Theorem ringchom

Description: Set of arrows of the category of unital rings (in a universe). (Contributed by AV, 14-Feb-2020)

	Ref	Expression
Hypotheses	ringcbas.c	$$⊢C=\mathrm{RingCat}\left(U\right)$$
	ringcbas.b	$$⊢B={\mathrm{Base}}_C$$
	ringcbas.u	$$⊢\phi \to U\in V$$
	ringchomfval.h	$$⊢H=\mathrm{Hom}\left(C\right)$$
	ringchom.x	$$⊢\phi \to X\in B$$
	ringchom.y	$$⊢\phi \to Y\in B$$
Assertion	ringchom	$$⊢\phi \to XHY=X\mathrm{RingHom}Y$$

Proof

Step	Hyp	Ref	Expression
1		ringcbas.c	$$⊢C=\mathrm{RingCat}\left(U\right)$$
2		ringcbas.b	$$⊢B={\mathrm{Base}}_C$$
3		ringcbas.u	$$⊢\phi \to U\in V$$
4		ringchomfval.h	$$⊢H=\mathrm{Hom}\left(C\right)$$
5		ringchom.x	$$⊢\phi \to X\in B$$
6		ringchom.y	$$⊢\phi \to Y\in B$$
7	1 2 3 4	ringchomfval	$$⊢\phi \to H={\mathrm{RingHom}\upharpoonright }_{\left(B\times B\right)}$$
8	7	oveqd	$$⊢\phi \to XHY=X\left({\mathrm{RingHom}\upharpoonright }_{\left(B\times B\right)}\right)Y$$
9	5 6	ovresd	$$⊢\phi \to X\left({\mathrm{RingHom}\upharpoonright }_{\left(B\times B\right)}\right)Y=X\mathrm{RingHom}Y$$
10	8 9	eqtrd	$$⊢\phi \to XHY=X\mathrm{RingHom}Y$$