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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 6906 | . . . . 5 ⊢ (𝑓 = 𝐾 → (dist‘𝑓) = (dist‘𝐾)) | |
| 2 | fveq2 6906 | . . . . . . 7 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾)) | |
| 3 | isms.x | . . . . . . 7 ⊢ 𝑋 = (Base‘𝐾) | |
| 4 | 2, 3 | eqtr4di 2795 | . . . . . 6 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = 𝑋) |
| 5 | 4 | sqxpeqd 5717 | . . . . 5 ⊢ (𝑓 = 𝐾 → ((Base‘𝑓) × (Base‘𝑓)) = (𝑋 × 𝑋)) |
| 6 | 1, 5 | reseq12d 5998 | . . . 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 2795 | . . 3 ⊢ (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = 𝐷) |
| 9 | 4 | fveq2d 6910 | . . 3 ⊢ (𝑓 = 𝐾 → (Met‘(Base‘𝑓)) = (Met‘𝑋)) |
| 10 | 8, 9 | eleq12d 2835 | . 2 ⊢ (𝑓 = 𝐾 → (((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓)) ↔ 𝐷 ∈ (Met‘𝑋))) |
| 11 | df-ms 24331 | . 2 ⊢ MetSp = {𝑓 ∈ ∞MetSp ∣ ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))} | |
| 12 | 10, 11 | elrab2 3695 | 1 ⊢ (𝐾 ∈ MetSp ↔ (𝐾 ∈ ∞MetSp ∧ 𝐷 ∈ (Met‘𝑋))) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 × cxp 5683 ↾ cres 5687 ‘cfv 6561 Basecbs 17247 distcds 17306 TopOpenctopn 17466 Metcmet 21350 ∞MetSpcxms 24327 MetSpcms 24328 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-res 5697 df-iota 6514 df-fv 6569 df-ms 24331 |
| This theorem is referenced by: isms2 24460 msxms 24464 mspropd 24484 setsms 24492 tmsms 24500 imasf1oms 24503 ressms 24539 prdsms 24544 |
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