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 ⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
rngcco.o ⊢ · = ( comp ‘ 𝐶 )
Assertion rngccofval ( 𝜑 → · = ( comp ‘ ( ExtStrCat ‘ 𝑈 ) ) )

Proof

Step Hyp Ref Expression
1 rngcco.c ⊢ 𝐶 = ( RngCat ‘ 𝑈 )
2 rngcco.u ⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
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 ) = ( RngHom ↾ ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) ) ) )
8 1 2 5 7 rngcval ⊢ ( 𝜑 → 𝐶 = ( ( 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 ⊢ ( 𝜑 → 𝑈 = ( 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 ) ) )