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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 | β’ π½ = (TopOpenβπΊ) |
Ref | Expression |
---|---|
tgpt1 | β’ (πΊ β TopGrp β (π½ β Haus β π½ β Fre)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | haust1 23255 | . 2 β’ (π½ β Haus β π½ β Fre) | |
2 | tgpgrp 23981 | . . . . . 6 β’ (πΊ β TopGrp β πΊ β Grp) | |
3 | eqid 2728 | . . . . . . 7 β’ (BaseβπΊ) = (BaseβπΊ) | |
4 | eqid 2728 | . . . . . . 7 β’ (0gβπΊ) = (0gβπΊ) | |
5 | 3, 4 | grpidcl 18921 | . . . . . 6 β’ (πΊ β Grp β (0gβπΊ) β (BaseβπΊ)) |
6 | 2, 5 | syl 17 | . . . . 5 β’ (πΊ β TopGrp β (0gβπΊ) β (BaseβπΊ)) |
7 | tgpt1.j | . . . . . . 7 β’ π½ = (TopOpenβπΊ) | |
8 | 7, 3 | tgptopon 23985 | . . . . . 6 β’ (πΊ β TopGrp β π½ β (TopOnβ(BaseβπΊ))) |
9 | toponuni 22815 | . . . . . 6 β’ (π½ β (TopOnβ(BaseβπΊ)) β (BaseβπΊ) = βͺ π½) | |
10 | 8, 9 | syl 17 | . . . . 5 β’ (πΊ β TopGrp β (BaseβπΊ) = βͺ π½) |
11 | 6, 10 | eleqtrd 2831 | . . . 4 β’ (πΊ β TopGrp β (0gβπΊ) β βͺ π½) |
12 | eqid 2728 | . . . . . 6 β’ βͺ π½ = βͺ π½ | |
13 | 12 | t1sncld 23229 | . . . . 5 β’ ((π½ β Fre β§ (0gβπΊ) β βͺ π½) β {(0gβπΊ)} β (Clsdβπ½)) |
14 | 13 | expcom 413 | . . . 4 β’ ((0gβπΊ) β βͺ π½ β (π½ β Fre β {(0gβπΊ)} β (Clsdβπ½))) |
15 | 11, 14 | syl 17 | . . 3 β’ (πΊ β TopGrp β (π½ β Fre β {(0gβπΊ)} β (Clsdβπ½))) |
16 | 4, 7 | tgphaus 24020 | . . 3 β’ (πΊ β TopGrp β (π½ β Haus β {(0gβπΊ)} β (Clsdβπ½))) |
17 | 15, 16 | sylibrd 259 | . 2 β’ (πΊ β TopGrp β (π½ β Fre β π½ β Haus)) |
18 | 1, 17 | impbid2 225 | 1 β’ (πΊ β TopGrp β (π½ β Haus β π½ β Fre)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1534 β wcel 2099 {csn 4629 βͺ cuni 4908 βcfv 6548 Basecbs 17179 TopOpenctopn 17402 0gc0g 17420 Grpcgrp 18889 TopOnctopon 22811 Clsdccld 22919 Frect1 23210 Hauscha 23211 TopGrpctgp 23974 |
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 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fo 6554 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-map 8846 df-0g 17422 df-topgen 17424 df-plusf 18598 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18892 df-minusg 18893 df-sbg 18894 df-top 22795 df-topon 22812 df-topsp 22834 df-bases 22848 df-cld 22922 df-cn 23130 df-t1 23217 df-haus 23218 df-tx 23465 df-tmd 23975 df-tgp 23976 |
This theorem is referenced by: tgpt0 24022 |
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