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Mirrors > Home > MPE Home > Th. List > tgpsubcn | Structured version Visualization version GIF version |
Description: In a topological group, the "subtraction" (or "division") is continuous. Axiom GT' of [BourbakiTop1] p. III.1. (Contributed by FL, 21-Jun-2010.) (Revised by Mario Carneiro, 19-Mar-2015.) |
Ref | Expression |
---|---|
tgpsubcn.2 | β’ π½ = (TopOpenβπΊ) |
tgpsubcn.3 | β’ β = (-gβπΊ) |
Ref | Expression |
---|---|
tgpsubcn | β’ (πΊ β TopGrp β β β ((π½ Γt π½) Cn π½)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . 3 β’ (BaseβπΊ) = (BaseβπΊ) | |
2 | eqid 2726 | . . 3 β’ (+gβπΊ) = (+gβπΊ) | |
3 | eqid 2726 | . . 3 β’ (invgβπΊ) = (invgβπΊ) | |
4 | tgpsubcn.3 | . . 3 β’ β = (-gβπΊ) | |
5 | 1, 2, 3, 4 | grpsubfval 18910 | . 2 β’ β = (π₯ β (BaseβπΊ), π¦ β (BaseβπΊ) β¦ (π₯(+gβπΊ)((invgβπΊ)βπ¦))) |
6 | tgpsubcn.2 | . . 3 β’ π½ = (TopOpenβπΊ) | |
7 | tgptmd 23933 | . . 3 β’ (πΊ β TopGrp β πΊ β TopMnd) | |
8 | 6, 1 | tgptopon 23936 | . . 3 β’ (πΊ β TopGrp β π½ β (TopOnβ(BaseβπΊ))) |
9 | 8, 8 | cnmpt1st 23522 | . . 3 β’ (πΊ β TopGrp β (π₯ β (BaseβπΊ), π¦ β (BaseβπΊ) β¦ π₯) β ((π½ Γt π½) Cn π½)) |
10 | 8, 8 | cnmpt2nd 23523 | . . . 4 β’ (πΊ β TopGrp β (π₯ β (BaseβπΊ), π¦ β (BaseβπΊ) β¦ π¦) β ((π½ Γt π½) Cn π½)) |
11 | 6, 3 | tgpinv 23939 | . . . 4 β’ (πΊ β TopGrp β (invgβπΊ) β (π½ Cn π½)) |
12 | 8, 8, 10, 11 | cnmpt21f 23526 | . . 3 β’ (πΊ β TopGrp β (π₯ β (BaseβπΊ), π¦ β (BaseβπΊ) β¦ ((invgβπΊ)βπ¦)) β ((π½ Γt π½) Cn π½)) |
13 | 6, 2, 7, 8, 8, 9, 12 | cnmpt2plusg 23942 | . 2 β’ (πΊ β TopGrp β (π₯ β (BaseβπΊ), π¦ β (BaseβπΊ) β¦ (π₯(+gβπΊ)((invgβπΊ)βπ¦))) β ((π½ Γt π½) Cn π½)) |
14 | 5, 13 | eqeltrid 2831 | 1 β’ (πΊ β TopGrp β β β ((π½ Γt π½) Cn π½)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βcfv 6536 (class class class)co 7404 β cmpo 7406 Basecbs 17150 +gcplusg 17203 TopOpenctopn 17373 invgcminusg 18861 -gcsg 18862 Cn ccn 23078 Γt ctx 23414 TopGrpctgp 23925 |
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 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fo 6542 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-map 8821 df-topgen 17395 df-plusf 18569 df-sbg 18865 df-top 22746 df-topon 22763 df-topsp 22785 df-bases 22799 df-cn 23081 df-tx 23416 df-tmd 23926 df-tgp 23927 |
This theorem is referenced by: istgp2 23945 clssubg 23963 clsnsg 23964 tgphaus 23971 tgpt0 23973 qustgplem 23975 |
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