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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 17801 | . . . . . . 7 ⊢ (𝐹 ∈ 𝐵 → (𝐹 ∈ 𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐴)) |
4 | 3 | ibi 256 | . . . . . 6 ⊢ (𝐹 ∈ 𝐵 → 𝐹:𝐴–1-1-onto→𝐴) |
5 | f1ocnv 6149 | . . . . . 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 7112 | . . . . . 6 ⊢ (𝐹 ∈ 𝐵 → ◡𝐹 ∈ V) | |
8 | 1, 2 | elsymgbas2 17801 | . . . . . 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 247 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → ◡𝐹 ∈ 𝐵) |
11 | eqid 2622 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
12 | 1, 2, 11 | symgov 17810 | . . . 4 ⊢ ((𝐹 ∈ 𝐵 ∧ ◡𝐹 ∈ 𝐵) → (𝐹(+g‘𝐺)◡𝐹) = (𝐹 ∘ ◡𝐹)) |
13 | 10, 12 | mpdan 702 | . . 3 ⊢ (𝐹 ∈ 𝐵 → (𝐹(+g‘𝐺)◡𝐹) = (𝐹 ∘ ◡𝐹)) |
14 | f1ococnv2 6163 | . . . 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 15920 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → 𝐴 ∈ V) |
17 | 1 | symgid 17821 | . . . 4 ⊢ (𝐴 ∈ V → ( I ↾ 𝐴) = (0g‘𝐺)) |
18 | 16, 17 | syl 17 | . . 3 ⊢ (𝐹 ∈ 𝐵 → ( I ↾ 𝐴) = (0g‘𝐺)) |
19 | 13, 15, 18 | 3eqtrd 2660 | . 2 ⊢ (𝐹 ∈ 𝐵 → (𝐹(+g‘𝐺)◡𝐹) = (0g‘𝐺)) |
20 | 1 | symggrp 17820 | . . . 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 2622 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
24 | symginv.3 | . . . 4 ⊢ 𝑁 = (invg‘𝐺) | |
25 | 2, 11, 23, 24 | grpinvid1 17470 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐹 ∈ 𝐵 ∧ ◡𝐹 ∈ 𝐵) → ((𝑁‘𝐹) = ◡𝐹 ↔ (𝐹(+g‘𝐺)◡𝐹) = (0g‘𝐺))) |
26 | 21, 22, 10, 25 | syl3anc 1326 | . 2 ⊢ (𝐹 ∈ 𝐵 → ((𝑁‘𝐹) = ◡𝐹 ↔ (𝐹(+g‘𝐺)◡𝐹) = (0g‘𝐺))) |
27 | 19, 26 | mpbird 247 | 1 ⊢ (𝐹 ∈ 𝐵 → (𝑁‘𝐹) = ◡𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 ∈ wcel 1990 Vcvv 3200 I cid 5023 ◡ccnv 5113 ↾ cres 5116 ∘ ccom 5118 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 0gc0g 16100 Grpcgrp 17422 invgcminusg 17423 SymGrpcsymg 17797 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-tset 15960 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-symg 17798 |
This theorem is referenced by: symgsssg 17887 symgfisg 17888 symgtrinv 17892 psgninv 19928 zrhpsgninv 19931 evpmodpmf1o 19942 mdetleib2 20394 symgtgp 21905 symgfcoeu 29845 madjusmdetlem3 29895 madjusmdetlem4 29896 |
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