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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 22258 | . 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽))) |
4 | 3 | simp3bi 1181 | 1 ⊢ (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽 Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 ‘cfv 6127 (class class class)co 6910 TopOpenctopn 16442 Grpcgrp 17783 invgcminusg 17784 Cn ccn 21406 TopMndctmd 22251 TopGrpctgp 22252 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-nul 5015 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-iota 6090 df-fv 6135 df-ov 6913 df-tgp 22254 |
This theorem is referenced by: grpinvhmeo 22267 tgpsubcn 22271 tgpmulg 22274 oppgtgp 22279 subgtgp 22286 prdstgpd 22305 tsmsinv 22328 invrcn2 22360 |
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