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Metamath Proof Explorer

Theorem ovresd

Description: Lemma for converting metric theorems to metric space theorems. (Contributed by Mario Carneiro, 2-Oct-2015)

Ref Expression
Hypotheses ovresd.1 ⊢ ( 𝜑 → 𝐴 ∈ 𝑋 )
ovresd.2 ⊢ ( 𝜑 → 𝐵 ∈ 𝑋 )
Assertion ovresd ( 𝜑 → ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐵 ) = ( 𝐴 𝐷 𝐵 ) )

Proof

Step Hyp Ref Expression
1 ovresd.1 ⊢ ( 𝜑 → 𝐴 ∈ 𝑋 )
2 ovresd.2 ⊢ ( 𝜑 → 𝐵 ∈ 𝑋 )
3 ovres ⊢ ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐵 ) = ( 𝐴 𝐷 𝐵 ) )
4 1 2 3 syl2anc ⊢ ( 𝜑 → ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐵 ) = ( 𝐴 𝐷 𝐵 ) )