Step |
Hyp |
Ref |
Expression |
1 |
|
rngchomrnghmresALTV.c |
|- C = ( RngCatALTV ` U ) |
2 |
|
rngchomrnghmresALTV.b |
|- B = ( Rng i^i U ) |
3 |
|
rngchomrnghmresALTV.u |
|- ( ph -> U e. V ) |
4 |
|
rngchomrnghmresALTV.f |
|- F = ( Homf ` C ) |
5 |
|
eqid |
|- ( Base ` C ) = ( Base ` C ) |
6 |
1 5 3
|
rngcbasALTV |
|- ( ph -> ( Base ` C ) = ( U i^i Rng ) ) |
7 |
|
inss2 |
|- ( U i^i Rng ) C_ Rng |
8 |
6 7
|
eqsstrdi |
|- ( ph -> ( Base ` C ) C_ Rng ) |
9 |
|
resmpo |
|- ( ( ( Base ` C ) C_ Rng /\ ( Base ` C ) C_ Rng ) -> ( ( x e. Rng , y e. Rng |-> ( x RngHomo y ) ) |` ( ( Base ` C ) X. ( Base ` C ) ) ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( x RngHomo y ) ) ) |
10 |
8 8 9
|
syl2anc |
|- ( ph -> ( ( x e. Rng , y e. Rng |-> ( x RngHomo y ) ) |` ( ( Base ` C ) X. ( Base ` C ) ) ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( x RngHomo y ) ) ) |
11 |
|
df-rnghomo |
|- RngHomo = ( r e. Rng , s e. Rng |-> [_ ( Base ` r ) / v ]_ [_ ( Base ` s ) / w ]_ { f e. ( w ^m v ) | A. x e. v A. y e. v ( ( f ` ( x ( +g ` r ) y ) ) = ( ( f ` x ) ( +g ` s ) ( f ` y ) ) /\ ( f ` ( x ( .r ` r ) y ) ) = ( ( f ` x ) ( .r ` s ) ( f ` y ) ) ) } ) |
12 |
|
ovex |
|- ( w ^m v ) e. _V |
13 |
12
|
rabex |
|- { f e. ( w ^m v ) | A. x e. v A. y e. v ( ( f ` ( x ( +g ` r ) y ) ) = ( ( f ` x ) ( +g ` s ) ( f ` y ) ) /\ ( f ` ( x ( .r ` r ) y ) ) = ( ( f ` x ) ( .r ` s ) ( f ` y ) ) ) } e. _V |
14 |
13
|
csbex |
|- [_ ( Base ` s ) / w ]_ { f e. ( w ^m v ) | A. x e. v A. y e. v ( ( f ` ( x ( +g ` r ) y ) ) = ( ( f ` x ) ( +g ` s ) ( f ` y ) ) /\ ( f ` ( x ( .r ` r ) y ) ) = ( ( f ` x ) ( .r ` s ) ( f ` y ) ) ) } e. _V |
15 |
14
|
csbex |
|- [_ ( Base ` r ) / v ]_ [_ ( Base ` s ) / w ]_ { f e. ( w ^m v ) | A. x e. v A. y e. v ( ( f ` ( x ( +g ` r ) y ) ) = ( ( f ` x ) ( +g ` s ) ( f ` y ) ) /\ ( f ` ( x ( .r ` r ) y ) ) = ( ( f ` x ) ( .r ` s ) ( f ` y ) ) ) } e. _V |
16 |
11 15
|
fnmpoi |
|- RngHomo Fn ( Rng X. Rng ) |
17 |
16
|
a1i |
|- ( ph -> RngHomo Fn ( Rng X. Rng ) ) |
18 |
|
fnov |
|- ( RngHomo Fn ( Rng X. Rng ) <-> RngHomo = ( x e. Rng , y e. Rng |-> ( x RngHomo y ) ) ) |
19 |
17 18
|
sylib |
|- ( ph -> RngHomo = ( x e. Rng , y e. Rng |-> ( x RngHomo y ) ) ) |
20 |
|
incom |
|- ( U i^i Rng ) = ( Rng i^i U ) |
21 |
20
|
a1i |
|- ( ph -> ( U i^i Rng ) = ( Rng i^i U ) ) |
22 |
2
|
a1i |
|- ( ph -> B = ( Rng i^i U ) ) |
23 |
21 6 22
|
3eqtr4rd |
|- ( ph -> B = ( Base ` C ) ) |
24 |
23
|
sqxpeqd |
|- ( ph -> ( B X. B ) = ( ( Base ` C ) X. ( Base ` C ) ) ) |
25 |
19 24
|
reseq12d |
|- ( ph -> ( RngHomo |` ( B X. B ) ) = ( ( x e. Rng , y e. Rng |-> ( x RngHomo y ) ) |` ( ( Base ` C ) X. ( Base ` C ) ) ) ) |
26 |
1 5 3 4
|
rngchomffvalALTV |
|- ( ph -> F = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( x RngHomo y ) ) ) |
27 |
10 25 26
|
3eqtr4rd |
|- ( ph -> F = ( RngHomo |` ( B X. B ) ) ) |