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 ) )