Theorem setsmsds

Description: The distance function of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015)

	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		9re	⊢ 9 ∈ ℝ
6		1nn	⊢ 1 ∈ ℕ
7		2nn0	⊢ 2 ∈ ℕ0
8		9nn0	⊢ 9 ∈ ℕ0
9		9lt10	⊢ 9 < ; 1 0
10	6 7 8 9	declti	⊢ 9 < ; 1 2
11	5 10	gtneii	⊢ ; 1 2 ≠ 9
12		dsndx	⊢ ( dist ‘ ndx ) = ; 1 2
13		tsetndx	⊢ ( TopSet ‘ ndx ) = 9
14	12 13	neeq12i	⊢ ( ( dist ‘ ndx ) ≠ ( TopSet ‘ ndx ) ↔ ; 1 2 ≠ 9 )
15	11 14	mpbir	⊢ ( dist ‘ ndx ) ≠ ( TopSet ‘ ndx )
16	4 15	setsnid	⊢ ( dist ‘ 𝑀 ) = ( dist ‘ ( 𝑀 sSet ⟨ ( TopSet ‘ ndx ) , ( MetOpen ‘ 𝐷 ) ⟩ ) )
17	3	fveq2d	⊢ ( 𝜑 → ( dist ‘ 𝐾 ) = ( dist ‘ ( 𝑀 sSet ⟨ ( TopSet ‘ ndx ) , ( MetOpen ‘ 𝐷 ) ⟩ ) ) )
18	16 17	eqtr4id	⊢ ( 𝜑 → ( dist ‘ 𝑀 ) = ( dist ‘ 𝐾 ) )