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Metamath Proof Explorer

Theorem setsmsds

Description: The distance function of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015) (Proof shortened by AV, 11-Nov-2024)

Ref Expression
Hypotheses setsms.x ⊢ ( 𝜑 → 𝑋 = ( Base ‘ 𝑀 ) )
setsms.d ⊢ ( 𝜑 → 𝐷 = ( ( dist ‘ 𝑀 ) ↾ ( 𝑋 × 𝑋 ) ) )
setsms.k ⊢ ( 𝜑 → 𝐾 = ( 𝑀 sSet ⟨ ( TopSet ‘ ndx ) , ( MetOpen ‘ 𝐷 ) ⟩ ) )
Assertion setsmsds ( 𝜑 → ( dist ‘ 𝑀 ) = ( dist ‘ 𝐾 ) )

Proof

Step Hyp Ref Expression
1 setsms.x ⊢ ( 𝜑 → 𝑋 = ( Base ‘ 𝑀 ) )
2 setsms.d ⊢ ( 𝜑 → 𝐷 = ( ( dist ‘ 𝑀 ) ↾ ( 𝑋 × 𝑋 ) ) )
3 setsms.k ⊢ ( 𝜑 → 𝐾 = ( 𝑀 sSet ⟨ ( TopSet ‘ ndx ) , ( MetOpen ‘ 𝐷 ) ⟩ ) )
4 dsid ⊢ dist = Slot ( dist ‘ ndx )
5 dsndxntsetndx ⊢ ( dist ‘ ndx ) ≠ ( TopSet ‘ ndx )
6 4 5 setsnid ⊢ ( dist ‘ 𝑀 ) = ( dist ‘ ( 𝑀 sSet ⟨ ( TopSet ‘ ndx ) , ( MetOpen ‘ 𝐷 ) ⟩ ) )
7 3 fveq2d ⊢ ( 𝜑 → ( dist ‘ 𝐾 ) = ( dist ‘ ( 𝑀 sSet ⟨ ( TopSet ‘ ndx ) , ( MetOpen ‘ 𝐷 ) ⟩ ) ) )
8 6 7 eqtr4id ⊢ ( 𝜑 → ( dist ‘ 𝑀 ) = ( dist ‘ 𝐾 ) )