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Mirrors > Home > MPE Home > Th. List > symgtrinv | Structured version Visualization version GIF version |
Description: To invert a permutation represented as a sequence of transpositions, reverse the sequence. (Contributed by Stefan O'Rear, 27-Aug-2015.) |
Ref | Expression |
---|---|
symgtrinv.t | ⊢ 𝑇 = ran (pmTrsp‘𝐷) |
symgtrinv.g | ⊢ 𝐺 = (SymGrp‘𝐷) |
symgtrinv.i | ⊢ 𝐼 = (invg‘𝐺) |
Ref | Expression |
---|---|
symgtrinv | ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → (𝐼‘(𝐺 Σg 𝑊)) = (𝐺 Σg (reverse‘𝑊))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | symgtrinv.g | . . . . 5 ⊢ 𝐺 = (SymGrp‘𝐷) | |
2 | 1 | symggrp 18764 | . . . 4 ⊢ (𝐷 ∈ 𝑉 → 𝐺 ∈ Grp) |
3 | eqid 2734 | . . . . 5 ⊢ (oppg‘𝐺) = (oppg‘𝐺) | |
4 | symgtrinv.i | . . . . 5 ⊢ 𝐼 = (invg‘𝐺) | |
5 | 3, 4 | invoppggim 18724 | . . . 4 ⊢ (𝐺 ∈ Grp → 𝐼 ∈ (𝐺 GrpIso (oppg‘𝐺))) |
6 | gimghm 18640 | . . . 4 ⊢ (𝐼 ∈ (𝐺 GrpIso (oppg‘𝐺)) → 𝐼 ∈ (𝐺 GrpHom (oppg‘𝐺))) | |
7 | ghmmhm 18604 | . . . 4 ⊢ (𝐼 ∈ (𝐺 GrpHom (oppg‘𝐺)) → 𝐼 ∈ (𝐺 MndHom (oppg‘𝐺))) | |
8 | 2, 5, 6, 7 | 4syl 19 | . . 3 ⊢ (𝐷 ∈ 𝑉 → 𝐼 ∈ (𝐺 MndHom (oppg‘𝐺))) |
9 | symgtrinv.t | . . . . . 6 ⊢ 𝑇 = ran (pmTrsp‘𝐷) | |
10 | eqid 2734 | . . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
11 | 9, 1, 10 | symgtrf 18833 | . . . . 5 ⊢ 𝑇 ⊆ (Base‘𝐺) |
12 | sswrd 14060 | . . . . 5 ⊢ (𝑇 ⊆ (Base‘𝐺) → Word 𝑇 ⊆ Word (Base‘𝐺)) | |
13 | 11, 12 | ax-mp 5 | . . . 4 ⊢ Word 𝑇 ⊆ Word (Base‘𝐺) |
14 | 13 | sseli 3887 | . . 3 ⊢ (𝑊 ∈ Word 𝑇 → 𝑊 ∈ Word (Base‘𝐺)) |
15 | 10 | gsumwmhm 18244 | . . 3 ⊢ ((𝐼 ∈ (𝐺 MndHom (oppg‘𝐺)) ∧ 𝑊 ∈ Word (Base‘𝐺)) → (𝐼‘(𝐺 Σg 𝑊)) = ((oppg‘𝐺) Σg (𝐼 ∘ 𝑊))) |
16 | 8, 14, 15 | syl2an 599 | . 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → (𝐼‘(𝐺 Σg 𝑊)) = ((oppg‘𝐺) Σg (𝐼 ∘ 𝑊))) |
17 | 10, 4 | grpinvf 18386 | . . . . . . 7 ⊢ (𝐺 ∈ Grp → 𝐼:(Base‘𝐺)⟶(Base‘𝐺)) |
18 | 2, 17 | syl 17 | . . . . . 6 ⊢ (𝐷 ∈ 𝑉 → 𝐼:(Base‘𝐺)⟶(Base‘𝐺)) |
19 | wrdf 14057 | . . . . . . . 8 ⊢ (𝑊 ∈ Word 𝑇 → 𝑊:(0..^(♯‘𝑊))⟶𝑇) | |
20 | 19 | adantl 485 | . . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → 𝑊:(0..^(♯‘𝑊))⟶𝑇) |
21 | fss 6551 | . . . . . . 7 ⊢ ((𝑊:(0..^(♯‘𝑊))⟶𝑇 ∧ 𝑇 ⊆ (Base‘𝐺)) → 𝑊:(0..^(♯‘𝑊))⟶(Base‘𝐺)) | |
22 | 20, 11, 21 | sylancl 589 | . . . . . 6 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → 𝑊:(0..^(♯‘𝑊))⟶(Base‘𝐺)) |
23 | fco 6558 | . . . . . 6 ⊢ ((𝐼:(Base‘𝐺)⟶(Base‘𝐺) ∧ 𝑊:(0..^(♯‘𝑊))⟶(Base‘𝐺)) → (𝐼 ∘ 𝑊):(0..^(♯‘𝑊))⟶(Base‘𝐺)) | |
24 | 18, 22, 23 | syl2an2r 685 | . . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → (𝐼 ∘ 𝑊):(0..^(♯‘𝑊))⟶(Base‘𝐺)) |
25 | 24 | ffnd 6535 | . . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → (𝐼 ∘ 𝑊) Fn (0..^(♯‘𝑊))) |
26 | 20 | ffnd 6535 | . . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → 𝑊 Fn (0..^(♯‘𝑊))) |
27 | fvco2 6797 | . . . . . 6 ⊢ ((𝑊 Fn (0..^(♯‘𝑊)) ∧ 𝑥 ∈ (0..^(♯‘𝑊))) → ((𝐼 ∘ 𝑊)‘𝑥) = (𝐼‘(𝑊‘𝑥))) | |
28 | 26, 27 | sylan 583 | . . . . 5 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) ∧ 𝑥 ∈ (0..^(♯‘𝑊))) → ((𝐼 ∘ 𝑊)‘𝑥) = (𝐼‘(𝑊‘𝑥))) |
29 | 20 | ffvelrnda 6893 | . . . . . . 7 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) ∧ 𝑥 ∈ (0..^(♯‘𝑊))) → (𝑊‘𝑥) ∈ 𝑇) |
30 | 11, 29 | sseldi 3889 | . . . . . 6 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) ∧ 𝑥 ∈ (0..^(♯‘𝑊))) → (𝑊‘𝑥) ∈ (Base‘𝐺)) |
31 | 1, 10, 4 | symginv 18766 | . . . . . 6 ⊢ ((𝑊‘𝑥) ∈ (Base‘𝐺) → (𝐼‘(𝑊‘𝑥)) = ◡(𝑊‘𝑥)) |
32 | 30, 31 | syl 17 | . . . . 5 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) ∧ 𝑥 ∈ (0..^(♯‘𝑊))) → (𝐼‘(𝑊‘𝑥)) = ◡(𝑊‘𝑥)) |
33 | eqid 2734 | . . . . . . 7 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘𝐷) | |
34 | 33, 9 | pmtrfcnv 18828 | . . . . . 6 ⊢ ((𝑊‘𝑥) ∈ 𝑇 → ◡(𝑊‘𝑥) = (𝑊‘𝑥)) |
35 | 29, 34 | syl 17 | . . . . 5 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) ∧ 𝑥 ∈ (0..^(♯‘𝑊))) → ◡(𝑊‘𝑥) = (𝑊‘𝑥)) |
36 | 28, 32, 35 | 3eqtrd 2778 | . . . 4 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) ∧ 𝑥 ∈ (0..^(♯‘𝑊))) → ((𝐼 ∘ 𝑊)‘𝑥) = (𝑊‘𝑥)) |
37 | 25, 26, 36 | eqfnfvd 6844 | . . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → (𝐼 ∘ 𝑊) = 𝑊) |
38 | 37 | oveq2d 7218 | . 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → ((oppg‘𝐺) Σg (𝐼 ∘ 𝑊)) = ((oppg‘𝐺) Σg 𝑊)) |
39 | 2 | grpmndd 18349 | . . 3 ⊢ (𝐷 ∈ 𝑉 → 𝐺 ∈ Mnd) |
40 | 10, 3 | gsumwrev 18730 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑊 ∈ Word (Base‘𝐺)) → ((oppg‘𝐺) Σg 𝑊) = (𝐺 Σg (reverse‘𝑊))) |
41 | 39, 14, 40 | syl2an 599 | . 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → ((oppg‘𝐺) Σg 𝑊) = (𝐺 Σg (reverse‘𝑊))) |
42 | 16, 38, 41 | 3eqtrd 2778 | 1 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → (𝐼‘(𝐺 Σg 𝑊)) = (𝐺 Σg (reverse‘𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ⊆ wss 3857 ◡ccnv 5539 ran crn 5541 ∘ ccom 5544 Fn wfn 6364 ⟶wf 6365 ‘cfv 6369 (class class class)co 7202 0cc0 10712 ..^cfzo 13221 ♯chash 13879 Word cword 14052 reversecreverse 14306 Basecbs 16684 Σg cgsu 16917 Mndcmnd 18145 MndHom cmhm 18188 Grpcgrp 18337 invgcminusg 18338 GrpHom cghm 18591 GrpIso cgim 18633 oppgcoppg 18709 SymGrpcsymg 18731 pmTrspcpmtr 18805 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-tpos 7957 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-2o 8192 df-er 8380 df-map 8499 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-xnn0 12146 df-z 12160 df-uz 12422 df-fz 13079 df-fzo 13222 df-seq 13558 df-hash 13880 df-word 14053 df-lsw 14101 df-concat 14109 df-s1 14136 df-substr 14189 df-pfx 14219 df-reverse 14307 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-tset 16786 df-0g 16918 df-gsum 16919 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-mhm 18190 df-submnd 18191 df-efmnd 18268 df-grp 18340 df-minusg 18341 df-ghm 18592 df-gim 18635 df-oppg 18710 df-symg 18732 df-pmtr 18806 |
This theorem is referenced by: psgnuni 18863 |
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