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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 22726 | . 2 β’ (π½ β Haus β π½ β Fre) | |
2 | tgpgrp 23452 | . . . . . 6 β’ (πΊ β TopGrp β πΊ β Grp) | |
3 | eqid 2733 | . . . . . . 7 β’ (BaseβπΊ) = (BaseβπΊ) | |
4 | eqid 2733 | . . . . . . 7 β’ (0gβπΊ) = (0gβπΊ) | |
5 | 3, 4 | grpidcl 18786 | . . . . . 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 23456 | . . . . . 6 β’ (πΊ β TopGrp β π½ β (TopOnβ(BaseβπΊ))) |
9 | toponuni 22286 | . . . . . 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 2836 | . . . 4 β’ (πΊ β TopGrp β (0gβπΊ) β βͺ π½) |
12 | eqid 2733 | . . . . . 6 β’ βͺ π½ = βͺ π½ | |
13 | 12 | t1sncld 22700 | . . . . 5 β’ ((π½ β Fre β§ (0gβπΊ) β βͺ π½) β {(0gβπΊ)} β (Clsdβπ½)) |
14 | 13 | expcom 415 | . . . 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 23491 | . . 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 1542 β wcel 2107 {csn 4590 βͺ cuni 4869 βcfv 6500 Basecbs 17091 TopOpenctopn 17311 0gc0g 17329 Grpcgrp 18756 TopOnctopon 22282 Clsdccld 22390 Frect1 22681 Hauscha 22682 TopGrpctgp 23445 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-fo 6506 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-1st 7925 df-2nd 7926 df-map 8773 df-0g 17331 df-topgen 17333 df-plusf 18504 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-grp 18759 df-minusg 18760 df-sbg 18761 df-top 22266 df-topon 22283 df-topsp 22305 df-bases 22319 df-cld 22393 df-cn 22601 df-t1 22688 df-haus 22689 df-tx 22936 df-tmd 23446 df-tgp 23447 |
This theorem is referenced by: tgpt0 23493 |
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