| Step |
Hyp |
Ref |
Expression |
| 1 |
|
rngcrescrhm.u |
|- ( ph -> U e. V ) |
| 2 |
|
rngcrescrhm.c |
|- C = ( RngCat ` U ) |
| 3 |
|
rngcrescrhm.r |
|- ( ph -> R = ( Ring i^i U ) ) |
| 4 |
|
rngcrescrhm.h |
|- H = ( RingHom |` ( R X. R ) ) |
| 5 |
3
|
eleq2d |
|- ( ph -> ( x e. R <-> x e. ( Ring i^i U ) ) ) |
| 6 |
|
elinel1 |
|- ( x e. ( Ring i^i U ) -> x e. Ring ) |
| 7 |
5 6
|
syl6bi |
|- ( ph -> ( x e. R -> x e. Ring ) ) |
| 8 |
7
|
imp |
|- ( ( ph /\ x e. R ) -> x e. Ring ) |
| 9 |
|
eqid |
|- ( Base ` x ) = ( Base ` x ) |
| 10 |
9
|
idrhm |
|- ( x e. Ring -> ( _I |` ( Base ` x ) ) e. ( x RingHom x ) ) |
| 11 |
8 10
|
syl |
|- ( ( ph /\ x e. R ) -> ( _I |` ( Base ` x ) ) e. ( x RingHom x ) ) |
| 12 |
|
eqid |
|- ( Base ` C ) = ( Base ` C ) |
| 13 |
2
|
eqcomi |
|- ( RngCat ` U ) = C |
| 14 |
13
|
fveq2i |
|- ( Id ` ( RngCat ` U ) ) = ( Id ` C ) |
| 15 |
1
|
adantr |
|- ( ( ph /\ x e. R ) -> U e. V ) |
| 16 |
|
incom |
|- ( Ring i^i U ) = ( U i^i Ring ) |
| 17 |
|
ringssrng |
|- Ring C_ Rng |
| 18 |
|
sslin |
|- ( Ring C_ Rng -> ( U i^i Ring ) C_ ( U i^i Rng ) ) |
| 19 |
17 18
|
mp1i |
|- ( ph -> ( U i^i Ring ) C_ ( U i^i Rng ) ) |
| 20 |
16 19
|
eqsstrid |
|- ( ph -> ( Ring i^i U ) C_ ( U i^i Rng ) ) |
| 21 |
2 12 1
|
rngcbas |
|- ( ph -> ( Base ` C ) = ( U i^i Rng ) ) |
| 22 |
20 3 21
|
3sstr4d |
|- ( ph -> R C_ ( Base ` C ) ) |
| 23 |
22
|
sselda |
|- ( ( ph /\ x e. R ) -> x e. ( Base ` C ) ) |
| 24 |
2 12 14 15 23 9
|
rngcid |
|- ( ( ph /\ x e. R ) -> ( ( Id ` ( RngCat ` U ) ) ` x ) = ( _I |` ( Base ` x ) ) ) |
| 25 |
1 2 3 4
|
rhmsubclem2 |
|- ( ( ph /\ x e. R /\ x e. R ) -> ( x H x ) = ( x RingHom x ) ) |
| 26 |
25
|
3anidm23 |
|- ( ( ph /\ x e. R ) -> ( x H x ) = ( x RingHom x ) ) |
| 27 |
11 24 26
|
3eltr4d |
|- ( ( ph /\ x e. R ) -> ( ( Id ` ( RngCat ` U ) ) ` x ) e. ( x H x ) ) |