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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 21534 | . 2 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Fre) | |
2 | tgpgrp 22259 | . . . . . 6 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp) | |
3 | eqid 2825 | . . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
4 | eqid 2825 | . . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
5 | 3, 4 | grpidcl 17811 | . . . . . 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 22263 | . . . . . 6 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺))) |
9 | toponuni 21096 | . . . . . 6 ⊢ (𝐽 ∈ (TopOn‘(Base‘𝐺)) → (Base‘𝐺) = ∪ 𝐽) | |
10 | 8, 9 | syl 17 | . . . . 5 ⊢ (𝐺 ∈ TopGrp → (Base‘𝐺) = ∪ 𝐽) |
11 | 6, 10 | eleqtrd 2908 | . . . 4 ⊢ (𝐺 ∈ TopGrp → (0g‘𝐺) ∈ ∪ 𝐽) |
12 | eqid 2825 | . . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
13 | 12 | t1sncld 21508 | . . . . 5 ⊢ ((𝐽 ∈ Fre ∧ (0g‘𝐺) ∈ ∪ 𝐽) → {(0g‘𝐺)} ∈ (Clsd‘𝐽)) |
14 | 13 | expcom 404 | . . . 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 22297 | . . 3 ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ {(0g‘𝐺)} ∈ (Clsd‘𝐽))) |
17 | 15, 16 | sylibrd 251 | . 2 ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Fre → 𝐽 ∈ Haus)) |
18 | 1, 17 | impbid2 218 | 1 ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ 𝐽 ∈ Fre)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1656 ∈ wcel 2164 {csn 4399 ∪ cuni 4660 ‘cfv 6127 Basecbs 16229 TopOpenctopn 16442 0gc0g 16460 Grpcgrp 17783 TopOnctopon 21092 Clsdccld 21198 Frect1 21489 Hauscha 21490 TopGrpctgp 22252 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-1st 7433 df-2nd 7434 df-map 8129 df-0g 16462 df-topgen 16464 df-plusf 17601 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-grp 17786 df-minusg 17787 df-sbg 17788 df-top 21076 df-topon 21093 df-topsp 21115 df-bases 21128 df-cld 21201 df-cn 21409 df-t1 21496 df-haus 21497 df-tx 21743 df-tmd 22253 df-tgp 22254 |
This theorem is referenced by: tgpt0 22299 |
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