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Mirrors > Home > ILE Home > Th. List > xmstopn | GIF version |
Description: The topology component of an extended metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.) |
Ref | Expression |
---|---|
isms.j | ⊢ 𝐽 = (TopOpen‘𝐾) |
isms.x | ⊢ 𝑋 = (Base‘𝐾) |
isms.d | ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋)) |
Ref | Expression |
---|---|
xmstopn | ⊢ (𝐾 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isms.j | . . 3 ⊢ 𝐽 = (TopOpen‘𝐾) | |
2 | isms.x | . . 3 ⊢ 𝑋 = (Base‘𝐾) | |
3 | isms.d | . . 3 ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋)) | |
4 | 1, 2, 3 | isxms 13330 | . 2 ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷))) |
5 | 4 | simprbi 273 | 1 ⊢ (𝐾 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1349 ∈ wcel 2142 × cxp 4610 ↾ cres 4614 ‘cfv 5200 Basecbs 12420 distcds 12493 TopOpenctopn 12584 MetOpencmopn 12864 TopSpctps 12907 ∞MetSpcxms 13215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 705 ax-5 1441 ax-7 1442 ax-gen 1443 ax-ie1 1487 ax-ie2 1488 ax-8 1498 ax-10 1499 ax-11 1500 ax-i12 1501 ax-bndl 1503 ax-4 1504 ax-17 1520 ax-i9 1524 ax-ial 1528 ax-i5r 1529 ax-ext 2153 |
This theorem depends on definitions: df-bi 116 df-3an 976 df-tru 1352 df-nf 1455 df-sb 1757 df-clab 2158 df-cleq 2164 df-clel 2167 df-nfc 2302 df-rex 2455 df-rab 2458 df-v 2733 df-un 3126 df-in 3128 df-sn 3590 df-pr 3591 df-op 3593 df-uni 3798 df-br 3991 df-opab 4052 df-xp 4618 df-res 4624 df-iota 5162 df-fv 5208 df-xms 13218 |
This theorem is referenced by: (None) |
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