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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 2622 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2622 | . . . . 5 ⊢ (UnifSt‘𝑊) = (UnifSt‘𝑊) | |
3 | uspreg.1 | . . . . 5 ⊢ 𝐽 = (TopOpen‘𝑊) | |
4 | 1, 2, 3 | isusp 22065 | . . . 4 ⊢ (𝑊 ∈ UnifSp ↔ ((UnifSt‘𝑊) ∈ (UnifOn‘(Base‘𝑊)) ∧ 𝐽 = (unifTop‘(UnifSt‘𝑊)))) |
5 | 4 | simprbi 480 | . . 3 ⊢ (𝑊 ∈ UnifSp → 𝐽 = (unifTop‘(UnifSt‘𝑊))) |
6 | 5 | adantr 481 | . 2 ⊢ ((𝑊 ∈ UnifSp ∧ 𝐽 ∈ Haus) → 𝐽 = (unifTop‘(UnifSt‘𝑊))) |
7 | 4 | simplbi 476 | . . . 4 ⊢ (𝑊 ∈ UnifSp → (UnifSt‘𝑊) ∈ (UnifOn‘(Base‘𝑊))) |
8 | 7 | adantr 481 | . . 3 ⊢ ((𝑊 ∈ UnifSp ∧ 𝐽 ∈ Haus) → (UnifSt‘𝑊) ∈ (UnifOn‘(Base‘𝑊))) |
9 | simpr 477 | . . . 4 ⊢ ((𝑊 ∈ UnifSp ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Haus) | |
10 | 6, 9 | eqeltrrd 2702 | . . 3 ⊢ ((𝑊 ∈ UnifSp ∧ 𝐽 ∈ Haus) → (unifTop‘(UnifSt‘𝑊)) ∈ Haus) |
11 | eqid 2622 | . . . 4 ⊢ (unifTop‘(UnifSt‘𝑊)) = (unifTop‘(UnifSt‘𝑊)) | |
12 | 11 | utopreg 22056 | . . 3 ⊢ (((UnifSt‘𝑊) ∈ (UnifOn‘(Base‘𝑊)) ∧ (unifTop‘(UnifSt‘𝑊)) ∈ Haus) → (unifTop‘(UnifSt‘𝑊)) ∈ Reg) |
13 | 8, 10, 12 | syl2anc 693 | . 2 ⊢ ((𝑊 ∈ UnifSp ∧ 𝐽 ∈ Haus) → (unifTop‘(UnifSt‘𝑊)) ∈ Reg) |
14 | 6, 13 | eqeltrd 2701 | 1 ⊢ ((𝑊 ∈ UnifSp ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Reg) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 Basecbs 15857 TopOpenctopn 16082 Hauscha 21112 Regcreg 21113 UnifOncust 22003 unifTopcutop 22034 UnifStcuss 22057 UnifSpcusp 22058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-fin 7959 df-fi 8317 df-topgen 16104 df-top 20699 df-topon 20716 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-cn 21031 df-cnp 21032 df-reg 21120 df-tx 21365 df-ust 22004 df-utop 22035 df-usp 22061 |
This theorem is referenced by: cnextucn 22107 rrhre 30065 |
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