Metamath Proof Explorer


Theorem rngccofval

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 ( 𝜑𝑈𝑉 )
rngcco.o · = ( comp ‘ 𝐶 )
Assertion rngccofval ( 𝜑· = ( comp ‘ ( ExtStrCat ‘ 𝑈 ) ) )

Proof

Step Hyp Ref Expression
1 rngcco.c 𝐶 = ( RngCat ‘ 𝑈 )
2 rngcco.u ( 𝜑𝑈𝑉 )
3 rngcco.o · = ( comp ‘ 𝐶 )
4 eqid ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 )
5 1 4 2 rngcbas ( 𝜑 → ( Base ‘ 𝐶 ) = ( 𝑈 ∩ Rng ) )
6 eqid ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 )
7 1 4 2 6 rngchomfval ( 𝜑 → ( Hom ‘ 𝐶 ) = ( RngHomo ↾ ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) ) ) )
8 1 2 5 7 rngcval ( 𝜑𝐶 = ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( Hom ‘ 𝐶 ) ) )
9 8 fveq2d ( 𝜑 → ( comp ‘ 𝐶 ) = ( comp ‘ ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( Hom ‘ 𝐶 ) ) ) )
10 3 a1i ( 𝜑· = ( comp ‘ 𝐶 ) )
11 eqid ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( Hom ‘ 𝐶 ) ) = ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( Hom ‘ 𝐶 ) )
12 eqid ( Base ‘ ( ExtStrCat ‘ 𝑈 ) ) = ( Base ‘ ( ExtStrCat ‘ 𝑈 ) )
13 fvexd ( 𝜑 → ( ExtStrCat ‘ 𝑈 ) ∈ V )
14 5 7 rnghmresfn ( 𝜑 → ( Hom ‘ 𝐶 ) Fn ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) ) )
15 inss1 ( 𝑈 ∩ Rng ) ⊆ 𝑈
16 15 a1i ( 𝜑 → ( 𝑈 ∩ Rng ) ⊆ 𝑈 )
17 eqid ( ExtStrCat ‘ 𝑈 ) = ( ExtStrCat ‘ 𝑈 )
18 17 2 estrcbas ( 𝜑𝑈 = ( Base ‘ ( ExtStrCat ‘ 𝑈 ) ) )
19 18 eqcomd ( 𝜑 → ( Base ‘ ( ExtStrCat ‘ 𝑈 ) ) = 𝑈 )
20 16 5 19 3sstr4d ( 𝜑 → ( Base ‘ 𝐶 ) ⊆ ( Base ‘ ( ExtStrCat ‘ 𝑈 ) ) )
21 eqid ( comp ‘ ( ExtStrCat ‘ 𝑈 ) ) = ( comp ‘ ( ExtStrCat ‘ 𝑈 ) )
22 11 12 13 14 20 21 rescco ( 𝜑 → ( comp ‘ ( ExtStrCat ‘ 𝑈 ) ) = ( comp ‘ ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( Hom ‘ 𝐶 ) ) ) )
23 9 10 22 3eqtr4d ( 𝜑· = ( comp ‘ ( ExtStrCat ‘ 𝑈 ) ) )