Step |
Hyp |
Ref |
Expression |
1 |
|
rhmsscrnghm.u |
|- ( ph -> U e. V ) |
2 |
|
rhmsscrnghm.r |
|- ( ph -> R = ( Ring i^i U ) ) |
3 |
|
rhmsscrnghm.s |
|- ( ph -> S = ( Rng i^i U ) ) |
4 |
|
ringrng |
|- ( r e. Ring -> r e. Rng ) |
5 |
4
|
a1i |
|- ( ph -> ( r e. Ring -> r e. Rng ) ) |
6 |
5
|
ssrdv |
|- ( ph -> Ring C_ Rng ) |
7 |
6
|
ssrind |
|- ( ph -> ( Ring i^i U ) C_ ( Rng i^i U ) ) |
8 |
7 2 3
|
3sstr4d |
|- ( ph -> R C_ S ) |
9 |
|
ovres |
|- ( ( x e. R /\ y e. R ) -> ( x ( RingHom |` ( R X. R ) ) y ) = ( x RingHom y ) ) |
10 |
9
|
adantl |
|- ( ( ph /\ ( x e. R /\ y e. R ) ) -> ( x ( RingHom |` ( R X. R ) ) y ) = ( x RingHom y ) ) |
11 |
10
|
eleq2d |
|- ( ( ph /\ ( x e. R /\ y e. R ) ) -> ( h e. ( x ( RingHom |` ( R X. R ) ) y ) <-> h e. ( x RingHom y ) ) ) |
12 |
|
rhmisrnghm |
|- ( h e. ( x RingHom y ) -> h e. ( x RngHomo y ) ) |
13 |
8
|
sseld |
|- ( ph -> ( x e. R -> x e. S ) ) |
14 |
8
|
sseld |
|- ( ph -> ( y e. R -> y e. S ) ) |
15 |
13 14
|
anim12d |
|- ( ph -> ( ( x e. R /\ y e. R ) -> ( x e. S /\ y e. S ) ) ) |
16 |
15
|
imp |
|- ( ( ph /\ ( x e. R /\ y e. R ) ) -> ( x e. S /\ y e. S ) ) |
17 |
|
ovres |
|- ( ( x e. S /\ y e. S ) -> ( x ( RngHomo |` ( S X. S ) ) y ) = ( x RngHomo y ) ) |
18 |
16 17
|
syl |
|- ( ( ph /\ ( x e. R /\ y e. R ) ) -> ( x ( RngHomo |` ( S X. S ) ) y ) = ( x RngHomo y ) ) |
19 |
18
|
eleq2d |
|- ( ( ph /\ ( x e. R /\ y e. R ) ) -> ( h e. ( x ( RngHomo |` ( S X. S ) ) y ) <-> h e. ( x RngHomo y ) ) ) |
20 |
12 19
|
syl5ibr |
|- ( ( ph /\ ( x e. R /\ y e. R ) ) -> ( h e. ( x RingHom y ) -> h e. ( x ( RngHomo |` ( S X. S ) ) y ) ) ) |
21 |
11 20
|
sylbid |
|- ( ( ph /\ ( x e. R /\ y e. R ) ) -> ( h e. ( x ( RingHom |` ( R X. R ) ) y ) -> h e. ( x ( RngHomo |` ( S X. S ) ) y ) ) ) |
22 |
21
|
ssrdv |
|- ( ( ph /\ ( x e. R /\ y e. R ) ) -> ( x ( RingHom |` ( R X. R ) ) y ) C_ ( x ( RngHomo |` ( S X. S ) ) y ) ) |
23 |
22
|
ralrimivva |
|- ( ph -> A. x e. R A. y e. R ( x ( RingHom |` ( R X. R ) ) y ) C_ ( x ( RngHomo |` ( S X. S ) ) y ) ) |
24 |
|
inss1 |
|- ( Ring i^i U ) C_ Ring |
25 |
2 24
|
eqsstrdi |
|- ( ph -> R C_ Ring ) |
26 |
|
xpss12 |
|- ( ( R C_ Ring /\ R C_ Ring ) -> ( R X. R ) C_ ( Ring X. Ring ) ) |
27 |
25 25 26
|
syl2anc |
|- ( ph -> ( R X. R ) C_ ( Ring X. Ring ) ) |
28 |
|
rhmfn |
|- RingHom Fn ( Ring X. Ring ) |
29 |
|
fnssresb |
|- ( RingHom Fn ( Ring X. Ring ) -> ( ( RingHom |` ( R X. R ) ) Fn ( R X. R ) <-> ( R X. R ) C_ ( Ring X. Ring ) ) ) |
30 |
28 29
|
mp1i |
|- ( ph -> ( ( RingHom |` ( R X. R ) ) Fn ( R X. R ) <-> ( R X. R ) C_ ( Ring X. Ring ) ) ) |
31 |
27 30
|
mpbird |
|- ( ph -> ( RingHom |` ( R X. R ) ) Fn ( R X. R ) ) |
32 |
|
inss1 |
|- ( Rng i^i U ) C_ Rng |
33 |
3 32
|
eqsstrdi |
|- ( ph -> S C_ Rng ) |
34 |
|
xpss12 |
|- ( ( S C_ Rng /\ S C_ Rng ) -> ( S X. S ) C_ ( Rng X. Rng ) ) |
35 |
33 33 34
|
syl2anc |
|- ( ph -> ( S X. S ) C_ ( Rng X. Rng ) ) |
36 |
|
rnghmfn |
|- RngHomo Fn ( Rng X. Rng ) |
37 |
|
fnssresb |
|- ( RngHomo Fn ( Rng X. Rng ) -> ( ( RngHomo |` ( S X. S ) ) Fn ( S X. S ) <-> ( S X. S ) C_ ( Rng X. Rng ) ) ) |
38 |
36 37
|
mp1i |
|- ( ph -> ( ( RngHomo |` ( S X. S ) ) Fn ( S X. S ) <-> ( S X. S ) C_ ( Rng X. Rng ) ) ) |
39 |
35 38
|
mpbird |
|- ( ph -> ( RngHomo |` ( S X. S ) ) Fn ( S X. S ) ) |
40 |
|
incom |
|- ( Rng i^i U ) = ( U i^i Rng ) |
41 |
|
inex1g |
|- ( U e. V -> ( U i^i Rng ) e. _V ) |
42 |
40 41
|
eqeltrid |
|- ( U e. V -> ( Rng i^i U ) e. _V ) |
43 |
1 42
|
syl |
|- ( ph -> ( Rng i^i U ) e. _V ) |
44 |
3 43
|
eqeltrd |
|- ( ph -> S e. _V ) |
45 |
31 39 44
|
isssc |
|- ( ph -> ( ( RingHom |` ( R X. R ) ) C_cat ( RngHomo |` ( S X. S ) ) <-> ( R C_ S /\ A. x e. R A. y e. R ( x ( RingHom |` ( R X. R ) ) y ) C_ ( x ( RngHomo |` ( S X. S ) ) y ) ) ) ) |
46 |
8 23 45
|
mpbir2and |
|- ( ph -> ( RingHom |` ( R X. R ) ) C_cat ( RngHomo |` ( S X. S ) ) ) |