![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > tgpinv | Structured version Visualization version GIF version |
Description: In a topological group, the inverse function is continuous. (Contributed by FL, 21-Jun-2010.) (Revised by FL, 27-Jun-2014.) |
Ref | Expression |
---|---|
tgpcn.j | ⊢ 𝐽 = (TopOpen‘𝐺) |
tgpinv.5 | ⊢ 𝐼 = (invg‘𝐺) |
Ref | Expression |
---|---|
tgpinv | ⊢ (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽 Cn 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgpcn.j | . . 3 ⊢ 𝐽 = (TopOpen‘𝐺) | |
2 | tgpinv.5 | . . 3 ⊢ 𝐼 = (invg‘𝐺) | |
3 | 1, 2 | istgp 23891 | . 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽))) |
4 | 3 | simp3bi 1144 | 1 ⊢ (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽 Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ‘cfv 6533 (class class class)co 7401 TopOpenctopn 17363 Grpcgrp 18850 invgcminusg 18851 Cn ccn 23038 TopMndctmd 23884 TopGrpctgp 23885 |
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-ext 2695 ax-nul 5296 |
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-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-rab 3425 df-v 3468 df-sbc 3770 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-iota 6485 df-fv 6541 df-ov 7404 df-tgp 23887 |
This theorem is referenced by: grpinvhmeo 23900 tgpsubcn 23904 tgpmulg 23907 oppgtgp 23912 subgtgp 23919 prdstgpd 23939 tsmsinv 23962 invrcn2 23994 |
Copyright terms: Public domain | W3C validator |