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Metamath Proof Explorer

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 ) ) )