Theorem isxms

Description: Express the predicate " <. X , D >. is an extended metric space" with underlying set X and distance function D . (Contributed by Mario Carneiro, 2-Sep-2015)

	Ref	Expression
Hypotheses	isms.j	⊢ 𝐽 = ( TopOpen ‘ 𝐾 )
	isms.x	⊢ 𝑋 = ( Base ‘ 𝐾 )
	isms.d	⊢ 𝐷 = ( ( dist ‘ 𝐾 ) ↾ ( 𝑋 × 𝑋 ) )
Assertion	isxms	⊢ ( 𝐾 ∈ ∞MetSp ↔ ( 𝐾 ∈ TopSp ∧ 𝐽 = ( MetOpen ‘ 𝐷 ) ) )

Proof

Step	Hyp	Ref	Expression
1		isms.j	⊢ 𝐽 = ( TopOpen ‘ 𝐾 )
2		isms.x	⊢ 𝑋 = ( Base ‘ 𝐾 )
3		isms.d	⊢ 𝐷 = ( ( dist ‘ 𝐾 ) ↾ ( 𝑋 × 𝑋 ) )
4		fveq2	⊢ ( 𝑓 = 𝐾 → ( TopOpen ‘ 𝑓 ) = ( TopOpen ‘ 𝐾 ) )
5	4 1	eqtr4di	⊢ ( 𝑓 = 𝐾 → ( TopOpen ‘ 𝑓 ) = 𝐽 )
6		fveq2	⊢ ( 𝑓 = 𝐾 → ( dist ‘ 𝑓 ) = ( dist ‘ 𝐾 ) )
7		fveq2	⊢ ( 𝑓 = 𝐾 → ( Base ‘ 𝑓 ) = ( Base ‘ 𝐾 ) )
8	7 2	eqtr4di	⊢ ( 𝑓 = 𝐾 → ( Base ‘ 𝑓 ) = 𝑋 )
9	8	sqxpeqd	⊢ ( 𝑓 = 𝐾 → ( ( Base ‘ 𝑓 ) × ( Base ‘ 𝑓 ) ) = ( 𝑋 × 𝑋 ) )
10	6 9	reseq12d	⊢ ( 𝑓 = 𝐾 → ( ( dist ‘ 𝑓 ) ↾ ( ( Base ‘ 𝑓 ) × ( Base ‘ 𝑓 ) ) ) = ( ( dist ‘ 𝐾 ) ↾ ( 𝑋 × 𝑋 ) ) )
11	10 3	eqtr4di	⊢ ( 𝑓 = 𝐾 → ( ( dist ‘ 𝑓 ) ↾ ( ( Base ‘ 𝑓 ) × ( Base ‘ 𝑓 ) ) ) = 𝐷 )
12	11	fveq2d	⊢ ( 𝑓 = 𝐾 → ( MetOpen ‘ ( ( dist ‘ 𝑓 ) ↾ ( ( Base ‘ 𝑓 ) × ( Base ‘ 𝑓 ) ) ) ) = ( MetOpen ‘ 𝐷 ) )
13	5 12	eqeq12d	⊢ ( 𝑓 = 𝐾 → ( ( TopOpen ‘ 𝑓 ) = ( MetOpen ‘ ( ( dist ‘ 𝑓 ) ↾ ( ( Base ‘ 𝑓 ) × ( Base ‘ 𝑓 ) ) ) ) ↔ 𝐽 = ( MetOpen ‘ 𝐷 ) ) )
14		df-xms	⊢ ∞MetSp = { 𝑓 ∈ TopSp ∣ ( TopOpen ‘ 𝑓 ) = ( MetOpen ‘ ( ( dist ‘ 𝑓 ) ↾ ( ( Base ‘ 𝑓 ) × ( Base ‘ 𝑓 ) ) ) ) }
15	13 14	elrab2	⊢ ( 𝐾 ∈ ∞MetSp ↔ ( 𝐾 ∈ TopSp ∧ 𝐽 = ( MetOpen ‘ 𝐷 ) ) )