Step |
Hyp |
Ref |
Expression |
1 |
|
rngcco.c |
|- C = ( RngCat ` U ) |
2 |
|
rngcco.u |
|- ( ph -> U e. V ) |
3 |
|
rngcco.o |
|- .x. = ( comp ` C ) |
4 |
|
eqid |
|- ( Base ` C ) = ( Base ` C ) |
5 |
1 4 2
|
rngcbas |
|- ( ph -> ( Base ` C ) = ( U i^i Rng ) ) |
6 |
|
eqid |
|- ( Hom ` C ) = ( Hom ` C ) |
7 |
1 4 2 6
|
rngchomfval |
|- ( ph -> ( Hom ` C ) = ( RngHomo |` ( ( Base ` C ) X. ( Base ` C ) ) ) ) |
8 |
1 2 5 7
|
rngcval |
|- ( ph -> C = ( ( ExtStrCat ` U ) |`cat ( Hom ` C ) ) ) |
9 |
8
|
fveq2d |
|- ( ph -> ( comp ` C ) = ( comp ` ( ( ExtStrCat ` U ) |`cat ( Hom ` C ) ) ) ) |
10 |
3
|
a1i |
|- ( ph -> .x. = ( comp ` C ) ) |
11 |
|
eqid |
|- ( ( ExtStrCat ` U ) |`cat ( Hom ` C ) ) = ( ( ExtStrCat ` U ) |`cat ( Hom ` C ) ) |
12 |
|
eqid |
|- ( Base ` ( ExtStrCat ` U ) ) = ( Base ` ( ExtStrCat ` U ) ) |
13 |
|
fvexd |
|- ( ph -> ( ExtStrCat ` U ) e. _V ) |
14 |
5 7
|
rnghmresfn |
|- ( ph -> ( Hom ` C ) Fn ( ( Base ` C ) X. ( Base ` C ) ) ) |
15 |
|
inss1 |
|- ( U i^i Rng ) C_ U |
16 |
15
|
a1i |
|- ( ph -> ( U i^i Rng ) C_ U ) |
17 |
|
eqid |
|- ( ExtStrCat ` U ) = ( ExtStrCat ` U ) |
18 |
17 2
|
estrcbas |
|- ( ph -> U = ( Base ` ( ExtStrCat ` U ) ) ) |
19 |
18
|
eqcomd |
|- ( ph -> ( Base ` ( ExtStrCat ` U ) ) = U ) |
20 |
16 5 19
|
3sstr4d |
|- ( ph -> ( Base ` C ) C_ ( Base ` ( ExtStrCat ` U ) ) ) |
21 |
|
eqid |
|- ( comp ` ( ExtStrCat ` U ) ) = ( comp ` ( ExtStrCat ` U ) ) |
22 |
11 12 13 14 20 21
|
rescco |
|- ( ph -> ( comp ` ( ExtStrCat ` U ) ) = ( comp ` ( ( ExtStrCat ` U ) |`cat ( Hom ` C ) ) ) ) |
23 |
9 10 22
|
3eqtr4d |
|- ( ph -> .x. = ( comp ` ( ExtStrCat ` U ) ) ) |