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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 19392 | . . . 4 ⊢ (𝐷 ∈ 𝑉 → 𝐺 ∈ Grp) |
3 | eqid 2726 | . . . . 5 ⊢ (oppg‘𝐺) = (oppg‘𝐺) | |
4 | symgtrinv.i | . . . . 5 ⊢ 𝐼 = (invg‘𝐺) | |
5 | 3, 4 | invoppggim 19351 | . . . 4 ⊢ (𝐺 ∈ Grp → 𝐼 ∈ (𝐺 GrpIso (oppg‘𝐺))) |
6 | gimghm 19252 | . . . 4 ⊢ (𝐼 ∈ (𝐺 GrpIso (oppg‘𝐺)) → 𝐼 ∈ (𝐺 GrpHom (oppg‘𝐺))) | |
7 | ghmmhm 19214 | . . . 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 2726 | . . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
11 | 9, 1, 10 | symgtrf 19461 | . . . . 5 ⊢ 𝑇 ⊆ (Base‘𝐺) |
12 | sswrd 14523 | . . . . 5 ⊢ (𝑇 ⊆ (Base‘𝐺) → Word 𝑇 ⊆ Word (Base‘𝐺)) | |
13 | 11, 12 | ax-mp 5 | . . . 4 ⊢ Word 𝑇 ⊆ Word (Base‘𝐺) |
14 | 13 | sseli 3975 | . . 3 ⊢ (𝑊 ∈ Word 𝑇 → 𝑊 ∈ Word (Base‘𝐺)) |
15 | 10 | gsumwmhm 18828 | . . 3 ⊢ ((𝐼 ∈ (𝐺 MndHom (oppg‘𝐺)) ∧ 𝑊 ∈ Word (Base‘𝐺)) → (𝐼‘(𝐺 Σg 𝑊)) = ((oppg‘𝐺) Σg (𝐼 ∘ 𝑊))) |
16 | 8, 14, 15 | syl2an 594 | . 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → (𝐼‘(𝐺 Σg 𝑊)) = ((oppg‘𝐺) Σg (𝐼 ∘ 𝑊))) |
17 | 10, 4 | grpinvf 18974 | . . . . . . 7 ⊢ (𝐺 ∈ Grp → 𝐼:(Base‘𝐺)⟶(Base‘𝐺)) |
18 | 2, 17 | syl 17 | . . . . . 6 ⊢ (𝐷 ∈ 𝑉 → 𝐼:(Base‘𝐺)⟶(Base‘𝐺)) |
19 | wrdf 14520 | . . . . . . . 8 ⊢ (𝑊 ∈ Word 𝑇 → 𝑊:(0..^(♯‘𝑊))⟶𝑇) | |
20 | 19 | adantl 480 | . . . . . . 7 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → 𝑊:(0..^(♯‘𝑊))⟶𝑇) |
21 | fss 6734 | . . . . . . 7 ⊢ ((𝑊:(0..^(♯‘𝑊))⟶𝑇 ∧ 𝑇 ⊆ (Base‘𝐺)) → 𝑊:(0..^(♯‘𝑊))⟶(Base‘𝐺)) | |
22 | 20, 11, 21 | sylancl 584 | . . . . . 6 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → 𝑊:(0..^(♯‘𝑊))⟶(Base‘𝐺)) |
23 | fco 6742 | . . . . . 6 ⊢ ((𝐼:(Base‘𝐺)⟶(Base‘𝐺) ∧ 𝑊:(0..^(♯‘𝑊))⟶(Base‘𝐺)) → (𝐼 ∘ 𝑊):(0..^(♯‘𝑊))⟶(Base‘𝐺)) | |
24 | 18, 22, 23 | syl2an2r 683 | . . . . 5 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → (𝐼 ∘ 𝑊):(0..^(♯‘𝑊))⟶(Base‘𝐺)) |
25 | 24 | ffnd 6719 | . . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → (𝐼 ∘ 𝑊) Fn (0..^(♯‘𝑊))) |
26 | 20 | ffnd 6719 | . . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → 𝑊 Fn (0..^(♯‘𝑊))) |
27 | fvco2 6989 | . . . . . 6 ⊢ ((𝑊 Fn (0..^(♯‘𝑊)) ∧ 𝑥 ∈ (0..^(♯‘𝑊))) → ((𝐼 ∘ 𝑊)‘𝑥) = (𝐼‘(𝑊‘𝑥))) | |
28 | 26, 27 | sylan 578 | . . . . 5 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) ∧ 𝑥 ∈ (0..^(♯‘𝑊))) → ((𝐼 ∘ 𝑊)‘𝑥) = (𝐼‘(𝑊‘𝑥))) |
29 | 20 | ffvelcdmda 7088 | . . . . . . 7 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) ∧ 𝑥 ∈ (0..^(♯‘𝑊))) → (𝑊‘𝑥) ∈ 𝑇) |
30 | 11, 29 | sselid 3977 | . . . . . 6 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) ∧ 𝑥 ∈ (0..^(♯‘𝑊))) → (𝑊‘𝑥) ∈ (Base‘𝐺)) |
31 | 1, 10, 4 | symginv 19394 | . . . . . 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 2726 | . . . . . . 7 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘𝐷) | |
34 | 33, 9 | pmtrfcnv 19456 | . . . . . 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 2770 | . . . 4 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) ∧ 𝑥 ∈ (0..^(♯‘𝑊))) → ((𝐼 ∘ 𝑊)‘𝑥) = (𝑊‘𝑥)) |
37 | 25, 26, 36 | eqfnfvd 7037 | . . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → (𝐼 ∘ 𝑊) = 𝑊) |
38 | 37 | oveq2d 7430 | . 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → ((oppg‘𝐺) Σg (𝐼 ∘ 𝑊)) = ((oppg‘𝐺) Σg 𝑊)) |
39 | 2 | grpmndd 18934 | . . 3 ⊢ (𝐷 ∈ 𝑉 → 𝐺 ∈ Mnd) |
40 | 10, 3 | gsumwrev 19357 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑊 ∈ Word (Base‘𝐺)) → ((oppg‘𝐺) Σg 𝑊) = (𝐺 Σg (reverse‘𝑊))) |
41 | 39, 14, 40 | syl2an 594 | . 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → ((oppg‘𝐺) Σg 𝑊) = (𝐺 Σg (reverse‘𝑊))) |
42 | 16, 38, 41 | 3eqtrd 2770 | 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 394 = wceq 1534 ∈ wcel 2099 ⊆ wss 3947 ◡ccnv 5672 ran crn 5674 ∘ ccom 5677 Fn wfn 6539 ⟶wf 6540 ‘cfv 6544 (class class class)co 7414 0cc0 11147 ..^cfzo 13673 ♯chash 14340 Word cword 14515 reversecreverse 14759 Basecbs 17206 Σg cgsu 17448 Mndcmnd 18720 MndHom cmhm 18764 Grpcgrp 18921 invgcminusg 18922 GrpHom cghm 19200 GrpIso cgim 19245 oppgcoppg 19333 SymGrpcsymg 19358 pmTrspcpmtr 19433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-cnex 11203 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4907 df-int 4948 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-1st 7993 df-2nd 7994 df-tpos 8231 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-2o 8487 df-er 8724 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9973 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-nn 12257 df-2 12319 df-3 12320 df-4 12321 df-5 12322 df-6 12323 df-7 12324 df-8 12325 df-9 12326 df-n0 12517 df-xnn0 12589 df-z 12603 df-uz 12867 df-fz 13531 df-fzo 13674 df-seq 14014 df-hash 14341 df-word 14516 df-lsw 14564 df-concat 14572 df-s1 14597 df-substr 14642 df-pfx 14672 df-reverse 14760 df-struct 17142 df-sets 17159 df-slot 17177 df-ndx 17189 df-base 17207 df-ress 17236 df-plusg 17272 df-tset 17278 df-0g 17449 df-gsum 17450 df-mgm 18626 df-sgrp 18705 df-mnd 18721 df-mhm 18766 df-submnd 18767 df-efmnd 18852 df-grp 18924 df-minusg 18925 df-ghm 19201 df-gim 19247 df-oppg 19334 df-symg 19359 df-pmtr 19434 |
This theorem is referenced by: psgnuni 19491 |
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