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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 2733 | . . 3 β’ (BaseβπΊ) = (BaseβπΊ) | |
2 | eqid 2733 | . . 3 β’ (+gβπΊ) = (+gβπΊ) | |
3 | eqid 2733 | . . 3 β’ (invgβπΊ) = (invgβπΊ) | |
4 | tgpsubcn.3 | . . 3 β’ β = (-gβπΊ) | |
5 | 1, 2, 3, 4 | grpsubfval 18802 | . 2 β’ β = (π₯ β (BaseβπΊ), π¦ β (BaseβπΊ) β¦ (π₯(+gβπΊ)((invgβπΊ)βπ¦))) |
6 | tgpsubcn.2 | . . 3 β’ π½ = (TopOpenβπΊ) | |
7 | tgptmd 23453 | . . 3 β’ (πΊ β TopGrp β πΊ β TopMnd) | |
8 | 6, 1 | tgptopon 23456 | . . 3 β’ (πΊ β TopGrp β π½ β (TopOnβ(BaseβπΊ))) |
9 | 8, 8 | cnmpt1st 23042 | . . 3 β’ (πΊ β TopGrp β (π₯ β (BaseβπΊ), π¦ β (BaseβπΊ) β¦ π₯) β ((π½ Γt π½) Cn π½)) |
10 | 8, 8 | cnmpt2nd 23043 | . . . 4 β’ (πΊ β TopGrp β (π₯ β (BaseβπΊ), π¦ β (BaseβπΊ) β¦ π¦) β ((π½ Γt π½) Cn π½)) |
11 | 6, 3 | tgpinv 23459 | . . . 4 β’ (πΊ β TopGrp β (invgβπΊ) β (π½ Cn π½)) |
12 | 8, 8, 10, 11 | cnmpt21f 23046 | . . 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 23462 | . 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 2838 | 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 1542 β wcel 2107 βcfv 6500 (class class class)co 7361 β cmpo 7363 Basecbs 17091 +gcplusg 17141 TopOpenctopn 17311 invgcminusg 18757 -gcsg 18758 Cn ccn 22598 Γt ctx 22934 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-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-ov 7364 df-oprab 7365 df-mpo 7366 df-1st 7925 df-2nd 7926 df-map 8773 df-topgen 17333 df-plusf 18504 df-sbg 18761 df-top 22266 df-topon 22283 df-topsp 22305 df-bases 22319 df-cn 22601 df-tx 22936 df-tmd 23446 df-tgp 23447 |
This theorem is referenced by: istgp2 23465 clssubg 23483 clsnsg 23484 tgphaus 23491 tgpt0 23493 qustgplem 23495 |
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