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| Mirrors > Home > MPE Home > Th. List > isxms | Structured version Visualization version GIF version | ||
| 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 | 
|---|---|
| isms.j | ⊢ 𝐽 = (TopOpen‘𝐾) | 
| isms.x | ⊢ 𝑋 = (Base‘𝐾) | 
| isms.d | ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋)) | 
| Ref | Expression | 
|---|---|
| isxms | ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fveq2 6905 | . . . 4 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = (TopOpen‘𝐾)) | |
| 2 | isms.j | . . . 4 ⊢ 𝐽 = (TopOpen‘𝐾) | |
| 3 | 1, 2 | eqtr4di 2794 | . . 3 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = 𝐽) | 
| 4 | fveq2 6905 | . . . . . 6 ⊢ (𝑓 = 𝐾 → (dist‘𝑓) = (dist‘𝐾)) | |
| 5 | fveq2 6905 | . . . . . . . 8 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾)) | |
| 6 | isms.x | . . . . . . . 8 ⊢ 𝑋 = (Base‘𝐾) | |
| 7 | 5, 6 | eqtr4di 2794 | . . . . . . 7 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = 𝑋) | 
| 8 | 7 | sqxpeqd 5716 | . . . . . 6 ⊢ (𝑓 = 𝐾 → ((Base‘𝑓) × (Base‘𝑓)) = (𝑋 × 𝑋)) | 
| 9 | 4, 8 | reseq12d 5997 | . . . . 5 ⊢ (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = ((dist‘𝐾) ↾ (𝑋 × 𝑋))) | 
| 10 | isms.d | . . . . 5 ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋)) | |
| 11 | 9, 10 | eqtr4di 2794 | . . . 4 ⊢ (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = 𝐷) | 
| 12 | 11 | fveq2d 6909 | . . 3 ⊢ (𝑓 = 𝐾 → (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓)))) = (MetOpen‘𝐷)) | 
| 13 | 3, 12 | eqeq12d 2752 | . 2 ⊢ (𝑓 = 𝐾 → ((TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓)))) ↔ 𝐽 = (MetOpen‘𝐷))) | 
| 14 | df-xms 24331 | . 2 ⊢ ∞MetSp = {𝑓 ∈ TopSp ∣ (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))} | |
| 15 | 13, 14 | elrab2 3694 | 1 ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 × cxp 5682 ↾ cres 5686 ‘cfv 6560 Basecbs 17248 distcds 17307 TopOpenctopn 17467 MetOpencmopn 21355 TopSpctps 22939 ∞MetSpcxms 24328 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-xp 5690 df-res 5696 df-iota 6513 df-fv 6568 df-xms 24331 | 
| This theorem is referenced by: isxms2 24459 xmstopn 24462 xmstps 24464 xmspropd 24484 | 
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