Article BCIIDEAL, MML version 4.99.1005
:: BCIIDEAL:th 1
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2, b3, b4, b5 being Element of the carrier of b1
st b2 <= b3
holds b5 \ (b4 \ b2) <= b5 \ (b4 \ b3);
:: BCIIDEAL:th 2
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2, b3, b4, b5 being Element of the carrier of b1 holds
(b2 \ (b3 \ b4)) \ (b2 \ (b3 \ b5)) <= b4 \ b5;
:: BCIIDEAL:th 3
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2, b3, b4, b5, b6 being Element of the carrier of b1 holds
(b2 \ (b3 \ (b4 \ b5))) \ (b2 \ (b3 \ (b4 \ b6))) <= b6 \ b5;
:: BCIIDEAL: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 holds
((0. b1) \ (b2 \ b3)) \ (b3 \ b2) = 0. b1;
:: BCIIDEAL:funcnot 1 => BCIIDEAL:func 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;
func initial_section A2 -> set equals
{b1 where b1 is Element of the carrier of a1: b1 <= a2};
end;
:: BCIIDEAL:def 1
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
initial_section b2 = {b3 where b3 is Element of the carrier of b1: b3 <= b2};
:: BCIIDEAL:th 5
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 Element of the carrier of b1
for b4 being Element of b2
st b3 <= b4
holds b3 in b2;
:: BCIIDEAL:th 6
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2, b3, b4 being Element of AtomSet b1
st b2 is Element of BranchV b4
holds b3 \ b2 = b3 \ b4;
:: BCIIDEAL:th 7
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 AtomSet b1
st b3 is Element of BranchV b4
holds b2 \ b3 = b2 \ b4;
:: BCIIDEAL:th 8
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 Element of b2 holds
initial_section b3 c= b2;
:: BCIIDEAL:th 9
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
st AtomSet b1 is Ideal of b1
for b2 being Element of BCK-part b1
for b3 being Element of AtomSet b1
st b2 \ b3 in AtomSet b1
holds b2 = 0. b1;
:: BCIIDEAL:th 10
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
st AtomSet b1 is Ideal of b1
holds AtomSet b1 is closed Ideal of b1;
:: BCIIDEAL:attrnot 1 => BCIIDEAL:attr 1
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 positive means
for b1 being Element of a2 holds
b1 is positive(a1);
end;
:: BCIIDEAL:dfs 2
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 positive
it is sufficient to prove
thus for b1 being Element of a2 holds
b1 is positive(a1);
:: BCIIDEAL:def 2
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 positive(b1)
iff
for b3 being Element of b2 holds
b3 is positive(b1);
:: BCIIDEAL:th 11
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0
for b2, b3 being Ideal of b1 holds
b2 /\ b3 = {0. b1}
iff
for b4 being Element of b2
for b5 being Element of b3 holds
b4 \ b5 = b4;
:: BCIIDEAL:th 12
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 associative BCIStr_0
for b2 being Ideal of b1 holds
b2 is closed(b1);
:: BCIIDEAL:th 13
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Ideal of b1
st b1 is quasi-associative
holds b2 is closed(b1);
:: BCIIDEAL:attrnot 2 => BCIIDEAL:attr 2
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 associative means
0. a1 in a2 &
(for b1, b2, b3 being Element of the carrier of a1
st b1 \ (b2 \ b3) in a2 & b2 \ b3 in a2
holds b1 in a2);
end;
:: BCIIDEAL:dfs 3
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 associative
it is sufficient to prove
thus 0. a1 in a2 &
(for b1, b2, b3 being Element of the carrier of a1
st b1 \ (b2 \ b3) in a2 & b2 \ b3 in a2
holds b1 in a2);
:: BCIIDEAL:def 3
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 associative(b1)
iff
0. b1 in b2 &
(for b3, b4, b5 being Element of the carrier of b1
st b3 \ (b4 \ b5) in b2 & b4 \ b5 in b2
holds b3 in b2);
:: BCIIDEAL:exreg 1
registration
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
cluster non empty associative Ideal of a1;
end;
:: BCIIDEAL:modenot 1 => BCIIDEAL:mode 1
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
mode associative-ideal of A1 -> non empty Element of bool the carrier of a1 means
0. a1 in it &
(for b1, b2, b3 being Element of the carrier of a1
st (b1 \ b2) \ b3 in it & b2 \ b3 in it
holds b1 in it);
end;
:: BCIIDEAL:dfs 4
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 associative-ideal of a1
it is sufficient to prove
thus 0. a1 in a2 &
(for b1, b2, b3 being Element of the carrier of a1
st (b1 \ b2) \ b3 in a2 & b2 \ b3 in a2
holds b1 in a2);
:: BCIIDEAL:def 4
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 associative-ideal of b1
iff
0. b1 in b2 &
(for b3, b4, b5 being Element of the carrier of b1
st (b3 \ b4) \ b5 in b2 & b4 \ b5 in b2
holds b3 in b2);
:: BCIIDEAL:th 14
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 associative-ideal of b1
holds b2 is Ideal of b1;
:: BCIIDEAL:th 15
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 associative-ideal of b1
iff
for b3, b4, b5 being Element of the carrier of b1
st (b3 \ b4) \ b5 in b2
holds b3 \ (b4 \ b5) in b2;
:: BCIIDEAL:th 16
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 is associative-ideal of b1
for b3 being Element of the carrier of b1 holds
b3 \ ((0. b1) \ b3) in b2;
:: BCIIDEAL:th 17
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0
for b2 being Ideal of b1
st for b3 being Element of the carrier of b1 holds
b3 \ ((0. b1) \ b3) in b2
holds b2 is closed Ideal of b1;
:: BCIIDEAL:modenot 2 => BCIIDEAL:mode 2
definition
let a1 be non empty being_B being_C being_I being_BCI-4 BCIStr_0;
mode p-ideal of A1 -> non empty Element of bool the carrier of a1 means
0. a1 in it &
(for b1, b2, b3 being Element of the carrier of a1
st (b1 \ b3) \ (b2 \ b3) in it & b2 in it
holds b1 in it);
end;
:: BCIIDEAL:dfs 5
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 p-ideal of a1
it is sufficient to prove
thus 0. a1 in a2 &
(for b1, b2, b3 being Element of the carrier of a1
st (b1 \ b3) \ (b2 \ b3) in a2 & b2 in a2
holds b1 in a2);
:: BCIIDEAL:def 5
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 p-ideal of b1
iff
0. b1 in b2 &
(for b3, b4, b5 being Element of the carrier of b1
st (b3 \ b5) \ (b4 \ b5) in b2 & b4 in b2
holds b3 in b2);
:: BCIIDEAL:th 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
st b2 is p-ideal of b1
holds b2 is Ideal of b1;
:: BCIIDEAL:th 19
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 is p-ideal of b1
holds BCK-part b1 c= b2;
:: BCIIDEAL:th 20
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 BCIStr_0 holds
BCK-part b1 is p-ideal of b1;
:: BCIIDEAL:th 21
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 p-ideal of b1
iff
for b3, b4 being Element of the carrier of b1
st b3 in b2 & b3 <= b4
holds b4 in b2;
:: BCIIDEAL:th 22
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 p-ideal of b1
iff
for b3, b4, b5 being Element of the carrier of b1
st (b3 \ b5) \ (b4 \ b5) in b2
holds b3 \ b4 in b2;
:: BCIIDEAL:attrnot 3 => BCIIDEAL:attr 3
definition
let a1 be non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0;
let a2 be Ideal of a1;
attr a2 is commutative means
for b1, b2, b3 being Element of the carrier of a1
st (b1 \ b2) \ b3 in a2 & b3 in a2
holds b1 \ (b2 \ (b2 \ b1)) in a2;
end;
:: BCIIDEAL:dfs 6
definiens
let a1 be non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0;
let a2 be Ideal of a1;
To prove
a2 is commutative
it is sufficient to prove
thus for b1, b2, b3 being Element of the carrier of a1
st (b1 \ b2) \ b3 in a2 & b3 in a2
holds b1 \ (b2 \ (b2 \ b1)) in a2;
:: BCIIDEAL:def 6
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0
for b2 being Ideal of b1 holds
b2 is commutative(b1)
iff
for b3, b4, b5 being Element of the carrier of b1
st (b3 \ b4) \ b5 in b2 & b5 in b2
holds b3 \ (b4 \ (b4 \ b3)) in b2;
:: BCIIDEAL:exreg 2
registration
let a1 be non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0;
cluster non empty commutative Ideal of a1;
end;
:: BCIIDEAL:th 23
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0 holds
BCK-part b1 is commutative Ideal of b1;
:: BCIIDEAL:th 24
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0
st b1 is non empty being_B being_C being_I being_BCI-4 p-Semisimple BCIStr_0
holds {0. b1} is commutative Ideal of b1;
:: BCIIDEAL:th 25
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0 holds
BCK-part b1 = the carrier of b1;
:: BCIIDEAL:th 26
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 the carrier of b1 = BCK-part b1;
:: BCIIDEAL:th 27
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 the carrier of b1 = BCK-part b1;
:: BCIIDEAL:th 28
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 the carrier of b1 = BCK-part b1;
:: BCIIDEAL:th 29
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 the carrier of b1 = BCK-part b1;
:: BCIIDEAL:th 30
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 the carrier of b1 = BCK-part b1;
:: BCIIDEAL:th 31
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 the carrier of b1 = BCK-part b1;
:: BCIIDEAL:th 32
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0 holds
the carrier of b1 is commutative Ideal of b1;
:: BCIIDEAL:th 33
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0
for b2 being Ideal of b1 holds
b2 is commutative Ideal of b1
iff
for b3, b4 being Element of the carrier of b1
st b3 \ b4 in b2
holds b3 \ (b4 \ (b4 \ b3)) in b2;
:: BCIIDEAL:th 34
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0
for b2, b3 being Ideal of b1
st b2 c= b3 & b2 is commutative Ideal of b1
holds b3 is commutative Ideal of b1;
:: BCIIDEAL:th 35
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0 holds
for b2 being Ideal of b1 holds
b2 is commutative Ideal of b1
iff
{0. b1} is commutative Ideal of b1;
:: BCIIDEAL:th 36
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0 holds
{0. b1} is commutative Ideal of b1
iff
b1 is non empty being_B being_C being_I being_BCI-4 being_BCK-5 commutative BCIStr_0;
:: BCIIDEAL:th 37
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0 holds
b1 is non empty being_B being_C being_I being_BCI-4 being_BCK-5 commutative BCIStr_0
iff
for b2 being Ideal of b1 holds
b2 is commutative Ideal of b1;
:: BCIIDEAL:th 38
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0 holds
{0. b1} is commutative Ideal of b1
iff
for b2 being Ideal of b1 holds
b2 is commutative Ideal of b1;
:: BCIIDEAL:th 39
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0
for b2 being Ideal of b1
for b3, b4 being Element of the carrier of b1
st b3 \ (b3 \ b4) in b2
holds b3 \ ((b3 \ b4) \ ((b3 \ b4) \ b3)) in b2 &
(b4 \ (b4 \ b3)) \ b3 in b2 &
(b4 \ (b4 \ b3)) \ (b3 \ b4) in b2;
:: BCIIDEAL:th 40
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0 holds
{0. b1} is commutative Ideal of b1
iff
for b2, b3 being Element of the carrier of b1 holds
b2 \ (b2 \ b3) <= b3 \ (b3 \ b2);
:: BCIIDEAL:th 41
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0 holds
{0. b1} is commutative Ideal of b1
iff
for b2, b3 being Element of the carrier of b1 holds
b2 \ b3 = b2 \ (b3 \ (b3 \ b2));
:: BCIIDEAL:th 42
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0 holds
{0. b1} is commutative Ideal of b1
iff
for b2, b3 being Element of the carrier of b1 holds
b2 \ (b2 \ b3) = b3 \ (b3 \ (b2 \ (b2 \ b3)));
:: BCIIDEAL:th 43
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0 holds
{0. b1} is commutative Ideal of b1
iff
for b2, b3 being Element of the carrier of b1
st b2 <= b3
holds b2 = b3 \ (b3 \ b2);
:: BCIIDEAL:th 44
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0
st {0. b1} is commutative Ideal of b1
holds (for b2, b3 being Element of the carrier of b1 holds
b2 \ b3 = b2
iff
b3 \ (b3 \ b2) = 0. b1) &
(for b2, b3 being Element of the carrier of b1
st b2 \ b3 = b2
holds b3 \ b2 = b3) &
(for b2, b3, b4 being Element of the carrier of b1
st b3 <= b4
holds (b4 \ b2) \ (b4 \ b3) = b3 \ b2) &
(for b2, b3 being Element of the carrier of b1 holds
b2 \ (b3 \ (b3 \ b2)) = b2 \ b3 &
(b2 \ b3) \ ((b2 \ b3) \ b2) = b2 \ b3) &
(for b2, b3, b4 being Element of the carrier of b1
st b2 <= b4
holds (b4 \ b3) \ ((b4 \ b3) \ (b4 \ b2)) = (b4 \ b3) \ (b2 \ b3));
:: BCIIDEAL:th 45
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0 holds
for b2 being Ideal of b1 holds
b2 is commutative Ideal of b1
iff
for b2, b3 being Element of the carrier of b1 holds
b2 \ (b2 \ b3) <= b3 \ (b3 \ b2);
:: BCIIDEAL:th 46
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0 holds
for b2 being Ideal of b1 holds
b2 is commutative Ideal of b1
iff
for b2, b3 being Element of the carrier of b1 holds
b2 \ b3 = b2 \ (b3 \ (b3 \ b2));
:: BCIIDEAL:th 47
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0 holds
for b2 being Ideal of b1 holds
b2 is commutative Ideal of b1
iff
for b2, b3 being Element of the carrier of b1 holds
b2 \ (b2 \ b3) = b3 \ (b3 \ (b2 \ (b2 \ b3)));
:: BCIIDEAL:th 48
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0 holds
for b2 being Ideal of b1 holds
b2 is commutative Ideal of b1
iff
for b2, b3 being Element of the carrier of b1
st b2 <= b3
holds b2 = b3 \ (b3 \ b2);
:: BCIIDEAL:th 49
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0
st for b2 being Ideal of b1 holds
b2 is commutative Ideal of b1
holds (for b2, b3 being Element of the carrier of b1 holds
b2 \ b3 = b2
iff
b3 \ (b3 \ b2) = 0. b1) &
(for b2, b3 being Element of the carrier of b1
st b2 \ b3 = b2
holds b3 \ b2 = b3) &
(for b2, b3, b4 being Element of the carrier of b1
st b3 <= b4
holds (b4 \ b2) \ (b4 \ b3) = b3 \ b2) &
(for b2, b3 being Element of the carrier of b1 holds
b2 \ (b3 \ (b3 \ b2)) = b2 \ b3 &
(b2 \ b3) \ ((b2 \ b3) \ b2) = b2 \ b3) &
(for b2, b3, b4 being Element of the carrier of b1
st b2 <= b4
holds (b4 \ b3) \ ((b4 \ b3) \ (b4 \ b2)) = (b4 \ b3) \ (b2 \ b3));
:: BCIIDEAL:modenot 3 => BCIIDEAL:mode 3
definition
let a1 be non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0;
mode implicative-ideal of A1 -> non empty Element of bool the carrier of a1 means
0. a1 in it &
(for b1, b2, b3 being Element of the carrier of a1
st (b1 \ (b2 \ b1)) \ b3 in it & b3 in it
holds b1 in it);
end;
:: BCIIDEAL:dfs 7
definiens
let a1 be non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0;
let a2 be non empty Element of bool the carrier of a1;
To prove
a2 is implicative-ideal of a1
it is sufficient to prove
thus 0. a1 in a2 &
(for b1, b2, b3 being Element of the carrier of a1
st (b1 \ (b2 \ b1)) \ b3 in a2 & b3 in a2
holds b1 in a2);
:: BCIIDEAL:def 7
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0
for b2 being non empty Element of bool the carrier of b1 holds
b2 is implicative-ideal of b1
iff
0. b1 in b2 &
(for b3, b4, b5 being Element of the carrier of b1
st (b3 \ (b4 \ b3)) \ b5 in b2 & b5 in b2
holds b3 in b2);
:: BCIIDEAL:th 50
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0
for b2 being Ideal of b1 holds
b2 is implicative-ideal of b1
iff
for b3, b4 being Element of the carrier of b1
st b3 \ (b4 \ b3) in b2
holds b3 in b2;
:: BCIIDEAL:modenot 4 => BCIIDEAL:mode 4
definition
let a1 be non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0;
mode positive-implicative-ideal of A1 -> non empty Element of bool the carrier of a1 means
0. a1 in it &
(for b1, b2, b3 being Element of the carrier of a1
st (b1 \ b2) \ b3 in it & b2 \ b3 in it
holds b1 \ b3 in it);
end;
:: BCIIDEAL:dfs 8
definiens
let a1 be non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0;
let a2 be non empty Element of bool the carrier of a1;
To prove
a2 is positive-implicative-ideal of a1
it is sufficient to prove
thus 0. a1 in a2 &
(for b1, b2, b3 being Element of the carrier of a1
st (b1 \ b2) \ b3 in a2 & b2 \ b3 in a2
holds b1 \ b3 in a2);
:: BCIIDEAL:def 8
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0
for b2 being non empty Element of bool the carrier of b1 holds
b2 is positive-implicative-ideal of b1
iff
0. b1 in b2 &
(for b3, b4, b5 being Element of the carrier of b1
st (b3 \ b4) \ b5 in b2 & b4 \ b5 in b2
holds b3 \ b5 in b2);
:: BCIIDEAL:th 51
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0
for b2 being Ideal of b1 holds
b2 is positive-implicative-ideal of b1
iff
for b3, b4 being Element of the carrier of b1
st (b3 \ b4) \ b4 in b2
holds b3 \ b4 in b2;
:: BCIIDEAL:th 52
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0
for b2 being Ideal of b1
st for b3, b4, b5 being Element of the carrier of b1
st (b3 \ b4) \ b5 in b2 & b4 \ b5 in b2
holds b3 \ b5 in b2
for b3, b4, b5 being Element of the carrier of b1
st (b3 \ b4) \ b5 in b2
holds (b3 \ b5) \ (b4 \ b5) in b2;
:: BCIIDEAL:th 53
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0
for b2 being Ideal of b1
st for b3, b4, b5 being Element of the carrier of b1
st (b3 \ b4) \ b5 in b2
holds (b3 \ b5) \ (b4 \ b5) in b2
holds b2 is positive-implicative-ideal of b1;
:: BCIIDEAL:th 54
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0
for b2 being Ideal of b1 holds
b2 is positive-implicative-ideal of b1
iff
for b3, b4, b5 being Element of the carrier of b1
st (b3 \ b4) \ b5 in b2 & b4 \ b5 in b2
holds b3 \ b5 in b2;
:: BCIIDEAL:th 55
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0
for b2 being Ideal of b1 holds
b2 is positive-implicative-ideal of b1
iff
for b3, b4, b5 being Element of the carrier of b1
st (b3 \ b4) \ b5 in b2
holds (b3 \ b5) \ (b4 \ b5) in b2;
:: BCIIDEAL:th 56
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0
for b2, b3 being Ideal of b1
st b2 c= b3 & b2 is positive-implicative-ideal of b1
holds b3 is positive-implicative-ideal of b1;
:: BCIIDEAL:th 57
theorem
for b1 being non empty being_B being_C being_I being_BCI-4 being_BCK-5 BCIStr_0
for b2 being Ideal of b1
st b2 is implicative-ideal of b1
holds b2 is commutative Ideal of b1 & b2 is positive-implicative-ideal of b1;