Article BVFUNC14, MML version 4.99.1005
:: BVFUNC14:th 1
theorem
for b1 being non empty set
for b2 being Element of b1
for b3, b4 being a_partition of b1 holds
EqClass(b2,b3 '/\' b4) = (EqClass(b2,b3)) /\ EqClass(b2,b4);
:: BVFUNC14:th 2
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4 being a_partition of b1
st b2 = {b3,b4} & b3 <> b4
holds '/\' b2 = b3 '/\' b4;
:: BVFUNC14:th 3
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5 being a_partition of b1
st b2 = {b3,b4,b5} & b3 <> b4 & b4 <> b5 & b5 <> b3
holds '/\' b2 = (b3 '/\' b4) '/\' b5;
:: BVFUNC14:th 4
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5 being a_partition of b1
st b2 = {b3,b4,b5} & b3 <> b4 & b5 <> b3
holds CompF(b3,b2) = b4 '/\' b5;
:: BVFUNC14:th 5
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5 being a_partition of b1
st b2 = {b3,b4,b5} & b3 <> b4 & b4 <> b5
holds CompF(b4,b2) = b5 '/\' b3;
:: BVFUNC14:th 6
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5 being a_partition of b1
st b2 = {b3,b4,b5} & b4 <> b5 & b5 <> b3
holds CompF(b5,b2) = b3 '/\' b4;
:: BVFUNC14:th 7
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5, b6 being a_partition of b1
st b2 = {b3,b4,b5,b6} & b3 <> b4 & b3 <> b5 & b3 <> b6
holds CompF(b3,b2) = (b4 '/\' b5) '/\' b6;
:: BVFUNC14:th 8
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5, b6 being a_partition of b1
st b2 = {b3,b4,b5,b6} & b3 <> b4 & b4 <> b5 & b4 <> b6
holds CompF(b4,b2) = (b3 '/\' b5) '/\' b6;
:: BVFUNC14:th 9
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5, b6 being a_partition of b1
st b2 = {b3,b4,b5,b6} & b3 <> b5 & b4 <> b5 & b5 <> b6
holds CompF(b5,b2) = (b3 '/\' b4) '/\' b6;
:: BVFUNC14:th 10
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5, b6 being a_partition of b1
st b2 = {b3,b4,b5,b6} & b3 <> b6 & b4 <> b6 & b5 <> b6
holds CompF(b6,b2) = (b3 '/\' b5) '/\' b4;
:: BVFUNC14:th 14
theorem
for b1, b2, b3, b4, b5, b6 being set holds
proj1 (((b1 .--> b4) +* (b2 .--> b5)) +* (b3 .--> b6)) = {b1,b2,b3};
:: BVFUNC14:th 15
theorem
for b1 being Relation-like Function-like set
for b2, b3, b4, b5 being set
st b2 <> b3
holds ((b1 +* (b2 .--> b4)) +* (b3 .--> b5)) . b2 = b4;
:: BVFUNC14:th 16
theorem
for b1, b2, b3, b4, b5, b6 being set
st b1 <> b2 & b3 <> b1
holds (((b1 .--> b4) +* (b2 .--> b5)) +* (b3 .--> b6)) . b1 = b4;
:: BVFUNC14:th 17
theorem
for b1, b2, b3, b4, b5, b6 being set
for b7 being Relation-like Function-like set
st b7 = ((b1 .--> b4) +* (b2 .--> b5)) +* (b3 .--> b6)
holds proj2 b7 = {b7 . b1,b7 . b2,b7 . b3};
:: BVFUNC14:th 18
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5, b6 being a_partition of b1
for b7 being Relation-like Function-like set
for b8, b9, b10, b11 being set
st b2 = {b3,b4,b5,b6} &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b4 <> b5 &
b4 <> b6 &
b5 <> b6 &
b7 = (((b4 .--> b9) +* (b5 .--> b10)) +* (b6 .--> b11)) +* (b3 .--> b8)
holds b7 . b4 = b9 & b7 . b5 = b10 & b7 . b6 = b11;
:: BVFUNC14:th 19
theorem
for b1, b2, b3, b4 being set
for b5 being Relation-like Function-like set
for b6, b7, b8, b9 being set
st b5 = (((b2 .--> b7) +* (b3 .--> b8)) +* (b4 .--> b9)) +* (b1 .--> b6)
holds proj1 b5 = {b1,b2,b3,b4};
:: BVFUNC14:th 20
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5, b6 being a_partition of b1
for b7 being Relation-like Function-like set
for b8, b9, b10, b11 being set
st b2 = {b3,b4,b5,b6} &
b7 = (((b4 .--> b9) +* (b5 .--> b10)) +* (b6 .--> b11)) +* (b3 .--> b8)
holds proj2 b7 = {b7 . b3,b7 . b4,b7 . b5,b7 . b6};
:: BVFUNC14:th 21
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5, b6 being a_partition of b1
for b7, b8 being Element of b1
for b9 being Relation-like Function-like set
st b2 is independent(b1) & b2 = {b3,b4,b5,b6} & b3 <> b4 & b3 <> b5 & b3 <> b6 & b4 <> b5 & b4 <> b6 & b5 <> b6
holds EqClass(b8,(b4 '/\' b5) '/\' b6) meets EqClass(b7,b3);
:: BVFUNC14:th 22
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5, b6 being a_partition of b1
for b7, b8 being Element of b1
st b2 is independent(b1) &
b2 = {b3,b4,b5,b6} &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b4 <> b5 &
b4 <> b6 &
b5 <> b6 &
EqClass(b7,b5 '/\' b6) = EqClass(b8,b5 '/\' b6)
holds EqClass(b8,CompF(b3,b2)) meets EqClass(b7,CompF(b4,b2));
:: BVFUNC14:th 23
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5 being a_partition of b1
for b6, b7 being Element of b1
st b2 is independent(b1) & b2 = {b3,b4,b5} & b3 <> b4 & b4 <> b5 & b5 <> b3 & EqClass(b6,b5) = EqClass(b7,b5)
holds EqClass(b7,CompF(b3,b2)) meets EqClass(b6,CompF(b4,b2));
:: BVFUNC14:th 24
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5, b6, b7 being a_partition of b1
st b2 = {b3,b4,b5,b6,b7} & b3 <> b4 & b3 <> b5 & b3 <> b6 & b3 <> b7
holds CompF(b3,b2) = ((b4 '/\' b5) '/\' b6) '/\' b7;
:: BVFUNC14:th 25
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5, b6, b7 being a_partition of b1
st b2 is independent(b1) & b2 = {b3,b4,b5,b6,b7} & b3 <> b4 & b3 <> b5 & b3 <> b6 & b3 <> b7 & b4 <> b5 & b4 <> b6 & b4 <> b7 & b5 <> b6 & b5 <> b7 & b6 <> b7
holds CompF(b4,b2) = ((b3 '/\' b5) '/\' b6) '/\' b7;
:: BVFUNC14:th 26
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5, b6, b7 being a_partition of b1
st b2 is independent(b1) & b2 = {b3,b4,b5,b6,b7} & b3 <> b4 & b3 <> b5 & b3 <> b6 & b3 <> b7 & b4 <> b5 & b4 <> b6 & b4 <> b7 & b5 <> b6 & b5 <> b7 & b6 <> b7
holds CompF(b5,b2) = ((b3 '/\' b4) '/\' b6) '/\' b7;
:: BVFUNC14:th 27
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5, b6, b7 being a_partition of b1
st b2 is independent(b1) & b2 = {b3,b4,b5,b6,b7} & b3 <> b4 & b3 <> b5 & b3 <> b6 & b3 <> b7 & b4 <> b5 & b4 <> b6 & b4 <> b7 & b5 <> b6 & b5 <> b7 & b6 <> b7
holds CompF(b6,b2) = ((b3 '/\' b4) '/\' b5) '/\' b7;
:: BVFUNC14:th 28
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5, b6, b7 being a_partition of b1
st b2 is independent(b1) & b2 = {b3,b4,b5,b6,b7} & b3 <> b4 & b3 <> b5 & b3 <> b6 & b3 <> b7 & b4 <> b5 & b4 <> b6 & b4 <> b7 & b5 <> b6 & b5 <> b7 & b6 <> b7
holds CompF(b7,b2) = ((b3 '/\' b4) '/\' b5) '/\' b6;
:: BVFUNC14:th 29
theorem
for b1, b2, b3, b4, b5 being set
for b6 being Relation-like Function-like set
for b7, b8, b9, b10, b11 being set
st b1 <> b2 &
b1 <> b3 &
b1 <> b4 &
b1 <> b5 &
b2 <> b3 &
b2 <> b4 &
b2 <> b5 &
b3 <> b4 &
b3 <> b5 &
b4 <> b5 &
b6 = ((((b2 .--> b8) +* (b3 .--> b9)) +* (b4 .--> b10)) +* (b5 .--> b11)) +* (b1 .--> b7)
holds b6 . b1 = b7 & b6 . b2 = b8 & b6 . b3 = b9 & b6 . b4 = b10 & b6 . b5 = b11;
:: BVFUNC14:th 30
theorem
for b1, b2, b3, b4, b5 being set
for b6 being Relation-like Function-like set
for b7, b8, b9, b10, b11 being set
st b6 = ((((b2 .--> b8) +* (b3 .--> b9)) +* (b4 .--> b10)) +* (b5 .--> b11)) +* (b1 .--> b7)
holds proj1 b6 = {b1,b2,b3,b4,b5};
:: BVFUNC14:th 31
theorem
for b1, b2, b3, b4, b5 being set
for b6 being Relation-like Function-like set
for b7, b8, b9, b10, b11 being set
st b6 = ((((b2 .--> b8) +* (b3 .--> b9)) +* (b4 .--> b10)) +* (b5 .--> b11)) +* (b1 .--> b7)
holds proj2 b6 = {b6 . b1,b6 . b2,b6 . b3,b6 . b4,b6 . b5};
:: BVFUNC14:th 32
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5, b6, b7 being a_partition of b1
for b8, b9 being Element of b1
for b10 being Relation-like Function-like set
st b2 is independent(b1) & b2 = {b3,b4,b5,b6,b7} & b3 <> b4 & b3 <> b5 & b3 <> b6 & b3 <> b7 & b4 <> b5 & b4 <> b6 & b4 <> b7 & b5 <> b6 & b5 <> b7 & b6 <> b7
holds EqClass(b9,((b4 '/\' b5) '/\' b6) '/\' b7) meets EqClass(b8,b3);
:: BVFUNC14:th 33
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5, b6, b7 being a_partition of b1
for b8, b9 being Element of b1
st b2 is independent(b1) &
b2 = {b3,b4,b5,b6,b7} &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b5 <> b6 &
b5 <> b7 &
b6 <> b7 &
EqClass(b8,(b5 '/\' b6) '/\' b7) = EqClass(b9,(b5 '/\' b6) '/\' b7)
holds EqClass(b9,CompF(b3,b2)) meets EqClass(b8,CompF(b4,b2));
:: BVFUNC14:th 34
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5, b6, b7, b8 being a_partition of b1
st b2 = {b3,b4,b5,b6,b7,b8} & b3 <> b4 & b3 <> b5 & b3 <> b6 & b3 <> b7 & b3 <> b8 & b4 <> b5 & b4 <> b6 & b4 <> b7 & b4 <> b8 & b5 <> b6 & b5 <> b7 & b5 <> b8 & b6 <> b7 & b6 <> b8 & b7 <> b8
holds CompF(b3,b2) = (((b4 '/\' b5) '/\' b6) '/\' b7) '/\' b8;
:: BVFUNC14:th 35
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5, b6, b7, b8 being a_partition of b1
st b2 is independent(b1) & b2 = {b3,b4,b5,b6,b7,b8} & b3 <> b4 & b3 <> b5 & b3 <> b6 & b3 <> b7 & b3 <> b8 & b4 <> b5 & b4 <> b6 & b4 <> b7 & b4 <> b8 & b5 <> b6 & b5 <> b7 & b5 <> b8 & b6 <> b7 & b6 <> b8 & b7 <> b8
holds CompF(b4,b2) = (((b3 '/\' b5) '/\' b6) '/\' b7) '/\' b8;
:: BVFUNC14:th 36
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5, b6, b7, b8 being a_partition of b1
st b2 is independent(b1) & b2 = {b3,b4,b5,b6,b7,b8} & b3 <> b4 & b3 <> b5 & b3 <> b6 & b3 <> b7 & b3 <> b8 & b4 <> b5 & b4 <> b6 & b4 <> b7 & b4 <> b8 & b5 <> b6 & b5 <> b7 & b5 <> b8 & b6 <> b7 & b6 <> b8 & b7 <> b8
holds CompF(b5,b2) = (((b3 '/\' b4) '/\' b6) '/\' b7) '/\' b8;
:: BVFUNC14:th 37
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5, b6, b7, b8 being a_partition of b1
st b2 is independent(b1) & b2 = {b3,b4,b5,b6,b7,b8} & b3 <> b4 & b3 <> b5 & b3 <> b6 & b3 <> b7 & b3 <> b8 & b4 <> b5 & b4 <> b6 & b4 <> b7 & b4 <> b8 & b5 <> b6 & b5 <> b7 & b5 <> b8 & b6 <> b7 & b6 <> b8 & b7 <> b8
holds CompF(b6,b2) = (((b3 '/\' b4) '/\' b5) '/\' b7) '/\' b8;
:: BVFUNC14:th 38
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5, b6, b7, b8 being a_partition of b1
st b2 is independent(b1) & b2 = {b3,b4,b5,b6,b7,b8} & b3 <> b4 & b3 <> b5 & b3 <> b6 & b3 <> b7 & b3 <> b8 & b4 <> b5 & b4 <> b6 & b4 <> b7 & b4 <> b8 & b5 <> b6 & b5 <> b7 & b5 <> b8 & b6 <> b7 & b6 <> b8 & b7 <> b8
holds CompF(b7,b2) = (((b3 '/\' b4) '/\' b5) '/\' b6) '/\' b8;
:: BVFUNC14:th 39
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5, b6, b7, b8 being a_partition of b1
st b2 is independent(b1) & b2 = {b3,b4,b5,b6,b7,b8} & b3 <> b4 & b3 <> b5 & b3 <> b6 & b3 <> b7 & b3 <> b8 & b4 <> b5 & b4 <> b6 & b4 <> b7 & b4 <> b8 & b5 <> b6 & b5 <> b7 & b5 <> b8 & b6 <> b7 & b6 <> b8 & b7 <> b8
holds CompF(b8,b2) = (((b3 '/\' b4) '/\' b5) '/\' b6) '/\' b7;
:: BVFUNC14:th 40
theorem
for b1, b2, b3, b4, b5, b6 being set
for b7 being Relation-like Function-like set
for b8, b9, b10, b11, b12, b13 being set
st b1 <> b2 &
b1 <> b3 &
b1 <> b4 &
b1 <> b5 &
b1 <> b6 &
b2 <> b3 &
b2 <> b4 &
b2 <> b5 &
b2 <> b6 &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b4 <> b5 &
b4 <> b6 &
b5 <> b6 &
b7 = (((((b2 .--> b9) +* (b3 .--> b10)) +* (b4 .--> b11)) +* (b5 .--> b12)) +* (b6 .--> b13)) +* (b1 .--> b8)
holds b7 . b1 = b8 & b7 . b2 = b9 & b7 . b3 = b10 & b7 . b4 = b11 & b7 . b5 = b12 & b7 . b6 = b13;
:: BVFUNC14:th 41
theorem
for b1, b2, b3, b4, b5, b6 being set
for b7 being Relation-like Function-like set
for b8, b9, b10, b11, b12, b13 being set
st b7 = (((((b2 .--> b9) +* (b3 .--> b10)) +* (b4 .--> b11)) +* (b5 .--> b12)) +* (b6 .--> b13)) +* (b1 .--> b8)
holds proj1 b7 = {b1,b2,b3,b4,b5,b6};
:: BVFUNC14:th 42
theorem
for b1, b2, b3, b4, b5, b6 being set
for b7 being Relation-like Function-like set
for b8, b9, b10, b11, b12, b13 being set
st b1 <> b2 &
b1 <> b3 &
b1 <> b4 &
b1 <> b5 &
b1 <> b6 &
b2 <> b3 &
b2 <> b4 &
b2 <> b5 &
b2 <> b6 &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b4 <> b5 &
b4 <> b6 &
b5 <> b6 &
b7 = (((((b2 .--> b9) +* (b3 .--> b10)) +* (b4 .--> b11)) +* (b5 .--> b12)) +* (b6 .--> b13)) +* (b1 .--> b8)
holds proj2 b7 = {b7 . b1,b7 . b2,b7 . b3,b7 . b4,b7 . b5,b7 . b6};
:: BVFUNC14:th 43
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5, b6, b7, b8 being a_partition of b1
for b9, b10 being Element of b1
for b11 being Relation-like Function-like set
st b2 is independent(b1) & b2 = {b3,b4,b5,b6,b7,b8} & b3 <> b4 & b3 <> b5 & b3 <> b6 & b3 <> b7 & b3 <> b8 & b4 <> b5 & b4 <> b6 & b4 <> b7 & b4 <> b8 & b5 <> b6 & b5 <> b7 & b5 <> b8 & b6 <> b7 & b6 <> b8 & b7 <> b8
holds EqClass(b10,(((b4 '/\' b5) '/\' b6) '/\' b7) '/\' b8) meets EqClass(b9,b3);
:: BVFUNC14:th 44
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5, b6, b7, b8 being a_partition of b1
for b9, b10 being Element of b1
for b11 being Relation-like Function-like set
st b2 is independent(b1) &
b2 = {b3,b4,b5,b6,b7,b8} &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b3 <> b8 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b4 <> b8 &
b5 <> b6 &
b5 <> b7 &
b5 <> b8 &
b6 <> b7 &
b6 <> b8 &
b7 <> b8 &
EqClass(b9,((b5 '/\' b6) '/\' b7) '/\' b8) = EqClass(b10,((b5 '/\' b6) '/\' b7) '/\' b8)
holds EqClass(b10,CompF(b3,b2)) meets EqClass(b9,CompF(b4,b2));