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Mirrors > Home > MPE Home > Th. List > oppgtgp | Structured version Visualization version GIF version |
Description: The opposite of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.) |
Ref | Expression |
---|---|
oppgtmd.1 | ⊢ 𝑂 = (oppg‘𝐺) |
Ref | Expression |
---|---|
oppgtgp | ⊢ (𝐺 ∈ TopGrp → 𝑂 ∈ TopGrp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgpgrp 22616 | . . 3 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp) | |
2 | oppgtmd.1 | . . . 4 ⊢ 𝑂 = (oppg‘𝐺) | |
3 | 2 | oppggrp 18425 | . . 3 ⊢ (𝐺 ∈ Grp → 𝑂 ∈ Grp) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝐺 ∈ TopGrp → 𝑂 ∈ Grp) |
5 | tgptmd 22617 | . . 3 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd) | |
6 | 2 | oppgtmd 22635 | . . 3 ⊢ (𝐺 ∈ TopMnd → 𝑂 ∈ TopMnd) |
7 | 5, 6 | syl 17 | . 2 ⊢ (𝐺 ∈ TopGrp → 𝑂 ∈ TopMnd) |
8 | eqid 2821 | . . . . 5 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
9 | 2, 8 | oppginv 18427 | . . . 4 ⊢ (𝐺 ∈ Grp → (invg‘𝐺) = (invg‘𝑂)) |
10 | 1, 9 | syl 17 | . . 3 ⊢ (𝐺 ∈ TopGrp → (invg‘𝐺) = (invg‘𝑂)) |
11 | eqid 2821 | . . . 4 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺) | |
12 | 11, 8 | tgpinv 22623 | . . 3 ⊢ (𝐺 ∈ TopGrp → (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))) |
13 | 10, 12 | eqeltrrd 2914 | . 2 ⊢ (𝐺 ∈ TopGrp → (invg‘𝑂) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))) |
14 | 2, 11 | oppgtopn 18421 | . . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝑂) |
15 | eqid 2821 | . . 3 ⊢ (invg‘𝑂) = (invg‘𝑂) | |
16 | 14, 15 | istgp 22615 | . 2 ⊢ (𝑂 ∈ TopGrp ↔ (𝑂 ∈ Grp ∧ 𝑂 ∈ TopMnd ∧ (invg‘𝑂) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))) |
17 | 4, 7, 13, 16 | syl3anbrc 1335 | 1 ⊢ (𝐺 ∈ TopGrp → 𝑂 ∈ TopGrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ‘cfv 6349 (class class class)co 7145 TopOpenctopn 16685 Grpcgrp 18043 invgcminusg 18044 oppgcoppg 18413 Cn ccn 21762 TopMndctmd 22608 TopGrpctgp 22609 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-tpos 7883 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-er 8279 df-map 8398 df-en 8499 df-dom 8500 df-sdom 8501 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11628 df-2 11689 df-3 11690 df-4 11691 df-5 11692 df-6 11693 df-7 11694 df-8 11695 df-9 11696 df-ndx 16476 df-slot 16477 df-base 16479 df-sets 16480 df-plusg 16568 df-tset 16574 df-rest 16686 df-topn 16687 df-0g 16705 df-topgen 16707 df-plusf 17841 df-mgm 17842 df-sgrp 17891 df-mnd 17902 df-grp 18046 df-minusg 18047 df-oppg 18414 df-top 21432 df-topon 21449 df-topsp 21471 df-bases 21484 df-cn 21765 df-tx 22100 df-tmd 22610 df-tgp 22611 |
This theorem is referenced by: tgpconncomp 22650 qustgpopn 22657 |
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