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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 18991 | . . . . . . 7 ⊢ (𝐹 ∈ 𝐵 → (𝐹 ∈ 𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐴)) |
4 | 3 | ibi 266 | . . . . . 6 ⊢ (𝐹 ∈ 𝐵 → 𝐹:𝐴–1-1-onto→𝐴) |
5 | f1ocnv 6726 | . . . . . 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 7766 | . . . . . 6 ⊢ (𝐹 ∈ 𝐵 → ◡𝐹 ∈ V) | |
8 | 1, 2 | elsymgbas2 18991 | . . . . . 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 256 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → ◡𝐹 ∈ 𝐵) |
11 | eqid 2740 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
12 | 1, 2, 11 | symgov 19002 | . . . 4 ⊢ ((𝐹 ∈ 𝐵 ∧ ◡𝐹 ∈ 𝐵) → (𝐹(+g‘𝐺)◡𝐹) = (𝐹 ∘ ◡𝐹)) |
13 | 10, 12 | mpdan 684 | . . 3 ⊢ (𝐹 ∈ 𝐵 → (𝐹(+g‘𝐺)◡𝐹) = (𝐹 ∘ ◡𝐹)) |
14 | f1ococnv2 6740 | . . . 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 16929 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → 𝐴 ∈ V) |
17 | 1 | symgid 19020 | . . . 4 ⊢ (𝐴 ∈ V → ( I ↾ 𝐴) = (0g‘𝐺)) |
18 | 16, 17 | syl 17 | . . 3 ⊢ (𝐹 ∈ 𝐵 → ( I ↾ 𝐴) = (0g‘𝐺)) |
19 | 13, 15, 18 | 3eqtrd 2784 | . 2 ⊢ (𝐹 ∈ 𝐵 → (𝐹(+g‘𝐺)◡𝐹) = (0g‘𝐺)) |
20 | 1 | symggrp 19019 | . . . 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 2740 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
24 | symginv.3 | . . . 4 ⊢ 𝑁 = (invg‘𝐺) | |
25 | 2, 11, 23, 24 | grpinvid1 18641 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐹 ∈ 𝐵 ∧ ◡𝐹 ∈ 𝐵) → ((𝑁‘𝐹) = ◡𝐹 ↔ (𝐹(+g‘𝐺)◡𝐹) = (0g‘𝐺))) |
26 | 21, 22, 10, 25 | syl3anc 1370 | . 2 ⊢ (𝐹 ∈ 𝐵 → ((𝑁‘𝐹) = ◡𝐹 ↔ (𝐹(+g‘𝐺)◡𝐹) = (0g‘𝐺))) |
27 | 19, 26 | mpbird 256 | 1 ⊢ (𝐹 ∈ 𝐵 → (𝑁‘𝐹) = ◡𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2110 Vcvv 3431 I cid 5489 ◡ccnv 5589 ↾ cres 5592 ∘ ccom 5594 –1-1-onto→wf1o 6431 ‘cfv 6432 (class class class)co 7272 Basecbs 16923 +gcplusg 16973 0gc0g 17161 Grpcgrp 18588 invgcminusg 18589 SymGrpcsymg 18985 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-cnex 10938 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-om 7708 df-1st 7825 df-2nd 7826 df-frecs 8089 df-wrecs 8120 df-recs 8194 df-rdg 8233 df-1o 8289 df-er 8490 df-map 8609 df-en 8726 df-dom 8727 df-sdom 8728 df-fin 8729 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-nn 11985 df-2 12047 df-3 12048 df-4 12049 df-5 12050 df-6 12051 df-7 12052 df-8 12053 df-9 12054 df-n0 12245 df-z 12331 df-uz 12594 df-fz 13251 df-struct 16859 df-sets 16876 df-slot 16894 df-ndx 16906 df-base 16924 df-ress 16953 df-plusg 16986 df-tset 16992 df-0g 17163 df-mgm 18337 df-sgrp 18386 df-mnd 18397 df-submnd 18442 df-efmnd 18519 df-grp 18591 df-minusg 18592 df-symg 18986 |
This theorem is referenced by: symgsssg 19086 symgfisg 19087 symgtrinv 19091 psgninv 20798 zrhpsgninv 20801 evpmodpmf1o 20812 mdetleib2 21748 symgtgp 23268 symgfcoeu 31360 symgsubg 31365 cycpmconjv 31418 madjusmdetlem3 31788 madjusmdetlem4 31789 |
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