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Mirrors > Home > MPE Home > Th. List > uspreg | Structured version Visualization version GIF version |
Description: If a uniform space is Hausdorff, it is regular. Proposition 3 of [BourbakiTop1] p. II.5. (Contributed by Thierry Arnoux, 4-Jan-2018.) |
Ref | Expression |
---|---|
uspreg.1 | ⊢ 𝐽 = (TopOpen‘𝑊) |
Ref | Expression |
---|---|
uspreg | ⊢ ((𝑊 ∈ UnifSp ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Reg) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2728 | . . . . 5 ⊢ (UnifSt‘𝑊) = (UnifSt‘𝑊) | |
3 | uspreg.1 | . . . . 5 ⊢ 𝐽 = (TopOpen‘𝑊) | |
4 | 1, 2, 3 | isusp 24186 | . . . 4 ⊢ (𝑊 ∈ UnifSp ↔ ((UnifSt‘𝑊) ∈ (UnifOn‘(Base‘𝑊)) ∧ 𝐽 = (unifTop‘(UnifSt‘𝑊)))) |
5 | 4 | simprbi 495 | . . 3 ⊢ (𝑊 ∈ UnifSp → 𝐽 = (unifTop‘(UnifSt‘𝑊))) |
6 | 5 | adantr 479 | . 2 ⊢ ((𝑊 ∈ UnifSp ∧ 𝐽 ∈ Haus) → 𝐽 = (unifTop‘(UnifSt‘𝑊))) |
7 | 4 | simplbi 496 | . . 3 ⊢ (𝑊 ∈ UnifSp → (UnifSt‘𝑊) ∈ (UnifOn‘(Base‘𝑊))) |
8 | simpr 483 | . . . 4 ⊢ ((𝑊 ∈ UnifSp ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Haus) | |
9 | 6, 8 | eqeltrrd 2830 | . . 3 ⊢ ((𝑊 ∈ UnifSp ∧ 𝐽 ∈ Haus) → (unifTop‘(UnifSt‘𝑊)) ∈ Haus) |
10 | eqid 2728 | . . . 4 ⊢ (unifTop‘(UnifSt‘𝑊)) = (unifTop‘(UnifSt‘𝑊)) | |
11 | 10 | utopreg 24177 | . . 3 ⊢ (((UnifSt‘𝑊) ∈ (UnifOn‘(Base‘𝑊)) ∧ (unifTop‘(UnifSt‘𝑊)) ∈ Haus) → (unifTop‘(UnifSt‘𝑊)) ∈ Reg) |
12 | 7, 9, 11 | syl2an2r 683 | . 2 ⊢ ((𝑊 ∈ UnifSp ∧ 𝐽 ∈ Haus) → (unifTop‘(UnifSt‘𝑊)) ∈ Reg) |
13 | 6, 12 | eqeltrd 2829 | 1 ⊢ ((𝑊 ∈ UnifSp ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Reg) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ‘cfv 6553 Basecbs 17187 TopOpenctopn 17410 Hauscha 23232 Regcreg 23233 UnifOncust 24124 unifTopcutop 24155 UnifStcuss 24178 UnifSpcusp 24179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-1o 8493 df-er 8731 df-map 8853 df-en 8971 df-fin 8974 df-fi 9442 df-topgen 17432 df-top 22816 df-topon 22833 df-bases 22869 df-cld 22943 df-ntr 22944 df-cls 22945 df-nei 23022 df-cn 23151 df-cnp 23152 df-reg 23240 df-tx 23486 df-ust 24125 df-utop 24156 df-usp 24182 |
This theorem is referenced by: cnextucn 24228 rrhre 33655 |
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