Step |
Hyp |
Ref |
Expression |
1 |
|
mulc1cncfg.1 |
|- F/_ x F |
2 |
|
mulc1cncfg.2 |
|- F/ x ph |
3 |
|
mulc1cncfg.3 |
|- ( ph -> F e. ( A -cn-> CC ) ) |
4 |
|
mulc1cncfg.4 |
|- ( ph -> B e. CC ) |
5 |
|
eqid |
|- ( x e. CC |-> ( B x. x ) ) = ( x e. CC |-> ( B x. x ) ) |
6 |
5
|
mulc1cncf |
|- ( B e. CC -> ( x e. CC |-> ( B x. x ) ) e. ( CC -cn-> CC ) ) |
7 |
4 6
|
syl |
|- ( ph -> ( x e. CC |-> ( B x. x ) ) e. ( CC -cn-> CC ) ) |
8 |
|
cncff |
|- ( ( x e. CC |-> ( B x. x ) ) e. ( CC -cn-> CC ) -> ( x e. CC |-> ( B x. x ) ) : CC --> CC ) |
9 |
7 8
|
syl |
|- ( ph -> ( x e. CC |-> ( B x. x ) ) : CC --> CC ) |
10 |
|
cncff |
|- ( F e. ( A -cn-> CC ) -> F : A --> CC ) |
11 |
3 10
|
syl |
|- ( ph -> F : A --> CC ) |
12 |
|
fcompt |
|- ( ( ( x e. CC |-> ( B x. x ) ) : CC --> CC /\ F : A --> CC ) -> ( ( x e. CC |-> ( B x. x ) ) o. F ) = ( t e. A |-> ( ( x e. CC |-> ( B x. x ) ) ` ( F ` t ) ) ) ) |
13 |
9 11 12
|
syl2anc |
|- ( ph -> ( ( x e. CC |-> ( B x. x ) ) o. F ) = ( t e. A |-> ( ( x e. CC |-> ( B x. x ) ) ` ( F ` t ) ) ) ) |
14 |
11
|
ffvelrnda |
|- ( ( ph /\ t e. A ) -> ( F ` t ) e. CC ) |
15 |
4
|
adantr |
|- ( ( ph /\ t e. A ) -> B e. CC ) |
16 |
15 14
|
mulcld |
|- ( ( ph /\ t e. A ) -> ( B x. ( F ` t ) ) e. CC ) |
17 |
|
nfcv |
|- F/_ x t |
18 |
1 17
|
nffv |
|- F/_ x ( F ` t ) |
19 |
|
nfcv |
|- F/_ x B |
20 |
|
nfcv |
|- F/_ x x. |
21 |
19 20 18
|
nfov |
|- F/_ x ( B x. ( F ` t ) ) |
22 |
|
oveq2 |
|- ( x = ( F ` t ) -> ( B x. x ) = ( B x. ( F ` t ) ) ) |
23 |
18 21 22 5
|
fvmptf |
|- ( ( ( F ` t ) e. CC /\ ( B x. ( F ` t ) ) e. CC ) -> ( ( x e. CC |-> ( B x. x ) ) ` ( F ` t ) ) = ( B x. ( F ` t ) ) ) |
24 |
14 16 23
|
syl2anc |
|- ( ( ph /\ t e. A ) -> ( ( x e. CC |-> ( B x. x ) ) ` ( F ` t ) ) = ( B x. ( F ` t ) ) ) |
25 |
24
|
mpteq2dva |
|- ( ph -> ( t e. A |-> ( ( x e. CC |-> ( B x. x ) ) ` ( F ` t ) ) ) = ( t e. A |-> ( B x. ( F ` t ) ) ) ) |
26 |
|
nfcv |
|- F/_ t B |
27 |
|
nfcv |
|- F/_ t x. |
28 |
|
nfcv |
|- F/_ t ( F ` x ) |
29 |
26 27 28
|
nfov |
|- F/_ t ( B x. ( F ` x ) ) |
30 |
|
fveq2 |
|- ( t = x -> ( F ` t ) = ( F ` x ) ) |
31 |
30
|
oveq2d |
|- ( t = x -> ( B x. ( F ` t ) ) = ( B x. ( F ` x ) ) ) |
32 |
21 29 31
|
cbvmpt |
|- ( t e. A |-> ( B x. ( F ` t ) ) ) = ( x e. A |-> ( B x. ( F ` x ) ) ) |
33 |
25 32
|
eqtrdi |
|- ( ph -> ( t e. A |-> ( ( x e. CC |-> ( B x. x ) ) ` ( F ` t ) ) ) = ( x e. A |-> ( B x. ( F ` x ) ) ) ) |
34 |
13 33
|
eqtrd |
|- ( ph -> ( ( x e. CC |-> ( B x. x ) ) o. F ) = ( x e. A |-> ( B x. ( F ` x ) ) ) ) |
35 |
3 7
|
cncfco |
|- ( ph -> ( ( x e. CC |-> ( B x. x ) ) o. F ) e. ( A -cn-> CC ) ) |
36 |
34 35
|
eqeltrrd |
|- ( ph -> ( x e. A |-> ( B x. ( F ` x ) ) ) e. ( A -cn-> CC ) ) |