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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 23200 | . 2 β’ (π½ β Haus β π½ β Fre) | |
2 | tgpgrp 23926 | . . . . . 6 β’ (πΊ β TopGrp β πΊ β Grp) | |
3 | eqid 2724 | . . . . . . 7 β’ (BaseβπΊ) = (BaseβπΊ) | |
4 | eqid 2724 | . . . . . . 7 β’ (0gβπΊ) = (0gβπΊ) | |
5 | 3, 4 | grpidcl 18891 | . . . . . 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 23930 | . . . . . 6 β’ (πΊ β TopGrp β π½ β (TopOnβ(BaseβπΊ))) |
9 | toponuni 22760 | . . . . . 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 2827 | . . . 4 β’ (πΊ β TopGrp β (0gβπΊ) β βͺ π½) |
12 | eqid 2724 | . . . . . 6 β’ βͺ π½ = βͺ π½ | |
13 | 12 | t1sncld 23174 | . . . . 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 23965 | . . 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 1533 β wcel 2098 {csn 4621 βͺ cuni 4900 βcfv 6534 Basecbs 17149 TopOpenctopn 17372 0gc0g 17390 Grpcgrp 18859 TopOnctopon 22756 Clsdccld 22864 Frect1 23155 Hauscha 23156 TopGrpctgp 23919 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-fo 6540 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-1st 7969 df-2nd 7970 df-map 8819 df-0g 17392 df-topgen 17394 df-plusf 18568 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-grp 18862 df-minusg 18863 df-sbg 18864 df-top 22740 df-topon 22757 df-topsp 22779 df-bases 22793 df-cld 22867 df-cn 23075 df-t1 23162 df-haus 23163 df-tx 23410 df-tmd 23920 df-tgp 23921 |
This theorem is referenced by: tgpt0 23967 |
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