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Mirrors > Home > MPE Home > Th. List > tgpt1 | Structured version Visualization version GIF version |
Description: Hausdorff and T1 are equivalent for topological groups. (Contributed by Mario Carneiro, 18-Sep-2015.) |
Ref | Expression |
---|---|
tgpt1.j | ⊢ 𝐽 = (TopOpen‘𝐺) |
Ref | Expression |
---|---|
tgpt1 | ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ 𝐽 ∈ Fre)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | haust1 21888 | . 2 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Fre) | |
2 | tgpgrp 22614 | . . . . . 6 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp) | |
3 | eqid 2818 | . . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
4 | eqid 2818 | . . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
5 | 3, 4 | grpidcl 18069 | . . . . . 6 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ (Base‘𝐺)) |
6 | 2, 5 | syl 17 | . . . . 5 ⊢ (𝐺 ∈ TopGrp → (0g‘𝐺) ∈ (Base‘𝐺)) |
7 | tgpt1.j | . . . . . . 7 ⊢ 𝐽 = (TopOpen‘𝐺) | |
8 | 7, 3 | tgptopon 22618 | . . . . . 6 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺))) |
9 | toponuni 21450 | . . . . . 6 ⊢ (𝐽 ∈ (TopOn‘(Base‘𝐺)) → (Base‘𝐺) = ∪ 𝐽) | |
10 | 8, 9 | syl 17 | . . . . 5 ⊢ (𝐺 ∈ TopGrp → (Base‘𝐺) = ∪ 𝐽) |
11 | 6, 10 | eleqtrd 2912 | . . . 4 ⊢ (𝐺 ∈ TopGrp → (0g‘𝐺) ∈ ∪ 𝐽) |
12 | eqid 2818 | . . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
13 | 12 | t1sncld 21862 | . . . . 5 ⊢ ((𝐽 ∈ Fre ∧ (0g‘𝐺) ∈ ∪ 𝐽) → {(0g‘𝐺)} ∈ (Clsd‘𝐽)) |
14 | 13 | expcom 414 | . . . 4 ⊢ ((0g‘𝐺) ∈ ∪ 𝐽 → (𝐽 ∈ Fre → {(0g‘𝐺)} ∈ (Clsd‘𝐽))) |
15 | 11, 14 | syl 17 | . . 3 ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Fre → {(0g‘𝐺)} ∈ (Clsd‘𝐽))) |
16 | 4, 7 | tgphaus 22652 | . . 3 ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ {(0g‘𝐺)} ∈ (Clsd‘𝐽))) |
17 | 15, 16 | sylibrd 260 | . 2 ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Fre → 𝐽 ∈ Haus)) |
18 | 1, 17 | impbid2 227 | 1 ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ 𝐽 ∈ Fre)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 {csn 4557 ∪ cuni 4830 ‘cfv 6348 Basecbs 16471 TopOpenctopn 16683 0gc0g 16701 Grpcgrp 18041 TopOnctopon 21446 Clsdccld 21552 Frect1 21843 Hauscha 21844 TopGrpctgp 22607 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fo 6354 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-map 8397 df-0g 16703 df-topgen 16705 df-plusf 17839 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-sbg 18046 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cld 21555 df-cn 21763 df-t1 21850 df-haus 21851 df-tx 22098 df-tmd 22608 df-tgp 22609 |
This theorem is referenced by: tgpt0 22654 |
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