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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 19284 | . . . . . . 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 6836 | . . . . . 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 7909 | . . . . . 6 ⊢ (𝐹 ∈ 𝐵 → ◡𝐹 ∈ V) | |
8 | 1, 2 | elsymgbas2 19284 | . . . . . 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 2724 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
12 | 1, 2, 11 | symgov 19295 | . . . 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 6851 | . . . 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 17151 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → 𝐴 ∈ V) |
17 | 1 | symgid 19313 | . . . 4 ⊢ (𝐴 ∈ V → ( I ↾ 𝐴) = (0g‘𝐺)) |
18 | 16, 17 | syl 17 | . . 3 ⊢ (𝐹 ∈ 𝐵 → ( I ↾ 𝐴) = (0g‘𝐺)) |
19 | 13, 15, 18 | 3eqtrd 2768 | . 2 ⊢ (𝐹 ∈ 𝐵 → (𝐹(+g‘𝐺)◡𝐹) = (0g‘𝐺)) |
20 | 1 | symggrp 19312 | . . . 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 2724 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
24 | symginv.3 | . . . 4 ⊢ 𝑁 = (invg‘𝐺) | |
25 | 2, 11, 23, 24 | grpinvid1 18913 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐹 ∈ 𝐵 ∧ ◡𝐹 ∈ 𝐵) → ((𝑁‘𝐹) = ◡𝐹 ↔ (𝐹(+g‘𝐺)◡𝐹) = (0g‘𝐺))) |
26 | 21, 22, 10, 25 | syl3anc 1368 | . 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 1533 ∈ wcel 2098 Vcvv 3466 I cid 5564 ◡ccnv 5666 ↾ cres 5669 ∘ ccom 5671 –1-1-onto→wf1o 6533 ‘cfv 6534 (class class class)co 7402 Basecbs 17145 +gcplusg 17198 0gc0g 17386 Grpcgrp 18855 invgcminusg 18856 SymGrpcsymg 19278 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12471 df-z 12557 df-uz 12821 df-fz 13483 df-struct 17081 df-sets 17098 df-slot 17116 df-ndx 17128 df-base 17146 df-ress 17175 df-plusg 17211 df-tset 17217 df-0g 17388 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-efmnd 18786 df-grp 18858 df-minusg 18859 df-symg 19279 |
This theorem is referenced by: symgsssg 19379 symgfisg 19380 symgtrinv 19384 psgninv 21445 zrhpsgninv 21448 evpmodpmf1o 21459 mdetleib2 22414 symgtgp 23934 symgfcoeu 32714 symgsubg 32719 cycpmconjv 32772 madjusmdetlem3 33301 madjusmdetlem4 33302 |
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