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 = ( RingCat ‘ 𝑈 )
	ringcbas.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	ringcbas.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
	ringchomfval.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
	ringchom.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	ringchom.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion	ringchom	⊢ ( 𝜑 → ( 𝑋 𝐻 𝑌 ) = ( 𝑋 RingHom 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		ringcbas.c	⊢ 𝐶 = ( RingCat ‘ 𝑈 )
2		ringcbas.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
3		ringcbas.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
4		ringchomfval.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
5		ringchom.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6		ringchom.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7	1 2 3 4	ringchomfval	⊢ ( 𝜑 → 𝐻 = ( RingHom ↾ ( 𝐵 × 𝐵 ) ) )
8	7	oveqd	⊢ ( 𝜑 → ( 𝑋 𝐻 𝑌 ) = ( 𝑋 ( RingHom ↾ ( 𝐵 × 𝐵 ) ) 𝑌 ) )
9	5 6	ovresd	⊢ ( 𝜑 → ( 𝑋 ( RingHom ↾ ( 𝐵 × 𝐵 ) ) 𝑌 ) = ( 𝑋 RingHom 𝑌 ) )
10	8 9	eqtrd	⊢ ( 𝜑 → ( 𝑋 𝐻 𝑌 ) = ( 𝑋 RingHom 𝑌 ) )