![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2728 | . . 3 β’ (BaseβπΊ) = (BaseβπΊ) | |
2 | eqid 2728 | . . 3 β’ (+gβπΊ) = (+gβπΊ) | |
3 | eqid 2728 | . . 3 β’ (invgβπΊ) = (invgβπΊ) | |
4 | tgpsubcn.3 | . . 3 β’ β = (-gβπΊ) | |
5 | 1, 2, 3, 4 | grpsubfval 18947 | . 2 β’ β = (π₯ β (BaseβπΊ), π¦ β (BaseβπΊ) β¦ (π₯(+gβπΊ)((invgβπΊ)βπ¦))) |
6 | tgpsubcn.2 | . . 3 β’ π½ = (TopOpenβπΊ) | |
7 | tgptmd 24003 | . . 3 β’ (πΊ β TopGrp β πΊ β TopMnd) | |
8 | 6, 1 | tgptopon 24006 | . . 3 β’ (πΊ β TopGrp β π½ β (TopOnβ(BaseβπΊ))) |
9 | 8, 8 | cnmpt1st 23592 | . . 3 β’ (πΊ β TopGrp β (π₯ β (BaseβπΊ), π¦ β (BaseβπΊ) β¦ π₯) β ((π½ Γt π½) Cn π½)) |
10 | 8, 8 | cnmpt2nd 23593 | . . . 4 β’ (πΊ β TopGrp β (π₯ β (BaseβπΊ), π¦ β (BaseβπΊ) β¦ π¦) β ((π½ Γt π½) Cn π½)) |
11 | 6, 3 | tgpinv 24009 | . . . 4 β’ (πΊ β TopGrp β (invgβπΊ) β (π½ Cn π½)) |
12 | 8, 8, 10, 11 | cnmpt21f 23596 | . . 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 24012 | . 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 2833 | 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 6553 (class class class)co 7426 β cmpo 7428 Basecbs 17187 +gcplusg 17240 TopOpenctopn 17410 invgcminusg 18898 -gcsg 18899 Cn ccn 23148 Γt ctx 23484 TopGrpctgp 23995 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fo 6559 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-map 8853 df-topgen 17432 df-plusf 18606 df-sbg 18902 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-cn 23151 df-tx 23486 df-tmd 23996 df-tgp 23997 |
This theorem is referenced by: istgp2 24015 clssubg 24033 clsnsg 24034 tgphaus 24041 tgpt0 24043 qustgplem 24045 |
Copyright terms: Public domain | W3C validator |