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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 2823 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2823 | . . . . 5 ⊢ (UnifSt‘𝑊) = (UnifSt‘𝑊) | |
3 | uspreg.1 | . . . . 5 ⊢ 𝐽 = (TopOpen‘𝑊) | |
4 | 1, 2, 3 | isusp 22872 | . . . 4 ⊢ (𝑊 ∈ UnifSp ↔ ((UnifSt‘𝑊) ∈ (UnifOn‘(Base‘𝑊)) ∧ 𝐽 = (unifTop‘(UnifSt‘𝑊)))) |
5 | 4 | simprbi 499 | . . 3 ⊢ (𝑊 ∈ UnifSp → 𝐽 = (unifTop‘(UnifSt‘𝑊))) |
6 | 5 | adantr 483 | . 2 ⊢ ((𝑊 ∈ UnifSp ∧ 𝐽 ∈ Haus) → 𝐽 = (unifTop‘(UnifSt‘𝑊))) |
7 | 4 | simplbi 500 | . . 3 ⊢ (𝑊 ∈ UnifSp → (UnifSt‘𝑊) ∈ (UnifOn‘(Base‘𝑊))) |
8 | simpr 487 | . . . 4 ⊢ ((𝑊 ∈ UnifSp ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Haus) | |
9 | 6, 8 | eqeltrrd 2916 | . . 3 ⊢ ((𝑊 ∈ UnifSp ∧ 𝐽 ∈ Haus) → (unifTop‘(UnifSt‘𝑊)) ∈ Haus) |
10 | eqid 2823 | . . . 4 ⊢ (unifTop‘(UnifSt‘𝑊)) = (unifTop‘(UnifSt‘𝑊)) | |
11 | 10 | utopreg 22863 | . . 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 2915 | 1 ⊢ ((𝑊 ∈ UnifSp ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Reg) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 Basecbs 16485 TopOpenctopn 16697 Hauscha 21918 Regcreg 21919 UnifOncust 22810 unifTopcutop 22841 UnifStcuss 22864 UnifSpcusp 22865 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-fin 8515 df-fi 8877 df-topgen 16719 df-top 21504 df-topon 21521 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-cn 21837 df-cnp 21838 df-reg 21926 df-tx 22172 df-ust 22811 df-utop 22842 df-usp 22868 |
This theorem is referenced by: cnextucn 22914 rrhre 31264 |
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