Step |
Hyp |
Ref |
Expression |
1 |
|
suppofssd.1 |
|- ( ph -> A e. V ) |
2 |
|
suppofssd.2 |
|- ( ph -> Z e. B ) |
3 |
|
suppofssd.3 |
|- ( ph -> F : A --> B ) |
4 |
|
suppofssd.4 |
|- ( ph -> G : A --> B ) |
5 |
|
suppofssd.5 |
|- ( ph -> ( Z X Z ) = Z ) |
6 |
|
ovexd |
|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x X y ) e. _V ) |
7 |
|
inidm |
|- ( A i^i A ) = A |
8 |
6 3 4 1 1 7
|
off |
|- ( ph -> ( F oF X G ) : A --> _V ) |
9 |
|
eldif |
|- ( k e. ( A \ ( ( F supp Z ) u. ( G supp Z ) ) ) <-> ( k e. A /\ -. k e. ( ( F supp Z ) u. ( G supp Z ) ) ) ) |
10 |
|
ioran |
|- ( -. ( k e. ( F supp Z ) \/ k e. ( G supp Z ) ) <-> ( -. k e. ( F supp Z ) /\ -. k e. ( G supp Z ) ) ) |
11 |
|
elun |
|- ( k e. ( ( F supp Z ) u. ( G supp Z ) ) <-> ( k e. ( F supp Z ) \/ k e. ( G supp Z ) ) ) |
12 |
10 11
|
xchnxbir |
|- ( -. k e. ( ( F supp Z ) u. ( G supp Z ) ) <-> ( -. k e. ( F supp Z ) /\ -. k e. ( G supp Z ) ) ) |
13 |
12
|
anbi2i |
|- ( ( k e. A /\ -. k e. ( ( F supp Z ) u. ( G supp Z ) ) ) <-> ( k e. A /\ ( -. k e. ( F supp Z ) /\ -. k e. ( G supp Z ) ) ) ) |
14 |
9 13
|
bitri |
|- ( k e. ( A \ ( ( F supp Z ) u. ( G supp Z ) ) ) <-> ( k e. A /\ ( -. k e. ( F supp Z ) /\ -. k e. ( G supp Z ) ) ) ) |
15 |
3
|
ffnd |
|- ( ph -> F Fn A ) |
16 |
|
elsuppfn |
|- ( ( F Fn A /\ A e. V /\ Z e. B ) -> ( k e. ( F supp Z ) <-> ( k e. A /\ ( F ` k ) =/= Z ) ) ) |
17 |
15 1 2 16
|
syl3anc |
|- ( ph -> ( k e. ( F supp Z ) <-> ( k e. A /\ ( F ` k ) =/= Z ) ) ) |
18 |
17
|
notbid |
|- ( ph -> ( -. k e. ( F supp Z ) <-> -. ( k e. A /\ ( F ` k ) =/= Z ) ) ) |
19 |
18
|
biimpd |
|- ( ph -> ( -. k e. ( F supp Z ) -> -. ( k e. A /\ ( F ` k ) =/= Z ) ) ) |
20 |
4
|
ffnd |
|- ( ph -> G Fn A ) |
21 |
|
elsuppfn |
|- ( ( G Fn A /\ A e. V /\ Z e. B ) -> ( k e. ( G supp Z ) <-> ( k e. A /\ ( G ` k ) =/= Z ) ) ) |
22 |
20 1 2 21
|
syl3anc |
|- ( ph -> ( k e. ( G supp Z ) <-> ( k e. A /\ ( G ` k ) =/= Z ) ) ) |
23 |
22
|
notbid |
|- ( ph -> ( -. k e. ( G supp Z ) <-> -. ( k e. A /\ ( G ` k ) =/= Z ) ) ) |
24 |
23
|
biimpd |
|- ( ph -> ( -. k e. ( G supp Z ) -> -. ( k e. A /\ ( G ` k ) =/= Z ) ) ) |
25 |
19 24
|
anim12d |
|- ( ph -> ( ( -. k e. ( F supp Z ) /\ -. k e. ( G supp Z ) ) -> ( -. ( k e. A /\ ( F ` k ) =/= Z ) /\ -. ( k e. A /\ ( G ` k ) =/= Z ) ) ) ) |
26 |
25
|
anim2d |
|- ( ph -> ( ( k e. A /\ ( -. k e. ( F supp Z ) /\ -. k e. ( G supp Z ) ) ) -> ( k e. A /\ ( -. ( k e. A /\ ( F ` k ) =/= Z ) /\ -. ( k e. A /\ ( G ` k ) =/= Z ) ) ) ) ) |
27 |
26
|
imp |
|- ( ( ph /\ ( k e. A /\ ( -. k e. ( F supp Z ) /\ -. k e. ( G supp Z ) ) ) ) -> ( k e. A /\ ( -. ( k e. A /\ ( F ` k ) =/= Z ) /\ -. ( k e. A /\ ( G ` k ) =/= Z ) ) ) ) |
28 |
|
pm3.2 |
|- ( k e. A -> ( ( F ` k ) =/= Z -> ( k e. A /\ ( F ` k ) =/= Z ) ) ) |
29 |
28
|
necon1bd |
|- ( k e. A -> ( -. ( k e. A /\ ( F ` k ) =/= Z ) -> ( F ` k ) = Z ) ) |
30 |
|
pm3.2 |
|- ( k e. A -> ( ( G ` k ) =/= Z -> ( k e. A /\ ( G ` k ) =/= Z ) ) ) |
31 |
30
|
necon1bd |
|- ( k e. A -> ( -. ( k e. A /\ ( G ` k ) =/= Z ) -> ( G ` k ) = Z ) ) |
32 |
29 31
|
anim12d |
|- ( k e. A -> ( ( -. ( k e. A /\ ( F ` k ) =/= Z ) /\ -. ( k e. A /\ ( G ` k ) =/= Z ) ) -> ( ( F ` k ) = Z /\ ( G ` k ) = Z ) ) ) |
33 |
32
|
imdistani |
|- ( ( k e. A /\ ( -. ( k e. A /\ ( F ` k ) =/= Z ) /\ -. ( k e. A /\ ( G ` k ) =/= Z ) ) ) -> ( k e. A /\ ( ( F ` k ) = Z /\ ( G ` k ) = Z ) ) ) |
34 |
15
|
adantr |
|- ( ( ph /\ ( k e. A /\ ( ( F ` k ) = Z /\ ( G ` k ) = Z ) ) ) -> F Fn A ) |
35 |
20
|
adantr |
|- ( ( ph /\ ( k e. A /\ ( ( F ` k ) = Z /\ ( G ` k ) = Z ) ) ) -> G Fn A ) |
36 |
1
|
adantr |
|- ( ( ph /\ ( k e. A /\ ( ( F ` k ) = Z /\ ( G ` k ) = Z ) ) ) -> A e. V ) |
37 |
|
simprl |
|- ( ( ph /\ ( k e. A /\ ( ( F ` k ) = Z /\ ( G ` k ) = Z ) ) ) -> k e. A ) |
38 |
|
fnfvof |
|- ( ( ( F Fn A /\ G Fn A ) /\ ( A e. V /\ k e. A ) ) -> ( ( F oF X G ) ` k ) = ( ( F ` k ) X ( G ` k ) ) ) |
39 |
34 35 36 37 38
|
syl22anc |
|- ( ( ph /\ ( k e. A /\ ( ( F ` k ) = Z /\ ( G ` k ) = Z ) ) ) -> ( ( F oF X G ) ` k ) = ( ( F ` k ) X ( G ` k ) ) ) |
40 |
|
oveq12 |
|- ( ( ( F ` k ) = Z /\ ( G ` k ) = Z ) -> ( ( F ` k ) X ( G ` k ) ) = ( Z X Z ) ) |
41 |
40
|
ad2antll |
|- ( ( ph /\ ( k e. A /\ ( ( F ` k ) = Z /\ ( G ` k ) = Z ) ) ) -> ( ( F ` k ) X ( G ` k ) ) = ( Z X Z ) ) |
42 |
5
|
adantr |
|- ( ( ph /\ ( k e. A /\ ( ( F ` k ) = Z /\ ( G ` k ) = Z ) ) ) -> ( Z X Z ) = Z ) |
43 |
39 41 42
|
3eqtrd |
|- ( ( ph /\ ( k e. A /\ ( ( F ` k ) = Z /\ ( G ` k ) = Z ) ) ) -> ( ( F oF X G ) ` k ) = Z ) |
44 |
33 43
|
sylan2 |
|- ( ( ph /\ ( k e. A /\ ( -. ( k e. A /\ ( F ` k ) =/= Z ) /\ -. ( k e. A /\ ( G ` k ) =/= Z ) ) ) ) -> ( ( F oF X G ) ` k ) = Z ) |
45 |
27 44
|
syldan |
|- ( ( ph /\ ( k e. A /\ ( -. k e. ( F supp Z ) /\ -. k e. ( G supp Z ) ) ) ) -> ( ( F oF X G ) ` k ) = Z ) |
46 |
14 45
|
sylan2b |
|- ( ( ph /\ k e. ( A \ ( ( F supp Z ) u. ( G supp Z ) ) ) ) -> ( ( F oF X G ) ` k ) = Z ) |
47 |
8 46
|
suppss |
|- ( ph -> ( ( F oF X G ) supp Z ) C_ ( ( F supp Z ) u. ( G supp Z ) ) ) |