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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 6304 | . . . 4 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = (TopOpen‘𝐾)) | |
2 | isms.j | . . . 4 ⊢ 𝐽 = (TopOpen‘𝐾) | |
3 | 1, 2 | syl6eqr 2776 | . . 3 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = 𝐽) |
4 | fveq2 6304 | . . . . . 6 ⊢ (𝑓 = 𝐾 → (dist‘𝑓) = (dist‘𝐾)) | |
5 | fveq2 6304 | . . . . . . . 8 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾)) | |
6 | isms.x | . . . . . . . 8 ⊢ 𝑋 = (Base‘𝐾) | |
7 | 5, 6 | syl6eqr 2776 | . . . . . . 7 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = 𝑋) |
8 | 7 | sqxpeqd 5250 | . . . . . 6 ⊢ (𝑓 = 𝐾 → ((Base‘𝑓) × (Base‘𝑓)) = (𝑋 × 𝑋)) |
9 | 4, 8 | reseq12d 5504 | . . . . 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 | syl6eqr 2776 | . . . 4 ⊢ (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = 𝐷) |
12 | 11 | fveq2d 6308 | . . 3 ⊢ (𝑓 = 𝐾 → (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓)))) = (MetOpen‘𝐷)) |
13 | 3, 12 | eqeq12d 2739 | . 2 ⊢ (𝑓 = 𝐾 → ((TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓)))) ↔ 𝐽 = (MetOpen‘𝐷))) |
14 | df-xms 22247 | . 2 ⊢ ∞MetSp = {𝑓 ∈ TopSp ∣ (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))} | |
15 | 13, 14 | elrab2 3472 | 1 ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 383 = wceq 1596 ∈ wcel 2103 × cxp 5216 ↾ cres 5220 ‘cfv 6001 Basecbs 15980 distcds 16073 TopOpenctopn 16205 MetOpencmopn 19859 TopSpctps 20859 ∞MetSpcxme 22244 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1835 ax-4 1850 ax-5 1952 ax-6 2018 ax-7 2054 ax-9 2112 ax-10 2132 ax-11 2147 ax-12 2160 ax-13 2355 ax-ext 2704 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1599 df-ex 1818 df-nf 1823 df-sb 2011 df-clab 2711 df-cleq 2717 df-clel 2720 df-nfc 2855 df-rex 3020 df-rab 3023 df-v 3306 df-dif 3683 df-un 3685 df-in 3687 df-ss 3694 df-nul 4024 df-if 4195 df-sn 4286 df-pr 4288 df-op 4292 df-uni 4545 df-br 4761 df-opab 4821 df-xp 5224 df-res 5230 df-iota 5964 df-fv 6009 df-xms 22247 |
This theorem is referenced by: isxms2 22375 xmstopn 22378 xmstps 22380 xmspropd 22400 |
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