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;