:: BVFUNC14 semantic presentation
theorem Th1: :: BVFUNC14:1
theorem Th2: :: BVFUNC14:2
Lemma2:
for b1, b2, b3, b4, b5, b6 being set holds dom (((b1 .--> b4) +* (b2 .--> b5)) +* (b3 .--> b6)) = {b1,b2,b3}
Lemma3:
for b1 being Function
for b2, b3, b4, b5 being set holds
( b2 <> b3 implies ((b1 +* (b2 .--> b4)) +* (b3 .--> b5)) . b2 = b4 )
Lemma4:
for b1, b2, b3, b4, b5, b6 being set holds
( b1 <> b2 & b3 <> b1 implies (((b1 .--> b4) +* (b2 .--> b5)) +* (b3 .--> b6)) . b1 = b4 )
Lemma5:
for b1, b2, b3, b4, b5, b6 being set
for b7 being Function holds
( b7 = ((b1 .--> b4) +* (b2 .--> b5)) +* (b3 .--> b6) implies rng b7 = {(b7 . b1),(b7 . b2),(b7 . b3)} )
theorem Th3: :: BVFUNC14:3
theorem Th4: :: BVFUNC14:4
theorem Th5: :: BVFUNC14:5
theorem Th6: :: BVFUNC14:6
theorem Th7: :: BVFUNC14:7
theorem Th8: :: BVFUNC14:8
theorem Th9: :: BVFUNC14:9
theorem Th10: :: BVFUNC14:10
theorem Th11: :: BVFUNC14:11
canceled;
theorem Th12: :: BVFUNC14:12
canceled;
theorem Th13: :: BVFUNC14:13
canceled;
theorem Th14: :: BVFUNC14:14
theorem Th15: :: BVFUNC14:15
theorem Th16: :: BVFUNC14:16
theorem Th17: :: BVFUNC14:17
theorem Th18: :: BVFUNC14:18
theorem Th19: :: BVFUNC14:19
theorem Th20: :: BVFUNC14:20
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6 being
a_partition of b
1for b
7 being
Functionfor b
8, b
9, b
10, b
11 being
set holds
( b
2 = {b3,b4,b5,b6} & b
7 = (((b4 .--> b9) +* (b5 .--> b10)) +* (b6 .--> b11)) +* (b3 .--> b8) implies
rng b
7 = {(b7 . b3),(b7 . b4),(b7 . b5),(b7 . b6)} )
theorem Th21: :: BVFUNC14:21
theorem Th22: :: BVFUNC14:22
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6 being
a_partition of b
1for b
7, b
8 being
Element of b
1 holds
not ( b
2 is
independent & b
2 = {b3,b4,b5,b6} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
4 <> b
5 & b
4 <> b
6 & b
5 <> b
6 &
EqClass b
7,
(b5 '/\' b6) = EqClass b
8,
(b5 '/\' b6) & not
EqClass b
8,
(CompF b3,b2) meets EqClass b
7,
(CompF b4,b2) )
theorem Th23: :: BVFUNC14:23
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5 being
a_partition of b
1for b
6, b
7 being
Element of b
1 holds
not ( b
2 is
independent & b
2 = {b3,b4,b5} & b
3 <> b
4 & b
4 <> b
5 & b
5 <> b
3 &
EqClass b
6,b
5 = EqClass b
7,b
5 & not
EqClass b
7,
(CompF b3,b2) meets EqClass b
6,
(CompF b4,b2) )
theorem Th24: :: BVFUNC14:24
theorem Th25: :: BVFUNC14:25
theorem Th26: :: BVFUNC14:26
theorem Th27: :: BVFUNC14:27
theorem Th28: :: BVFUNC14:28
theorem Th29: :: BVFUNC14:29
for b
1, b
2, b
3, b
4, b
5 being
set for b
6 being
Functionfor b
7, b
8, b
9, b
10, b
11 being
set holds
( b
1 <> b
2 & b
1 <> b
3 & b
1 <> b
4 & b
1 <> b
5 & b
2 <> b
3 & b
2 <> b
4 & b
2 <> b
5 & b
3 <> b
4 & b
3 <> b
5 & b
4 <> b
5 & b
6 = ((((b2 .--> b8) +* (b3 .--> b9)) +* (b4 .--> b10)) +* (b5 .--> b11)) +* (b1 .--> b7) implies ( b
6 . b
1 = b
7 & b
6 . b
2 = b
8 & b
6 . b
3 = b
9 & b
6 . b
4 = b
10 & b
6 . b
5 = b
11 ) )
theorem Th30: :: BVFUNC14:30
for b
1, b
2, b
3, b
4, b
5 being
set for b
6 being
Functionfor b
7, b
8, b
9, b
10, b
11 being
set holds
( b
6 = ((((b2 .--> b8) +* (b3 .--> b9)) +* (b4 .--> b10)) +* (b5 .--> b11)) +* (b1 .--> b7) implies
dom b
6 = {b1,b2,b3,b4,b5} )
theorem Th31: :: BVFUNC14:31
for b
1, b
2, b
3, b
4, b
5 being
set for b
6 being
Functionfor b
7, b
8, b
9, b
10, b
11 being
set holds
( b
6 = ((((b2 .--> b8) +* (b3 .--> b9)) +* (b4 .--> b10)) +* (b5 .--> b11)) +* (b1 .--> b7) implies
rng b
6 = {(b6 . b1),(b6 . b2),(b6 . b3),(b6 . b4),(b6 . b5)} )
theorem Th32: :: BVFUNC14:32
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7 being
a_partition of b
1for b
8, b
9 being
Element of b
1for b
10 being
Function holds
not ( b
2 is
independent & b
2 = {b3,b4,b5,b6,b7} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
5 <> b
6 & b
5 <> b
7 & b
6 <> b
7 & not
EqClass b
9,
(((b4 '/\' b5) '/\' b6) '/\' b7) meets EqClass b
8,b
3 )
theorem Th33: :: BVFUNC14:33
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7 being
a_partition of b
1for b
8, b
9 being
Element of b
1 holds
not ( b
2 is
independent & b
2 = {b3,b4,b5,b6,b7} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
5 <> b
6 & b
5 <> b
7 & b
6 <> b
7 &
EqClass b
8,
((b5 '/\' b6) '/\' b7) = EqClass b
9,
((b5 '/\' b6) '/\' b7) & not
EqClass b
9,
(CompF b3,b2) meets EqClass b
8,
(CompF b4,b2) )
theorem Th34: :: BVFUNC14:34
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8 being
a_partition of b
1 holds
( b
2 = {b3,b4,b5,b6,b7,b8} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
6 <> b
7 & b
6 <> b
8 & b
7 <> b
8 implies
CompF b
3,b
2 = (((b4 '/\' b5) '/\' b6) '/\' b7) '/\' b
8 )
theorem Th35: :: BVFUNC14:35
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8 being
a_partition of b
1 holds
( b
2 is
independent & b
2 = {b3,b4,b5,b6,b7,b8} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
6 <> b
7 & b
6 <> b
8 & b
7 <> b
8 implies
CompF b
4,b
2 = (((b3 '/\' b5) '/\' b6) '/\' b7) '/\' b
8 )
theorem Th36: :: BVFUNC14:36
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8 being
a_partition of b
1 holds
( b
2 is
independent & b
2 = {b3,b4,b5,b6,b7,b8} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
6 <> b
7 & b
6 <> b
8 & b
7 <> b
8 implies
CompF b
5,b
2 = (((b3 '/\' b4) '/\' b6) '/\' b7) '/\' b
8 )
theorem Th37: :: BVFUNC14:37
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8 being
a_partition of b
1 holds
( b
2 is
independent & b
2 = {b3,b4,b5,b6,b7,b8} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
6 <> b
7 & b
6 <> b
8 & b
7 <> b
8 implies
CompF b
6,b
2 = (((b3 '/\' b4) '/\' b5) '/\' b7) '/\' b
8 )
theorem Th38: :: BVFUNC14:38
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8 being
a_partition of b
1 holds
( b
2 is
independent & b
2 = {b3,b4,b5,b6,b7,b8} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
6 <> b
7 & b
6 <> b
8 & b
7 <> b
8 implies
CompF b
7,b
2 = (((b3 '/\' b4) '/\' b5) '/\' b6) '/\' b
8 )
theorem Th39: :: BVFUNC14:39
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8 being
a_partition of b
1 holds
( b
2 is
independent & b
2 = {b3,b4,b5,b6,b7,b8} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
6 <> b
7 & b
6 <> b
8 & b
7 <> b
8 implies
CompF b
8,b
2 = (((b3 '/\' b4) '/\' b5) '/\' b6) '/\' b
7 )
theorem Th40: :: BVFUNC14:40
for b
1, b
2, b
3, b
4, b
5, b
6 being
set for b
7 being
Functionfor b
8, b
9, b
10, b
11, b
12, b
13 being
set holds
( b
1 <> b
2 & b
1 <> b
3 & b
1 <> b
4 & b
1 <> b
5 & b
1 <> b
6 & b
2 <> b
3 & b
2 <> b
4 & b
2 <> b
5 & b
2 <> b
6 & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
4 <> b
5 & b
4 <> b
6 & b
5 <> b
6 & b
7 = (((((b2 .--> b9) +* (b3 .--> b10)) +* (b4 .--> b11)) +* (b5 .--> b12)) +* (b6 .--> b13)) +* (b1 .--> b8) implies ( b
7 . b
1 = b
8 & b
7 . b
2 = b
9 & b
7 . b
3 = b
10 & b
7 . b
4 = b
11 & b
7 . b
5 = b
12 & b
7 . b
6 = b
13 ) )
theorem Th41: :: BVFUNC14:41
for b
1, b
2, b
3, b
4, b
5, b
6 being
set for b
7 being
Functionfor b
8, b
9, b
10, b
11, b
12, b
13 being
set holds
( b
7 = (((((b2 .--> b9) +* (b3 .--> b10)) +* (b4 .--> b11)) +* (b5 .--> b12)) +* (b6 .--> b13)) +* (b1 .--> b8) implies
dom b
7 = {b1,b2,b3,b4,b5,b6} )
theorem Th42: :: BVFUNC14:42
for b
1, b
2, b
3, b
4, b
5, b
6 being
set for b
7 being
Functionfor b
8, b
9, b
10, b
11, b
12, b
13 being
set holds
( b
1 <> b
2 & b
1 <> b
3 & b
1 <> b
4 & b
1 <> b
5 & b
1 <> b
6 & b
2 <> b
3 & b
2 <> b
4 & b
2 <> b
5 & b
2 <> b
6 & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
4 <> b
5 & b
4 <> b
6 & b
5 <> b
6 & b
7 = (((((b2 .--> b9) +* (b3 .--> b10)) +* (b4 .--> b11)) +* (b5 .--> b12)) +* (b6 .--> b13)) +* (b1 .--> b8) implies
rng b
7 = {(b7 . b1),(b7 . b2),(b7 . b3),(b7 . b4),(b7 . b5),(b7 . b6)} )
theorem Th43: :: BVFUNC14:43
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8 being
a_partition of b
1for b
9, b
10 being
Element of b
1for b
11 being
Function holds
not ( b
2 is
independent & b
2 = {b3,b4,b5,b6,b7,b8} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
6 <> b
7 & b
6 <> b
8 & b
7 <> b
8 & not
EqClass b
10,
((((b4 '/\' b5) '/\' b6) '/\' b7) '/\' b8) meets EqClass b
9,b
3 )
theorem Th44: :: BVFUNC14:44
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8 being
a_partition of b
1for b
9, b
10 being
Element of b
1for b
11 being
Function holds
not ( b
2 is
independent & b
2 = {b3,b4,b5,b6,b7,b8} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
6 <> b
7 & b
6 <> b
8 & b
7 <> b
8 &
EqClass b
9,
(((b5 '/\' b6) '/\' b7) '/\' b8) = EqClass b
10,
(((b5 '/\' b6) '/\' b7) '/\' b8) & not
EqClass b
10,
(CompF b3,b2) meets EqClass b
9,
(CompF b4,b2) )