG_2:th 39
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Ideal of b1
for b3 being I-congruence of b1,b2 holds
b2 is closed(b1)
iff
b2 = Class(b3,0. b1);
:: BCIALG_2:th 40
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
for b4 being Congruence of b1
st [b2,b3] in b4
holds b2 \ b3 in Class(b4,0. b1) & b3 \ b2 in Class(b4,0. b1);
:: BCIALG_2:th 41
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2, b3 being Ideal of b1
for b4 being I-congruence of b1,b2
for b5 being I-congruence of b1,b3
st Class(b4,0. b1) = Class(b5,0. b1)
holds b4 = b5;
:: BCIALG_2:th 42
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
for b5 being Element of NAT
for b6 being Congruence of b1
st [b2,b3] in b6 & b4 in Class(b6,0. b1)
holds [b2,(b3,b4)to_power b5] in b6;
:: BCIALG_2:th 43
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
ex b5, b6, b7, b8 being Element of NAT st
((b3,b3 \ b4)to_power b5,b4 \ b3)to_power b6 = ((b4,b4 \ b3)to_power b7,b3 \ b4)to_power b8
for b2 being Congruence of b1
for b3 being Ideal of b1
st b3 = Class(b2,0. b1)
holds b2 is I-congruence of b1,b3;
:: BCIALG_2:th 44
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
IConSet b1 c= ConSet b1;
:: BCIALG_2:th 45
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
ConSet b1 c= LConSet b1;
:: BCIALG_2:th 46
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
ConSet b1 c= RConSet b1;
:: BCIALG_2:th 47
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
ConSet b1 = (LConSet b1) /\ RConSet b1;
:: BCIALG_2:th 48
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Ideal of b1
for b3 being I-congruence of b1,b2
for b4 being Congruence of b1
st for b5 being L-congruence of b1 holds
b5 is I-congruence of b1,b2
holds b4 = b3;
:: BCIALG_2:th 49
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Ideal of b1
for b3 being I-congruence of b1,b2
for b4 being Congruence of b1
st for b5 being R-congruence of b1 holds
b5 is I-congruence of b1,b2
holds b4 = b3;
:: BCIALG_2:th 50
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being L-congruence of b1 holds
Class(b2,0. b1) is closed Ideal of b1;
:: BCIALG_2:funcreg 3
registration
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
let a2 be Congruence of a1;
cluster Class a2 -> non empty;
end;
:: BCIALG_2:funcnot 7 => BCIALG_2:func 7
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
let a2 be Congruence of a1;
func EqClaOp A2 -> Function-like quasi_total Relation of [:Class a2,Class a2:],Class a2 means
for b1, b2 being Element of Class a2
for b3, b4 being Element of the carrier of a1
st b1 = Class(a2,b3) & b2 = Class(a2,b4)
holds it .(b1,b2) = Class(a2,b3 \ b4);
end;
:: BCIALG_2:def 17
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Congruence of b1
for b3 being Function-like quasi_total Relation of [:Class b2,Class b2:],Class b2 holds
b3 = EqClaOp b2
iff
for b4, b5 being Element of Class b2
for b6, b7 being Element of the carrier of b1
st b4 = Class(b2,b6) & b5 = Class(b2,b7)
holds b3 .(b4,b5) = Class(b2,b6 \ b7);
:: BCIALG_2:funcnot 8 => BCIALG_2:func 8
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
let a2 be Congruence of a1;
func zeroEqC A2 -> Element of Class a2 equals
Class(a2,0. a1);
end;
:: BCIALG_2:def 18
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Congruence of b1 holds
zeroEqC b2 = Class(b2,0. b1);
:: BCIALG_2:funcnot 9 => BCIALG_2:func 9
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
let a2 be Congruence of a1;
func A1 ./. A2 -> BCIStr_0 equals
BCIStr_0(#Class a2,EqClaOp a2,zeroEqC a2#);
end;
:: BCIALG_2:def 19
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Congruence of b1 holds
b1 ./. b2 = BCIStr_0(#Class b2,EqClaOp b2,zeroEqC b2#);
:: BCIALG_2:funcreg 4
registration
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
let a2 be Congruence of a1;
cluster a1 ./. a2 -> non empty;
end;
:: BCIALG_2:funcnot 10 => BCIALG_2:func 10
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
let a2 be Congruence of a1;
let a3, a4 be Element of Class a2;
func A3 \ A4 -> Element of Class a2 equals
(EqClaOp a2) .(a3,a4);
end;
:: BCIALG_2:def 20
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Congruence of b1
for b3, b4 being Element of Class b2 holds
b3 \ b4 = (EqClaOp b2) .(b3,b4);
:: BCIALG_2:th 51
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Ideal of b1
for b3 being I-congruence of b1,b2 holds
b1 ./. b3 is non empty being_B being_C being_I being_BCI-4 BCIStr_0;
:: BCIALG_2:funcreg 5
registration
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
let a2 be Ideal of a1;
let a3 be I-congruence of a1,a2;
cluster a1 ./. a3 -> strict being_B being_C being_I being_BCI-4;
end;
:: BCIALG_2:th 52
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Ideal of b1
st b2 = BCK-part b1
for b3 being I-congruence of b1,b2 holds
b1 ./. b3 is non empty being_B being_C being_I being_BCI-4 p-Semisimple BCIStr_0;