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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 6191 | . . . 4 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = (TopOpen‘𝐾)) | |
2 | isms.j | . . . 4 ⊢ 𝐽 = (TopOpen‘𝐾) | |
3 | 1, 2 | syl6eqr 2674 | . . 3 ⊢ (𝑓 = 𝐾 → (TopOpen‘𝑓) = 𝐽) |
4 | fveq2 6191 | . . . . . 6 ⊢ (𝑓 = 𝐾 → (dist‘𝑓) = (dist‘𝐾)) | |
5 | fveq2 6191 | . . . . . . . 8 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾)) | |
6 | isms.x | . . . . . . . 8 ⊢ 𝑋 = (Base‘𝐾) | |
7 | 5, 6 | syl6eqr 2674 | . . . . . . 7 ⊢ (𝑓 = 𝐾 → (Base‘𝑓) = 𝑋) |
8 | 7 | sqxpeqd 5141 | . . . . . 6 ⊢ (𝑓 = 𝐾 → ((Base‘𝑓) × (Base‘𝑓)) = (𝑋 × 𝑋)) |
9 | 4, 8 | reseq12d 5397 | . . . . 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 2674 | . . . 4 ⊢ (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = 𝐷) |
12 | 11 | fveq2d 6195 | . . 3 ⊢ (𝑓 = 𝐾 → (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓)))) = (MetOpen‘𝐷)) |
13 | 3, 12 | eqeq12d 2637 | . 2 ⊢ (𝑓 = 𝐾 → ((TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓)))) ↔ 𝐽 = (MetOpen‘𝐷))) |
14 | df-xms 22125 | . 2 ⊢ ∞MetSp = {𝑓 ∈ TopSp ∣ (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))} | |
15 | 13, 14 | elrab2 3366 | 1 ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 × cxp 5112 ↾ cres 5116 ‘cfv 5888 Basecbs 15857 distcds 15950 TopOpenctopn 16082 MetOpencmopn 19736 TopSpctps 20736 ∞MetSpcxme 22122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-res 5126 df-iota 5851 df-fv 5896 df-xms 22125 |
This theorem is referenced by: isxms2 22253 xmstopn 22256 xmstps 22258 xmspropd 22278 |
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