Theorem isms

Description: Express the predicate " <. X , D >. is a metric space" with underlying set X and distance function D . (Contributed by NM, 27-Aug-2006) (Revised by Mario Carneiro, 24-Aug-2015)

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

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 → ( dist ‘ 𝑓 ) = ( dist ‘ 𝐾 ) )
5		fveq2	⊢ ( 𝑓 = 𝐾 → ( Base ‘ 𝑓 ) = ( Base ‘ 𝐾 ) )
6	5 2	eqtr4di	⊢ ( 𝑓 = 𝐾 → ( Base ‘ 𝑓 ) = 𝑋 )
7	6	sqxpeqd	⊢ ( 𝑓 = 𝐾 → ( ( Base ‘ 𝑓 ) × ( Base ‘ 𝑓 ) ) = ( 𝑋 × 𝑋 ) )
8	4 7	reseq12d	⊢ ( 𝑓 = 𝐾 → ( ( dist ‘ 𝑓 ) ↾ ( ( Base ‘ 𝑓 ) × ( Base ‘ 𝑓 ) ) ) = ( ( dist ‘ 𝐾 ) ↾ ( 𝑋 × 𝑋 ) ) )
9	8 3	eqtr4di	⊢ ( 𝑓 = 𝐾 → ( ( dist ‘ 𝑓 ) ↾ ( ( Base ‘ 𝑓 ) × ( Base ‘ 𝑓 ) ) ) = 𝐷 )
10	6	fveq2d	⊢ ( 𝑓 = 𝐾 → ( Met ‘ ( Base ‘ 𝑓 ) ) = ( Met ‘ 𝑋 ) )
11	9 10	eleq12d	⊢ ( 𝑓 = 𝐾 → ( ( ( dist ‘ 𝑓 ) ↾ ( ( Base ‘ 𝑓 ) × ( Base ‘ 𝑓 ) ) ) ∈ ( Met ‘ ( Base ‘ 𝑓 ) ) ↔ 𝐷 ∈ ( Met ‘ 𝑋 ) ) )
12		df-ms	⊢ MetSp = { 𝑓 ∈ ∞MetSp ∣ ( ( dist ‘ 𝑓 ) ↾ ( ( Base ‘ 𝑓 ) × ( Base ‘ 𝑓 ) ) ) ∈ ( Met ‘ ( Base ‘ 𝑓 ) ) }
13	11 12	elrab2	⊢ ( 𝐾 ∈ MetSp ↔ ( 𝐾 ∈ ∞MetSp ∧ 𝐷 ∈ ( Met ‘ 𝑋 ) ) )