:: BVFUNC_3 semantic presentation
theorem Th1: :: BVFUNC_3:1
theorem Th2: :: BVFUNC_3:2
theorem Th3: :: BVFUNC_3:3
theorem Th4: :: BVFUNC_3:4
for
Y being non
empty set for
G being
Subset of
(PARTITIONS Y) for
a,
b being
Element of
Funcs Y,
BOOLEAN for
PA being
a_partition of
Y holds
'not' ((All a,PA,G) '&' (All b,PA,G)) = (Ex ('not' a),PA,G) 'or' (Ex ('not' b),PA,G)
theorem Th5: :: BVFUNC_3:5
for
Y being non
empty set for
G being
Subset of
(PARTITIONS Y) for
a,
b being
Element of
Funcs Y,
BOOLEAN for
PA being
a_partition of
Y holds
'not' ((Ex a,PA,G) '&' (Ex b,PA,G)) = (All ('not' a),PA,G) 'or' (All ('not' b),PA,G)
theorem Th6: :: BVFUNC_3:6
theorem Th7: :: BVFUNC_3:7
theorem Th8: :: BVFUNC_3:8
for
Y being non
empty set for
G being
Subset of
(PARTITIONS Y) for
a,
b being
Element of
Funcs Y,
BOOLEAN for
PA being
a_partition of
Y holds
a 'xor' b '<' ('not' ((Ex ('not' a),PA,G) 'xor' (Ex b,PA,G))) 'or' ('not' ((Ex a,PA,G) 'xor' (Ex ('not' b),PA,G)))
theorem Th9: :: BVFUNC_3:9
E42:
now
let Y be non
empty set ;
let a be
Element of
Funcs Y,
BOOLEAN ;
let b be
Element of
Funcs Y,
BOOLEAN ;
let G be
Subset of
(PARTITIONS Y);
let PA be
a_partition of
Y;
let z be
Element of
Y;
assume E18:
(All (a 'or' b),PA,G) . z = TRUE
;
assume
ex
x being
Element of
Y st
(
x in EqClass z,
(CompF PA,G) & not
(a 'or' b) . x = TRUE )
;
then
(B_INF (a 'or' b),(CompF PA,G)) . z = FALSE
by BVFUNC_1:def 19;
then
(All (a 'or' b),PA,G) . z = FALSE
by BVFUNC_2:def 9;
hence
contradiction
by ;
end;
theorem Th10: :: BVFUNC_3:10
theorem Th11: :: BVFUNC_3:11
theorem Th12: :: BVFUNC_3:12
theorem Th13: :: BVFUNC_3:13
E43:
now
let Y be non
empty set ;
let a be
Element of
Funcs Y,
BOOLEAN ;
let b be
Element of
Funcs Y,
BOOLEAN ;
let G be
Subset of
(PARTITIONS Y);
let PA be
a_partition of
Y;
let z be
Element of
Y;
assume E18:
(All (a 'imp' b),PA,G) . z = TRUE
;
assume
ex
x being
Element of
Y st
(
x in EqClass z,
(CompF PA,G) & not
(a 'imp' b) . x = TRUE )
;
then
(B_INF (a 'imp' b),(CompF PA,G)) . z = FALSE
by BVFUNC_1:def 19;
then
(All (a 'imp' b),PA,G) . z = FALSE
by BVFUNC_2:def 9;
hence
contradiction
by ;
end;
theorem Th14: :: BVFUNC_3:14
theorem Th15: :: BVFUNC_3:15
theorem Th16: :: BVFUNC_3:16
theorem Th17: :: BVFUNC_3:17
theorem Th18: :: BVFUNC_3:18
theorem Th19: :: BVFUNC_3:19
theorem Th20: :: BVFUNC_3:20
theorem Th21: :: BVFUNC_3:21
theorem Th22: :: BVFUNC_3:22
theorem Th23: :: BVFUNC_3:23
theorem Th24: :: BVFUNC_3:24
for
Y being non
empty set for
G being
Subset of
(PARTITIONS Y) for
a,
b being
Element of
Funcs Y,
BOOLEAN for
PA being
a_partition of
Y holds
(All a,PA,G) 'imp' (All b,PA,G) '<' (All a,PA,G) 'imp' (Ex b,PA,G)
theorem Th25: :: BVFUNC_3:25
for
Y being non
empty set for
G being
Subset of
(PARTITIONS Y) for
a,
b being
Element of
Funcs Y,
BOOLEAN for
PA being
a_partition of
Y holds
(Ex a,PA,G) 'imp' (Ex b,PA,G) '<' (All a,PA,G) 'imp' (Ex b,PA,G)
theorem Th26: :: BVFUNC_3:26
theorem Th27: :: BVFUNC_3:27
theorem Th28: :: BVFUNC_3:28
theorem Th29: :: BVFUNC_3:29
theorem Th30: :: BVFUNC_3:30
theorem Th31: :: BVFUNC_3:31
theorem Th32: :: BVFUNC_3:32
theorem Th33: :: BVFUNC_3:33
theorem Th34: :: BVFUNC_3:34
theorem Th35: :: BVFUNC_3:35
theorem Th36: :: BVFUNC_3:36
theorem Th37: :: BVFUNC_3:37
theorem Th38: :: BVFUNC_3:38
for
Y being non
empty set for
G being
Subset of
(PARTITIONS Y) for
c,
b,
a being
Element of
Funcs Y,
BOOLEAN for
PA being
a_partition of
Y holds
((Ex c,PA,G) '&' (All (c 'imp' b),PA,G)) '&' (All (c 'imp' a),PA,G) '<' Ex (a '&' b),
PA,
G
theorem Th39: :: BVFUNC_3:39
theorem Th40: :: BVFUNC_3:40
for
Y being non
empty set for
G being
Subset of
(PARTITIONS Y) for
b,
c,
a being
Element of
Funcs Y,
BOOLEAN for
PA being
a_partition of
Y holds
((Ex b,PA,G) '&' (All (b 'imp' c),PA,G)) '&' (All (c 'imp' a),PA,G) '<' Ex (a '&' b),
PA,
G
theorem Th41: :: BVFUNC_3:41
for
Y being non
empty set for
G being
Subset of
(PARTITIONS Y) for
c,
b,
a being
Element of
Funcs Y,
BOOLEAN for
PA being
a_partition of
Y holds
((Ex c,PA,G) '&' (All (b 'imp' ('not' c)),PA,G)) '&' (All (c 'imp' a),PA,G) '<' Ex (a '&' ('not' b)),
PA,
G