Metamath Proof Explorer


Theorem rngcco

Description: Composition in the category of non-unital rings. (Contributed by AV, 27-Feb-2020) (Revised by AV, 8-Mar-2020)

Ref Expression
Hypotheses rngcco.c
|- C = ( RngCat ` U )
rngcco.u
|- ( ph -> U e. V )
rngcco.o
|- .x. = ( comp ` C )
rngcco.x
|- ( ph -> X e. U )
rngcco.y
|- ( ph -> Y e. U )
rngcco.z
|- ( ph -> Z e. U )
rngcco.f
|- ( ph -> F : ( Base ` X ) --> ( Base ` Y ) )
rngcco.g
|- ( ph -> G : ( Base ` Y ) --> ( Base ` Z ) )
Assertion rngcco
|- ( ph -> ( G ( <. X , Y >. .x. Z ) F ) = ( G o. F ) )

Proof

Step Hyp Ref Expression
1 rngcco.c
 |-  C = ( RngCat ` U )
2 rngcco.u
 |-  ( ph -> U e. V )
3 rngcco.o
 |-  .x. = ( comp ` C )
4 rngcco.x
 |-  ( ph -> X e. U )
5 rngcco.y
 |-  ( ph -> Y e. U )
6 rngcco.z
 |-  ( ph -> Z e. U )
7 rngcco.f
 |-  ( ph -> F : ( Base ` X ) --> ( Base ` Y ) )
8 rngcco.g
 |-  ( ph -> G : ( Base ` Y ) --> ( Base ` Z ) )
9 1 2 3 rngccofval
 |-  ( ph -> .x. = ( comp ` ( ExtStrCat ` U ) ) )
10 9 oveqd
 |-  ( ph -> ( <. X , Y >. .x. Z ) = ( <. X , Y >. ( comp ` ( ExtStrCat ` U ) ) Z ) )
11 10 oveqd
 |-  ( ph -> ( G ( <. X , Y >. .x. Z ) F ) = ( G ( <. X , Y >. ( comp ` ( ExtStrCat ` U ) ) Z ) F ) )
12 eqid
 |-  ( ExtStrCat ` U ) = ( ExtStrCat ` U )
13 eqid
 |-  ( comp ` ( ExtStrCat ` U ) ) = ( comp ` ( ExtStrCat ` U ) )
14 eqid
 |-  ( Base ` X ) = ( Base ` X )
15 eqid
 |-  ( Base ` Y ) = ( Base ` Y )
16 eqid
 |-  ( Base ` Z ) = ( Base ` Z )
17 12 2 13 4 5 6 14 15 16 7 8 estrcco
 |-  ( ph -> ( G ( <. X , Y >. ( comp ` ( ExtStrCat ` U ) ) Z ) F ) = ( G o. F ) )
18 11 17 eqtrd
 |-  ( ph -> ( G ( <. X , Y >. .x. Z ) F ) = ( G o. F ) )