Theorem rngchomALTV

Description: Set of arrows of the category of non-unital rings (in a universe). (New usage is discouraged.) (Contributed by AV, 27-Feb-2020)

	Ref	Expression
Hypotheses	rngcbasALTV.c	$$⊢C=\mathrm{RngCatALTV}\left(U\right)$$
	rngcbasALTV.b	$$⊢B={\mathrm{Base}}_C$$
	rngcbasALTV.u	$$⊢\phi \to U\in V$$
	rngchomfvalALTV.h	$$⊢H=\mathrm{Hom}\left(C\right)$$
	rngchomALTV.x	$$⊢\phi \to X\in B$$
	rngchomALTV.y	$$⊢\phi \to Y\in B$$
Assertion	rngchomALTV	$$⊢\phi \to XHY=X\mathrm{RngHomo}Y$$

Proof

Step	Hyp	Ref	Expression
1		rngcbasALTV.c	$$⊢C=\mathrm{RngCatALTV}\left(U\right)$$
2		rngcbasALTV.b	$$⊢B={\mathrm{Base}}_C$$
3		rngcbasALTV.u	$$⊢\phi \to U\in V$$
4		rngchomfvalALTV.h	$$⊢H=\mathrm{Hom}\left(C\right)$$
5		rngchomALTV.x	$$⊢\phi \to X\in B$$
6		rngchomALTV.y	$$⊢\phi \to Y\in B$$
7	1 2 3 4	rngchomfvalALTV	$$⊢\phi \to H=\left(x\in B,y\in B⟼x\mathrm{RngHomo}y\right)$$
8		oveq12	$$⊢\left(x=X\wedge y=Y\right)\to x\mathrm{RngHomo}y=X\mathrm{RngHomo}Y$$
9	8	adantl	$$⊢\left(\phi \wedge \left(x=X\wedge y=Y\right)\right)\to x\mathrm{RngHomo}y=X\mathrm{RngHomo}Y$$
10		ovexd	$$⊢\phi \to X\mathrm{RngHomo}Y\in \mathrm{V}$$
11	7 9 5 6 10	ovmpod	$$⊢\phi \to XHY=X\mathrm{RngHomo}Y$$