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 ) ) )