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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 2795 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2795 | . . . . 5 ⊢ (UnifSt‘𝑊) = (UnifSt‘𝑊) | |
3 | uspreg.1 | . . . . 5 ⊢ 𝐽 = (TopOpen‘𝑊) | |
4 | 1, 2, 3 | isusp 22558 | . . . 4 ⊢ (𝑊 ∈ UnifSp ↔ ((UnifSt‘𝑊) ∈ (UnifOn‘(Base‘𝑊)) ∧ 𝐽 = (unifTop‘(UnifSt‘𝑊)))) |
5 | 4 | simprbi 497 | . . 3 ⊢ (𝑊 ∈ UnifSp → 𝐽 = (unifTop‘(UnifSt‘𝑊))) |
6 | 5 | adantr 481 | . 2 ⊢ ((𝑊 ∈ UnifSp ∧ 𝐽 ∈ Haus) → 𝐽 = (unifTop‘(UnifSt‘𝑊))) |
7 | 4 | simplbi 498 | . . 3 ⊢ (𝑊 ∈ UnifSp → (UnifSt‘𝑊) ∈ (UnifOn‘(Base‘𝑊))) |
8 | simpr 485 | . . . 4 ⊢ ((𝑊 ∈ UnifSp ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Haus) | |
9 | 6, 8 | eqeltrrd 2884 | . . 3 ⊢ ((𝑊 ∈ UnifSp ∧ 𝐽 ∈ Haus) → (unifTop‘(UnifSt‘𝑊)) ∈ Haus) |
10 | eqid 2795 | . . . 4 ⊢ (unifTop‘(UnifSt‘𝑊)) = (unifTop‘(UnifSt‘𝑊)) | |
11 | 10 | utopreg 22549 | . . 3 ⊢ (((UnifSt‘𝑊) ∈ (UnifOn‘(Base‘𝑊)) ∧ (unifTop‘(UnifSt‘𝑊)) ∈ Haus) → (unifTop‘(UnifSt‘𝑊)) ∈ Reg) |
12 | 7, 9, 11 | syl2an2r 681 | . 2 ⊢ ((𝑊 ∈ UnifSp ∧ 𝐽 ∈ Haus) → (unifTop‘(UnifSt‘𝑊)) ∈ Reg) |
13 | 6, 12 | eqeltrd 2883 | 1 ⊢ ((𝑊 ∈ UnifSp ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Reg) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ‘cfv 6230 Basecbs 16317 TopOpenctopn 16529 Hauscha 21605 Regcreg 21606 UnifOncust 22496 unifTopcutop 22527 UnifStcuss 22550 UnifSpcusp 22551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5086 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-int 4787 df-iun 4831 df-iin 4832 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-ov 7024 df-oprab 7025 df-mpo 7026 df-om 7442 df-1st 7550 df-2nd 7551 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-1o 7958 df-oadd 7962 df-er 8144 df-map 8263 df-en 8363 df-fin 8366 df-fi 8726 df-topgen 16551 df-top 21191 df-topon 21208 df-bases 21243 df-cld 21316 df-ntr 21317 df-cls 21318 df-nei 21395 df-cn 21524 df-cnp 21525 df-reg 21613 df-tx 21859 df-ust 22497 df-utop 22528 df-usp 22554 |
This theorem is referenced by: cnextucn 22600 rrhre 30884 |
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