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Mirrors > Home > MPE Home > Th. List > symginv | Structured version Visualization version GIF version |
Description: The group inverse in the symmetric group corresponds to the functional inverse. (Contributed by Stefan O'Rear, 24-Aug-2015.) (Revised by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
symggrp.1 | ⊢ 𝐺 = (SymGrp‘𝐴) |
symginv.2 | ⊢ 𝐵 = (Base‘𝐺) |
symginv.3 | ⊢ 𝑁 = (invg‘𝐺) |
Ref | Expression |
---|---|
symginv | ⊢ (𝐹 ∈ 𝐵 → (𝑁‘𝐹) = ◡𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | symggrp.1 | . . . . . . . 8 ⊢ 𝐺 = (SymGrp‘𝐴) | |
2 | symginv.2 | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝐺) | |
3 | 1, 2 | elsymgbas2 19321 | . . . . . . 7 ⊢ (𝐹 ∈ 𝐵 → (𝐹 ∈ 𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐴)) |
4 | 3 | ibi 267 | . . . . . 6 ⊢ (𝐹 ∈ 𝐵 → 𝐹:𝐴–1-1-onto→𝐴) |
5 | f1ocnv 6846 | . . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → ◡𝐹:𝐴–1-1-onto→𝐴) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝐹 ∈ 𝐵 → ◡𝐹:𝐴–1-1-onto→𝐴) |
7 | cnvexg 7927 | . . . . . 6 ⊢ (𝐹 ∈ 𝐵 → ◡𝐹 ∈ V) | |
8 | 1, 2 | elsymgbas2 19321 | . . . . . 6 ⊢ (◡𝐹 ∈ V → (◡𝐹 ∈ 𝐵 ↔ ◡𝐹:𝐴–1-1-onto→𝐴)) |
9 | 7, 8 | syl 17 | . . . . 5 ⊢ (𝐹 ∈ 𝐵 → (◡𝐹 ∈ 𝐵 ↔ ◡𝐹:𝐴–1-1-onto→𝐴)) |
10 | 6, 9 | mpbird 257 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → ◡𝐹 ∈ 𝐵) |
11 | eqid 2728 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
12 | 1, 2, 11 | symgov 19332 | . . . 4 ⊢ ((𝐹 ∈ 𝐵 ∧ ◡𝐹 ∈ 𝐵) → (𝐹(+g‘𝐺)◡𝐹) = (𝐹 ∘ ◡𝐹)) |
13 | 10, 12 | mpdan 686 | . . 3 ⊢ (𝐹 ∈ 𝐵 → (𝐹(+g‘𝐺)◡𝐹) = (𝐹 ∘ ◡𝐹)) |
14 | f1ococnv2 6861 | . . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐴 → (𝐹 ∘ ◡𝐹) = ( I ↾ 𝐴)) | |
15 | 4, 14 | syl 17 | . . 3 ⊢ (𝐹 ∈ 𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ 𝐴)) |
16 | 1, 2 | elbasfv 17180 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → 𝐴 ∈ V) |
17 | 1 | symgid 19350 | . . . 4 ⊢ (𝐴 ∈ V → ( I ↾ 𝐴) = (0g‘𝐺)) |
18 | 16, 17 | syl 17 | . . 3 ⊢ (𝐹 ∈ 𝐵 → ( I ↾ 𝐴) = (0g‘𝐺)) |
19 | 13, 15, 18 | 3eqtrd 2772 | . 2 ⊢ (𝐹 ∈ 𝐵 → (𝐹(+g‘𝐺)◡𝐹) = (0g‘𝐺)) |
20 | 1 | symggrp 19349 | . . . 4 ⊢ (𝐴 ∈ V → 𝐺 ∈ Grp) |
21 | 16, 20 | syl 17 | . . 3 ⊢ (𝐹 ∈ 𝐵 → 𝐺 ∈ Grp) |
22 | id 22 | . . 3 ⊢ (𝐹 ∈ 𝐵 → 𝐹 ∈ 𝐵) | |
23 | eqid 2728 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
24 | symginv.3 | . . . 4 ⊢ 𝑁 = (invg‘𝐺) | |
25 | 2, 11, 23, 24 | grpinvid1 18942 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐹 ∈ 𝐵 ∧ ◡𝐹 ∈ 𝐵) → ((𝑁‘𝐹) = ◡𝐹 ↔ (𝐹(+g‘𝐺)◡𝐹) = (0g‘𝐺))) |
26 | 21, 22, 10, 25 | syl3anc 1369 | . 2 ⊢ (𝐹 ∈ 𝐵 → ((𝑁‘𝐹) = ◡𝐹 ↔ (𝐹(+g‘𝐺)◡𝐹) = (0g‘𝐺))) |
27 | 19, 26 | mpbird 257 | 1 ⊢ (𝐹 ∈ 𝐵 → (𝑁‘𝐹) = ◡𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 Vcvv 3470 I cid 5570 ◡ccnv 5672 ↾ cres 5675 ∘ ccom 5677 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7415 Basecbs 17174 +gcplusg 17227 0gc0g 17415 Grpcgrp 18884 invgcminusg 18885 SymGrpcsymg 19315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-1st 7988 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-er 8719 df-map 8841 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-uz 12848 df-fz 13512 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-tset 17246 df-0g 17417 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-submnd 18735 df-efmnd 18815 df-grp 18887 df-minusg 18888 df-symg 19316 |
This theorem is referenced by: symgsssg 19416 symgfisg 19417 symgtrinv 19421 psgninv 21508 zrhpsgninv 21511 evpmodpmf1o 21522 mdetleib2 22484 symgtgp 24004 symgfcoeu 32800 symgsubg 32805 cycpmconjv 32858 madjusmdetlem3 33425 madjusmdetlem4 33426 |
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