Theorem dvrval

Description: Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014) (Revised by Mario Carneiro, 2-Dec-2014)

	Ref	Expression
Hypotheses	dvrval.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	dvrval.t	⊢ · = ( .r ‘ 𝑅 )
	dvrval.u	⊢ 𝑈 = ( Unit ‘ 𝑅 )
	dvrval.i	⊢ 𝐼 = ( invr ‘ 𝑅 )
	dvrval.d	⊢ / = ( /r ‘ 𝑅 )
Assertion	dvrval	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈 ) → ( 𝑋 / 𝑌 ) = ( 𝑋 · ( 𝐼 ‘ 𝑌 ) ) )

Proof

Step	Hyp	Ref	Expression
1		dvrval.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		dvrval.t	⊢ · = ( .r ‘ 𝑅 )
3		dvrval.u	⊢ 𝑈 = ( Unit ‘ 𝑅 )
4		dvrval.i	⊢ 𝐼 = ( invr ‘ 𝑅 )
5		dvrval.d	⊢ / = ( /r ‘ 𝑅 )
6		oveq1	⊢ ( 𝑥 = 𝑋 → ( 𝑥 · ( 𝐼 ‘ 𝑦 ) ) = ( 𝑋 · ( 𝐼 ‘ 𝑦 ) ) )
7		fveq2	⊢ ( 𝑦 = 𝑌 → ( 𝐼 ‘ 𝑦 ) = ( 𝐼 ‘ 𝑌 ) )
8	7	oveq2d	⊢ ( 𝑦 = 𝑌 → ( 𝑋 · ( 𝐼 ‘ 𝑦 ) ) = ( 𝑋 · ( 𝐼 ‘ 𝑌 ) ) )
9	1 2 3 4 5	dvrfval	⊢ / = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝑈 ↦ ( 𝑥 · ( 𝐼 ‘ 𝑦 ) ) )
10		ovex	⊢ ( 𝑋 · ( 𝐼 ‘ 𝑌 ) ) ∈ V
11	6 8 9 10	ovmpo	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈 ) → ( 𝑋 / 𝑌 ) = ( 𝑋 · ( 𝐼 ‘ 𝑌 ) ) )