Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 18501 | . . . . . . 7 ⊢ (𝐹 ∈ 𝐵 → (𝐹 ∈ 𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐴)) |
4 | 3 | ibi 269 | . . . . . 6 ⊢ (𝐹 ∈ 𝐵 → 𝐹:𝐴–1-1-onto→𝐴) |
5 | f1ocnv 6627 | . . . . . 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 7629 | . . . . . 6 ⊢ (𝐹 ∈ 𝐵 → ◡𝐹 ∈ V) | |
8 | 1, 2 | elsymgbas2 18501 | . . . . . 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 259 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → ◡𝐹 ∈ 𝐵) |
11 | eqid 2821 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
12 | 1, 2, 11 | symgov 18512 | . . . 4 ⊢ ((𝐹 ∈ 𝐵 ∧ ◡𝐹 ∈ 𝐵) → (𝐹(+g‘𝐺)◡𝐹) = (𝐹 ∘ ◡𝐹)) |
13 | 10, 12 | mpdan 685 | . . 3 ⊢ (𝐹 ∈ 𝐵 → (𝐹(+g‘𝐺)◡𝐹) = (𝐹 ∘ ◡𝐹)) |
14 | f1ococnv2 6641 | . . . 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 16544 | . . . 4 ⊢ (𝐹 ∈ 𝐵 → 𝐴 ∈ V) |
17 | 1 | symgid 18529 | . . . 4 ⊢ (𝐴 ∈ V → ( I ↾ 𝐴) = (0g‘𝐺)) |
18 | 16, 17 | syl 17 | . . 3 ⊢ (𝐹 ∈ 𝐵 → ( I ↾ 𝐴) = (0g‘𝐺)) |
19 | 13, 15, 18 | 3eqtrd 2860 | . 2 ⊢ (𝐹 ∈ 𝐵 → (𝐹(+g‘𝐺)◡𝐹) = (0g‘𝐺)) |
20 | 1 | symggrp 18528 | . . . 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 2821 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
24 | symginv.3 | . . . 4 ⊢ 𝑁 = (invg‘𝐺) | |
25 | 2, 11, 23, 24 | grpinvid1 18154 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐹 ∈ 𝐵 ∧ ◡𝐹 ∈ 𝐵) → ((𝑁‘𝐹) = ◡𝐹 ↔ (𝐹(+g‘𝐺)◡𝐹) = (0g‘𝐺))) |
26 | 21, 22, 10, 25 | syl3anc 1367 | . 2 ⊢ (𝐹 ∈ 𝐵 → ((𝑁‘𝐹) = ◡𝐹 ↔ (𝐹(+g‘𝐺)◡𝐹) = (0g‘𝐺))) |
27 | 19, 26 | mpbird 259 | 1 ⊢ (𝐹 ∈ 𝐵 → (𝑁‘𝐹) = ◡𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 Vcvv 3494 I cid 5459 ◡ccnv 5554 ↾ cres 5557 ∘ ccom 5559 –1-1-onto→wf1o 6354 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 0gc0g 16713 Grpcgrp 18103 invgcminusg 18104 SymGrpcsymg 18495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-tset 16584 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-efmnd 18034 df-grp 18106 df-minusg 18107 df-symg 18496 |
This theorem is referenced by: symgsssg 18595 symgfisg 18596 symgtrinv 18600 psgninv 20726 zrhpsgninv 20729 evpmodpmf1o 20740 mdetleib2 21197 symgtgp 22714 symgfcoeu 30726 symgsubg 30731 cycpmconjv 30784 madjusmdetlem3 31094 madjusmdetlem4 31095 |
Copyright terms: Public domain | W3C validator |