Metamath Proof Explorer

Theorem ringccofval

Description: Composition in the category of unital rings. (Contributed by AV, 14-Feb-2020) (Revised by AV, 8-Mar-2020)

Ref Expression
Hypotheses ringcco.c 
|- C = ( RingCat ` U )
ringcco.u 
|- ( ph -> U e. V )
ringcco.o 
|- .x. = ( comp ` C )
Assertion ringccofval 
|- ( ph -> .x. = ( comp ` ( ExtStrCat ` U ) ) )

Proof

Step Hyp Ref Expression
1 ringcco.c 
 |-  C = ( RingCat ` U )
2 ringcco.u 
 |-  ( ph -> U e. V )
3 ringcco.o 
 |-  .x. = ( comp ` C )
4 eqid 
 |-  ( Base ` C ) = ( Base ` C )
5 1 4 2 ringcbas 
 |-  ( ph -> ( Base ` C ) = ( U i^i Ring ) )
6 eqid 
 |-  ( Hom ` C ) = ( Hom ` C )
7 1 4 2 6 ringchomfval 
 |-  ( ph -> ( Hom ` C ) = ( RingHom |` ( ( Base ` C ) X. ( Base ` C ) ) ) )
8 1 2 5 7 ringcval 
 |-  ( 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 rhmresfn 
 |-  ( ph -> ( Hom ` C ) Fn ( ( Base ` C ) X. ( Base ` C ) ) )
15 inss1 
 |-  ( U i^i Ring ) C_ U
16 15 a1i 
 |-  ( ph -> ( U i^i Ring ) 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 ) ) )