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Mirrors > Home > MPE Home > Th. List > isms | Structured version Visualization version GIF version |
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 |
---|---|
isms.j | ⊢ 𝐽 = (TopOpen‘𝐾) |
isms.x | ⊢ 𝑋 = (Base‘𝐾) |
isms.d | ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋)) |
Ref | Expression |
---|---|
isms | ⊢ (𝐾 ∈ MetSp ↔ (𝐾 ∈ ∞MetSp ∧ 𝐷 ∈ (Met‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6907 | . . . . 5 ⊢ (𝑓 = 𝐾 → (dist‘𝑓) = (dist‘𝐾)) | |
2 | fveq2 6907 | . . . . . . 7 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾)) | |
3 | isms.x | . . . . . . 7 ⊢ 𝑋 = (Base‘𝐾) | |
4 | 2, 3 | eqtr4di 2793 | . . . . . 6 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = 𝑋) |
5 | 4 | sqxpeqd 5721 | . . . . 5 ⊢ (𝑓 = 𝐾 → ((Base‘𝑓) × (Base‘𝑓)) = (𝑋 × 𝑋)) |
6 | 1, 5 | reseq12d 6001 | . . . 4 ⊢ (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = ((dist‘𝐾) ↾ (𝑋 × 𝑋))) |
7 | isms.d | . . . 4 ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋)) | |
8 | 6, 7 | eqtr4di 2793 | . . 3 ⊢ (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = 𝐷) |
9 | 4 | fveq2d 6911 | . . 3 ⊢ (𝑓 = 𝐾 → (Met‘(Base‘𝑓)) = (Met‘𝑋)) |
10 | 8, 9 | eleq12d 2833 | . 2 ⊢ (𝑓 = 𝐾 → (((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓)) ↔ 𝐷 ∈ (Met‘𝑋))) |
11 | df-ms 24347 | . 2 ⊢ MetSp = {𝑓 ∈ ∞MetSp ∣ ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))} | |
12 | 10, 11 | elrab2 3698 | 1 ⊢ (𝐾 ∈ MetSp ↔ (𝐾 ∈ ∞MetSp ∧ 𝐷 ∈ (Met‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 × cxp 5687 ↾ cres 5691 ‘cfv 6563 Basecbs 17245 distcds 17307 TopOpenctopn 17468 Metcmet 21368 ∞MetSpcxms 24343 MetSpcms 24344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-res 5701 df-iota 6516 df-fv 6571 df-ms 24347 |
This theorem is referenced by: isms2 24476 msxms 24480 mspropd 24500 setsms 24508 tmsms 24516 imasf1oms 24519 ressms 24555 prdsms 24560 |
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