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Metamath Proof Explorer

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 ) ) )