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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 23172 | . 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽))) |
4 | 3 | simp3bi 1145 | 1 ⊢ (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽 Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ‘cfv 6423 (class class class)co 7260 TopOpenctopn 17076 Grpcgrp 18521 invgcminusg 18522 Cn ccn 22319 TopMndctmd 23165 TopGrpctgp 23166 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 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 2708 ax-nul 5230 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3429 df-sbc 3717 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4259 df-if 4462 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-br 5076 df-iota 6381 df-fv 6431 df-ov 7263 df-tgp 23168 |
This theorem is referenced by: grpinvhmeo 23181 tgpsubcn 23185 tgpmulg 23188 oppgtgp 23193 subgtgp 23200 prdstgpd 23220 tsmsinv 23243 invrcn2 23275 |
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