![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 23344 | . 2 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Fre) | |
2 | tgpgrp 24070 | . . . . . 6 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp) | |
3 | eqid 2726 | . . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
4 | eqid 2726 | . . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
5 | 3, 4 | grpidcl 18955 | . . . . . 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 24074 | . . . . . 6 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺))) |
9 | toponuni 22904 | . . . . . 6 ⊢ (𝐽 ∈ (TopOn‘(Base‘𝐺)) → (Base‘𝐺) = ∪ 𝐽) | |
10 | 8, 9 | syl 17 | . . . . 5 ⊢ (𝐺 ∈ TopGrp → (Base‘𝐺) = ∪ 𝐽) |
11 | 6, 10 | eleqtrd 2828 | . . . 4 ⊢ (𝐺 ∈ TopGrp → (0g‘𝐺) ∈ ∪ 𝐽) |
12 | eqid 2726 | . . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
13 | 12 | t1sncld 23318 | . . . . 5 ⊢ ((𝐽 ∈ Fre ∧ (0g‘𝐺) ∈ ∪ 𝐽) → {(0g‘𝐺)} ∈ (Clsd‘𝐽)) |
14 | 13 | expcom 412 | . . . 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 24109 | . . 3 ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ {(0g‘𝐺)} ∈ (Clsd‘𝐽))) |
17 | 15, 16 | sylibrd 258 | . 2 ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Fre → 𝐽 ∈ Haus)) |
18 | 1, 17 | impbid2 225 | 1 ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ 𝐽 ∈ Fre)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 {csn 4623 ∪ cuni 4905 ‘cfv 6546 Basecbs 17208 TopOpenctopn 17431 0gc0g 17449 Grpcgrp 18923 TopOnctopon 22900 Clsdccld 23008 Frect1 23299 Hauscha 23300 TopGrpctgp 24063 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-fo 6552 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7995 df-2nd 7996 df-map 8849 df-0g 17451 df-topgen 17453 df-plusf 18627 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-grp 18926 df-minusg 18927 df-sbg 18928 df-top 22884 df-topon 22901 df-topsp 22923 df-bases 22937 df-cld 23011 df-cn 23219 df-t1 23306 df-haus 23307 df-tx 23554 df-tmd 24064 df-tgp 24065 |
This theorem is referenced by: tgpt0 24111 |
Copyright terms: Public domain | W3C validator |