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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 21957 | . 2 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Fre) | |
2 | tgpgrp 22683 | . . . . . 6 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp) | |
3 | eqid 2798 | . . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
4 | eqid 2798 | . . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
5 | 3, 4 | grpidcl 18123 | . . . . . 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 22687 | . . . . . 6 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘(Base‘𝐺))) |
9 | toponuni 21519 | . . . . . 6 ⊢ (𝐽 ∈ (TopOn‘(Base‘𝐺)) → (Base‘𝐺) = ∪ 𝐽) | |
10 | 8, 9 | syl 17 | . . . . 5 ⊢ (𝐺 ∈ TopGrp → (Base‘𝐺) = ∪ 𝐽) |
11 | 6, 10 | eleqtrd 2892 | . . . 4 ⊢ (𝐺 ∈ TopGrp → (0g‘𝐺) ∈ ∪ 𝐽) |
12 | eqid 2798 | . . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
13 | 12 | t1sncld 21931 | . . . . 5 ⊢ ((𝐽 ∈ Fre ∧ (0g‘𝐺) ∈ ∪ 𝐽) → {(0g‘𝐺)} ∈ (Clsd‘𝐽)) |
14 | 13 | expcom 417 | . . . 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 22722 | . . 3 ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ {(0g‘𝐺)} ∈ (Clsd‘𝐽))) |
17 | 15, 16 | sylibrd 262 | . 2 ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Fre → 𝐽 ∈ Haus)) |
18 | 1, 17 | impbid2 229 | 1 ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ 𝐽 ∈ Fre)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 {csn 4525 ∪ cuni 4800 ‘cfv 6324 Basecbs 16475 TopOpenctopn 16687 0gc0g 16705 Grpcgrp 18095 TopOnctopon 21515 Clsdccld 21621 Frect1 21912 Hauscha 21913 TopGrpctgp 22676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fo 6330 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-map 8391 df-0g 16707 df-topgen 16709 df-plusf 17843 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-sbg 18100 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-cn 21832 df-t1 21919 df-haus 21920 df-tx 22167 df-tmd 22677 df-tgp 22678 |
This theorem is referenced by: tgpt0 22724 |
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