Metamath Proof Explorer

Theorem ringchomfval

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

Ref Expression
Hypotheses ringcbas.c 
|- C = ( RingCat ` U )
ringcbas.b 
|- B = ( Base ` C )
ringcbas.u 
|- ( ph -> U e. V )
ringchomfval.h 
|- H = ( Hom ` C )
Assertion ringchomfval 
|- ( ph -> H = ( RingHom |` ( B X. B ) ) )

Proof

Step Hyp Ref Expression
1 ringcbas.c 
 |-  C = ( RingCat ` U )
2 ringcbas.b 
 |-  B = ( Base ` C )
3 ringcbas.u 
 |-  ( ph -> U e. V )
4 ringchomfval.h 
 |-  H = ( Hom ` C )
5 1 2 3 ringcbas 
 |-  ( ph -> B = ( U i^i Ring ) )
6 eqidd 
 |-  ( ph -> ( RingHom |` ( B X. B ) ) = ( RingHom |` ( B X. B ) ) )
7 1 3 5 6 ringcval 
 |-  ( ph -> C = ( ( ExtStrCat ` U ) |`cat ( RingHom |` ( B X. B ) ) ) )
8 7 fveq2d 
 |-  ( ph -> ( Hom ` C ) = ( Hom ` ( ( ExtStrCat ` U ) |`cat ( RingHom |` ( B X. B ) ) ) ) )
9 4 8 syl5eq 
 |-  ( ph -> H = ( Hom ` ( ( ExtStrCat ` U ) |`cat ( RingHom |` ( B X. B ) ) ) ) )
10 eqid 
 |-  ( ( ExtStrCat ` U ) |`cat ( RingHom |` ( B X. B ) ) ) = ( ( ExtStrCat ` U ) |`cat ( RingHom |` ( B X. B ) ) )
11 eqid 
 |-  ( Base ` ( ExtStrCat ` U ) ) = ( Base ` ( ExtStrCat ` U ) )
12 fvexd 
 |-  ( ph -> ( ExtStrCat ` U ) e. _V )
13 5 6 rhmresfn 
 |-  ( ph -> ( RingHom |` ( B X. B ) ) Fn ( B X. B ) )
14 inss1 
 |-  ( U i^i Ring ) C_ U
15 14 a1i 
 |-  ( ph -> ( U i^i Ring ) C_ U )
16 eqid 
 |-  ( ExtStrCat ` U ) = ( ExtStrCat ` U )
17 16 3 estrcbas 
 |-  ( ph -> U = ( Base ` ( ExtStrCat ` U ) ) )
18 17 eqcomd 
 |-  ( ph -> ( Base ` ( ExtStrCat ` U ) ) = U )
19 15 5 18 3sstr4d 
 |-  ( ph -> B C_ ( Base ` ( ExtStrCat ` U ) ) )
20 10 11 12 13 19 reschom 
 |-  ( ph -> ( RingHom |` ( B X. B ) ) = ( Hom ` ( ( ExtStrCat ` U ) |`cat ( RingHom |` ( B X. B ) ) ) ) )
21 9 20 eqtr4d 
 |-  ( ph -> H = ( RingHom |` ( B X. B ) ) )