Step |
Hyp |
Ref |
Expression |
1 |
|
rhmsscmap.u |
|- ( ph -> U e. V ) |
2 |
|
rhmsscmap.r |
|- ( ph -> R = ( Ring i^i U ) ) |
3 |
|
inss2 |
|- ( Ring i^i U ) C_ U |
4 |
2 3
|
eqsstrdi |
|- ( ph -> R C_ U ) |
5 |
|
eqid |
|- ( Base ` a ) = ( Base ` a ) |
6 |
|
eqid |
|- ( Base ` b ) = ( Base ` b ) |
7 |
5 6
|
rhmf |
|- ( h e. ( a RingHom b ) -> h : ( Base ` a ) --> ( Base ` b ) ) |
8 |
|
simpr |
|- ( ( ( ph /\ ( a e. R /\ b e. R ) ) /\ h : ( Base ` a ) --> ( Base ` b ) ) -> h : ( Base ` a ) --> ( Base ` b ) ) |
9 |
|
fvex |
|- ( Base ` b ) e. _V |
10 |
|
fvex |
|- ( Base ` a ) e. _V |
11 |
9 10
|
pm3.2i |
|- ( ( Base ` b ) e. _V /\ ( Base ` a ) e. _V ) |
12 |
|
elmapg |
|- ( ( ( Base ` b ) e. _V /\ ( Base ` a ) e. _V ) -> ( h e. ( ( Base ` b ) ^m ( Base ` a ) ) <-> h : ( Base ` a ) --> ( Base ` b ) ) ) |
13 |
11 12
|
mp1i |
|- ( ( ( ph /\ ( a e. R /\ b e. R ) ) /\ h : ( Base ` a ) --> ( Base ` b ) ) -> ( h e. ( ( Base ` b ) ^m ( Base ` a ) ) <-> h : ( Base ` a ) --> ( Base ` b ) ) ) |
14 |
8 13
|
mpbird |
|- ( ( ( ph /\ ( a e. R /\ b e. R ) ) /\ h : ( Base ` a ) --> ( Base ` b ) ) -> h e. ( ( Base ` b ) ^m ( Base ` a ) ) ) |
15 |
14
|
ex |
|- ( ( ph /\ ( a e. R /\ b e. R ) ) -> ( h : ( Base ` a ) --> ( Base ` b ) -> h e. ( ( Base ` b ) ^m ( Base ` a ) ) ) ) |
16 |
7 15
|
syl5 |
|- ( ( ph /\ ( a e. R /\ b e. R ) ) -> ( h e. ( a RingHom b ) -> h e. ( ( Base ` b ) ^m ( Base ` a ) ) ) ) |
17 |
16
|
ssrdv |
|- ( ( ph /\ ( a e. R /\ b e. R ) ) -> ( a RingHom b ) C_ ( ( Base ` b ) ^m ( Base ` a ) ) ) |
18 |
|
ovres |
|- ( ( a e. R /\ b e. R ) -> ( a ( RingHom |` ( R X. R ) ) b ) = ( a RingHom b ) ) |
19 |
18
|
adantl |
|- ( ( ph /\ ( a e. R /\ b e. R ) ) -> ( a ( RingHom |` ( R X. R ) ) b ) = ( a RingHom b ) ) |
20 |
|
eqidd |
|- ( ( ph /\ ( a e. R /\ b e. R ) ) -> ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) = ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) ) |
21 |
|
fveq2 |
|- ( y = b -> ( Base ` y ) = ( Base ` b ) ) |
22 |
|
fveq2 |
|- ( x = a -> ( Base ` x ) = ( Base ` a ) ) |
23 |
21 22
|
oveqan12rd |
|- ( ( x = a /\ y = b ) -> ( ( Base ` y ) ^m ( Base ` x ) ) = ( ( Base ` b ) ^m ( Base ` a ) ) ) |
24 |
23
|
adantl |
|- ( ( ( ph /\ ( a e. R /\ b e. R ) ) /\ ( x = a /\ y = b ) ) -> ( ( Base ` y ) ^m ( Base ` x ) ) = ( ( Base ` b ) ^m ( Base ` a ) ) ) |
25 |
4
|
sseld |
|- ( ph -> ( a e. R -> a e. U ) ) |
26 |
25
|
com12 |
|- ( a e. R -> ( ph -> a e. U ) ) |
27 |
26
|
adantr |
|- ( ( a e. R /\ b e. R ) -> ( ph -> a e. U ) ) |
28 |
27
|
impcom |
|- ( ( ph /\ ( a e. R /\ b e. R ) ) -> a e. U ) |
29 |
4
|
sseld |
|- ( ph -> ( b e. R -> b e. U ) ) |
30 |
29
|
com12 |
|- ( b e. R -> ( ph -> b e. U ) ) |
31 |
30
|
adantl |
|- ( ( a e. R /\ b e. R ) -> ( ph -> b e. U ) ) |
32 |
31
|
impcom |
|- ( ( ph /\ ( a e. R /\ b e. R ) ) -> b e. U ) |
33 |
|
ovexd |
|- ( ( ph /\ ( a e. R /\ b e. R ) ) -> ( ( Base ` b ) ^m ( Base ` a ) ) e. _V ) |
34 |
20 24 28 32 33
|
ovmpod |
|- ( ( ph /\ ( a e. R /\ b e. R ) ) -> ( a ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) b ) = ( ( Base ` b ) ^m ( Base ` a ) ) ) |
35 |
17 19 34
|
3sstr4d |
|- ( ( ph /\ ( a e. R /\ b e. R ) ) -> ( a ( RingHom |` ( R X. R ) ) b ) C_ ( a ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) b ) ) |
36 |
35
|
ralrimivva |
|- ( ph -> A. a e. R A. b e. R ( a ( RingHom |` ( R X. R ) ) b ) C_ ( a ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) b ) ) |
37 |
|
rhmfn |
|- RingHom Fn ( Ring X. Ring ) |
38 |
37
|
a1i |
|- ( ph -> RingHom Fn ( Ring X. Ring ) ) |
39 |
|
inss1 |
|- ( Ring i^i U ) C_ Ring |
40 |
2 39
|
eqsstrdi |
|- ( ph -> R C_ Ring ) |
41 |
|
xpss12 |
|- ( ( R C_ Ring /\ R C_ Ring ) -> ( R X. R ) C_ ( Ring X. Ring ) ) |
42 |
40 40 41
|
syl2anc |
|- ( ph -> ( R X. R ) C_ ( Ring X. Ring ) ) |
43 |
|
fnssres |
|- ( ( RingHom Fn ( Ring X. Ring ) /\ ( R X. R ) C_ ( Ring X. Ring ) ) -> ( RingHom |` ( R X. R ) ) Fn ( R X. R ) ) |
44 |
38 42 43
|
syl2anc |
|- ( ph -> ( RingHom |` ( R X. R ) ) Fn ( R X. R ) ) |
45 |
|
eqid |
|- ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) = ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) |
46 |
|
ovex |
|- ( ( Base ` y ) ^m ( Base ` x ) ) e. _V |
47 |
45 46
|
fnmpoi |
|- ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) Fn ( U X. U ) |
48 |
47
|
a1i |
|- ( ph -> ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) Fn ( U X. U ) ) |
49 |
|
elex |
|- ( U e. V -> U e. _V ) |
50 |
1 49
|
syl |
|- ( ph -> U e. _V ) |
51 |
44 48 50
|
isssc |
|- ( ph -> ( ( RingHom |` ( R X. R ) ) C_cat ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) <-> ( R C_ U /\ A. a e. R A. b e. R ( a ( RingHom |` ( R X. R ) ) b ) C_ ( a ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) b ) ) ) ) |
52 |
4 36 51
|
mpbir2and |
|- ( ph -> ( RingHom |` ( R X. R ) ) C_cat ( x e. U , y e. U |-> ( ( Base ` y ) ^m ( Base ` x ) ) ) ) |