Step |
Hyp |
Ref |
Expression |
1 |
|
ringccat.c |
|- C = ( RingCat ` U ) |
2 |
|
ringcid.b |
|- B = ( Base ` C ) |
3 |
|
ringcid.o |
|- .1. = ( Id ` C ) |
4 |
|
ringcid.u |
|- ( ph -> U e. V ) |
5 |
|
ringcid.x |
|- ( ph -> X e. B ) |
6 |
|
ringcid.s |
|- S = ( Base ` X ) |
7 |
|
eqidd |
|- ( ph -> ( U i^i Ring ) = ( U i^i Ring ) ) |
8 |
|
eqidd |
|- ( ph -> ( RingHom |` ( ( U i^i Ring ) X. ( U i^i Ring ) ) ) = ( RingHom |` ( ( U i^i Ring ) X. ( U i^i Ring ) ) ) ) |
9 |
1 4 7 8
|
ringcval |
|- ( ph -> C = ( ( ExtStrCat ` U ) |`cat ( RingHom |` ( ( U i^i Ring ) X. ( U i^i Ring ) ) ) ) ) |
10 |
9
|
fveq2d |
|- ( ph -> ( Id ` C ) = ( Id ` ( ( ExtStrCat ` U ) |`cat ( RingHom |` ( ( U i^i Ring ) X. ( U i^i Ring ) ) ) ) ) ) |
11 |
3 10
|
syl5eq |
|- ( ph -> .1. = ( Id ` ( ( ExtStrCat ` U ) |`cat ( RingHom |` ( ( U i^i Ring ) X. ( U i^i Ring ) ) ) ) ) ) |
12 |
11
|
fveq1d |
|- ( ph -> ( .1. ` X ) = ( ( Id ` ( ( ExtStrCat ` U ) |`cat ( RingHom |` ( ( U i^i Ring ) X. ( U i^i Ring ) ) ) ) ) ` X ) ) |
13 |
|
eqid |
|- ( ( ExtStrCat ` U ) |`cat ( RingHom |` ( ( U i^i Ring ) X. ( U i^i Ring ) ) ) ) = ( ( ExtStrCat ` U ) |`cat ( RingHom |` ( ( U i^i Ring ) X. ( U i^i Ring ) ) ) ) |
14 |
|
eqid |
|- ( ExtStrCat ` U ) = ( ExtStrCat ` U ) |
15 |
|
incom |
|- ( U i^i Ring ) = ( Ring i^i U ) |
16 |
15
|
a1i |
|- ( ph -> ( U i^i Ring ) = ( Ring i^i U ) ) |
17 |
14 4 16 8
|
rhmsubcsetc |
|- ( ph -> ( RingHom |` ( ( U i^i Ring ) X. ( U i^i Ring ) ) ) e. ( Subcat ` ( ExtStrCat ` U ) ) ) |
18 |
7 8
|
rhmresfn |
|- ( ph -> ( RingHom |` ( ( U i^i Ring ) X. ( U i^i Ring ) ) ) Fn ( ( U i^i Ring ) X. ( U i^i Ring ) ) ) |
19 |
|
eqid |
|- ( Id ` ( ExtStrCat ` U ) ) = ( Id ` ( ExtStrCat ` U ) ) |
20 |
1 2 4
|
ringcbas |
|- ( ph -> B = ( U i^i Ring ) ) |
21 |
20
|
eleq2d |
|- ( ph -> ( X e. B <-> X e. ( U i^i Ring ) ) ) |
22 |
5 21
|
mpbid |
|- ( ph -> X e. ( U i^i Ring ) ) |
23 |
13 17 18 19 22
|
subcid |
|- ( ph -> ( ( Id ` ( ExtStrCat ` U ) ) ` X ) = ( ( Id ` ( ( ExtStrCat ` U ) |`cat ( RingHom |` ( ( U i^i Ring ) X. ( U i^i Ring ) ) ) ) ) ` X ) ) |
24 |
|
elinel1 |
|- ( X e. ( U i^i Ring ) -> X e. U ) |
25 |
21 24
|
syl6bi |
|- ( ph -> ( X e. B -> X e. U ) ) |
26 |
5 25
|
mpd |
|- ( ph -> X e. U ) |
27 |
14 19 4 26
|
estrcid |
|- ( ph -> ( ( Id ` ( ExtStrCat ` U ) ) ` X ) = ( _I |` ( Base ` X ) ) ) |
28 |
6
|
eqcomi |
|- ( Base ` X ) = S |
29 |
28
|
a1i |
|- ( ph -> ( Base ` X ) = S ) |
30 |
29
|
reseq2d |
|- ( ph -> ( _I |` ( Base ` X ) ) = ( _I |` S ) ) |
31 |
27 30
|
eqtrd |
|- ( ph -> ( ( Id ` ( ExtStrCat ` U ) ) ` X ) = ( _I |` S ) ) |
32 |
12 23 31
|
3eqtr2d |
|- ( ph -> ( .1. ` X ) = ( _I |` S ) ) |