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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 2738 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2738 | . . . . 5 ⊢ (UnifSt‘𝑊) = (UnifSt‘𝑊) | |
3 | uspreg.1 | . . . . 5 ⊢ 𝐽 = (TopOpen‘𝑊) | |
4 | 1, 2, 3 | isusp 23013 | . . . 4 ⊢ (𝑊 ∈ UnifSp ↔ ((UnifSt‘𝑊) ∈ (UnifOn‘(Base‘𝑊)) ∧ 𝐽 = (unifTop‘(UnifSt‘𝑊)))) |
5 | 4 | simprbi 500 | . . 3 ⊢ (𝑊 ∈ UnifSp → 𝐽 = (unifTop‘(UnifSt‘𝑊))) |
6 | 5 | adantr 484 | . 2 ⊢ ((𝑊 ∈ UnifSp ∧ 𝐽 ∈ Haus) → 𝐽 = (unifTop‘(UnifSt‘𝑊))) |
7 | 4 | simplbi 501 | . . 3 ⊢ (𝑊 ∈ UnifSp → (UnifSt‘𝑊) ∈ (UnifOn‘(Base‘𝑊))) |
8 | simpr 488 | . . . 4 ⊢ ((𝑊 ∈ UnifSp ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Haus) | |
9 | 6, 8 | eqeltrrd 2834 | . . 3 ⊢ ((𝑊 ∈ UnifSp ∧ 𝐽 ∈ Haus) → (unifTop‘(UnifSt‘𝑊)) ∈ Haus) |
10 | eqid 2738 | . . . 4 ⊢ (unifTop‘(UnifSt‘𝑊)) = (unifTop‘(UnifSt‘𝑊)) | |
11 | 10 | utopreg 23004 | . . 3 ⊢ (((UnifSt‘𝑊) ∈ (UnifOn‘(Base‘𝑊)) ∧ (unifTop‘(UnifSt‘𝑊)) ∈ Haus) → (unifTop‘(UnifSt‘𝑊)) ∈ Reg) |
12 | 7, 9, 11 | syl2an2r 685 | . 2 ⊢ ((𝑊 ∈ UnifSp ∧ 𝐽 ∈ Haus) → (unifTop‘(UnifSt‘𝑊)) ∈ Reg) |
13 | 6, 12 | eqeltrd 2833 | 1 ⊢ ((𝑊 ∈ UnifSp ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Reg) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ‘cfv 6339 Basecbs 16586 TopOpenctopn 16798 Hauscha 22059 Regcreg 22060 UnifOncust 22951 unifTopcutop 22982 UnifStcuss 23005 UnifSpcusp 23006 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-1o 8131 df-er 8320 df-map 8439 df-en 8556 df-fin 8559 df-fi 8948 df-topgen 16820 df-top 21645 df-topon 21662 df-bases 21697 df-cld 21770 df-ntr 21771 df-cls 21772 df-nei 21849 df-cn 21978 df-cnp 21979 df-reg 22067 df-tx 22313 df-ust 22952 df-utop 22983 df-usp 23009 |
This theorem is referenced by: cnextucn 23055 rrhre 31541 |
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