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Mirrors > Home > MPE Home > Th. List > psgnvalii | Structured version Visualization version GIF version |
Description: Any representation of a permutation is length matching the permutation sign. (Contributed by Stefan O'Rear, 28-Aug-2015.) |
Ref | Expression |
---|---|
psgnval.g | ⊢ 𝐺 = (SymGrp‘𝐷) |
psgnval.t | ⊢ 𝑇 = ran (pmTrsp‘𝐷) |
psgnval.n | ⊢ 𝑁 = (pmSgn‘𝐷) |
Ref | Expression |
---|---|
psgnvalii | ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → (𝑁‘(𝐺 Σg 𝑊)) = (-1↑(♯‘𝑊))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psgnval.g | . . . 4 ⊢ 𝐺 = (SymGrp‘𝐷) | |
2 | psgnval.t | . . . 4 ⊢ 𝑇 = ran (pmTrsp‘𝐷) | |
3 | psgnval.n | . . . 4 ⊢ 𝑁 = (pmSgn‘𝐷) | |
4 | 1, 2, 3 | psgneldm2i 19464 | . . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → (𝐺 Σg 𝑊) ∈ dom 𝑁) |
5 | 1, 2, 3 | psgnval 19466 | . . 3 ⊢ ((𝐺 Σg 𝑊) ∈ dom 𝑁 → (𝑁‘(𝐺 Σg 𝑊)) = (℩𝑠∃𝑤 ∈ Word 𝑇((𝐺 Σg 𝑊) = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))) |
6 | 4, 5 | syl 17 | . 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → (𝑁‘(𝐺 Σg 𝑊)) = (℩𝑠∃𝑤 ∈ Word 𝑇((𝐺 Σg 𝑊) = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))) |
7 | simpr 483 | . . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → 𝑊 ∈ Word 𝑇) | |
8 | eqidd 2726 | . . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → (𝐺 Σg 𝑊) = (𝐺 Σg 𝑊)) | |
9 | eqidd 2726 | . . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → (-1↑(♯‘𝑊)) = (-1↑(♯‘𝑊))) | |
10 | oveq2 7424 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑊)) | |
11 | 10 | eqeq2d 2736 | . . . . . 6 ⊢ (𝑤 = 𝑊 → ((𝐺 Σg 𝑊) = (𝐺 Σg 𝑤) ↔ (𝐺 Σg 𝑊) = (𝐺 Σg 𝑊))) |
12 | fveq2 6892 | . . . . . . . 8 ⊢ (𝑤 = 𝑊 → (♯‘𝑤) = (♯‘𝑊)) | |
13 | 12 | oveq2d 7432 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (-1↑(♯‘𝑤)) = (-1↑(♯‘𝑊))) |
14 | 13 | eqeq2d 2736 | . . . . . 6 ⊢ (𝑤 = 𝑊 → ((-1↑(♯‘𝑊)) = (-1↑(♯‘𝑤)) ↔ (-1↑(♯‘𝑊)) = (-1↑(♯‘𝑊)))) |
15 | 11, 14 | anbi12d 630 | . . . . 5 ⊢ (𝑤 = 𝑊 → (((𝐺 Σg 𝑊) = (𝐺 Σg 𝑤) ∧ (-1↑(♯‘𝑊)) = (-1↑(♯‘𝑤))) ↔ ((𝐺 Σg 𝑊) = (𝐺 Σg 𝑊) ∧ (-1↑(♯‘𝑊)) = (-1↑(♯‘𝑊))))) |
16 | 15 | rspcev 3601 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑇 ∧ ((𝐺 Σg 𝑊) = (𝐺 Σg 𝑊) ∧ (-1↑(♯‘𝑊)) = (-1↑(♯‘𝑊)))) → ∃𝑤 ∈ Word 𝑇((𝐺 Σg 𝑊) = (𝐺 Σg 𝑤) ∧ (-1↑(♯‘𝑊)) = (-1↑(♯‘𝑤)))) |
17 | 7, 8, 9, 16 | syl12anc 835 | . . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → ∃𝑤 ∈ Word 𝑇((𝐺 Σg 𝑊) = (𝐺 Σg 𝑤) ∧ (-1↑(♯‘𝑊)) = (-1↑(♯‘𝑤)))) |
18 | ovexd 7451 | . . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → (-1↑(♯‘𝑊)) ∈ V) | |
19 | 1, 2, 3 | psgneu 19465 | . . . . 5 ⊢ ((𝐺 Σg 𝑊) ∈ dom 𝑁 → ∃!𝑠∃𝑤 ∈ Word 𝑇((𝐺 Σg 𝑊) = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))) |
20 | 4, 19 | syl 17 | . . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → ∃!𝑠∃𝑤 ∈ Word 𝑇((𝐺 Σg 𝑊) = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))) |
21 | eqeq1 2729 | . . . . . . 7 ⊢ (𝑠 = (-1↑(♯‘𝑊)) → (𝑠 = (-1↑(♯‘𝑤)) ↔ (-1↑(♯‘𝑊)) = (-1↑(♯‘𝑤)))) | |
22 | 21 | anbi2d 628 | . . . . . 6 ⊢ (𝑠 = (-1↑(♯‘𝑊)) → (((𝐺 Σg 𝑊) = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) ↔ ((𝐺 Σg 𝑊) = (𝐺 Σg 𝑤) ∧ (-1↑(♯‘𝑊)) = (-1↑(♯‘𝑤))))) |
23 | 22 | rexbidv 3169 | . . . . 5 ⊢ (𝑠 = (-1↑(♯‘𝑊)) → (∃𝑤 ∈ Word 𝑇((𝐺 Σg 𝑊) = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) ↔ ∃𝑤 ∈ Word 𝑇((𝐺 Σg 𝑊) = (𝐺 Σg 𝑤) ∧ (-1↑(♯‘𝑊)) = (-1↑(♯‘𝑤))))) |
24 | 23 | adantl 480 | . . . 4 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) ∧ 𝑠 = (-1↑(♯‘𝑊))) → (∃𝑤 ∈ Word 𝑇((𝐺 Σg 𝑊) = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) ↔ ∃𝑤 ∈ Word 𝑇((𝐺 Σg 𝑊) = (𝐺 Σg 𝑤) ∧ (-1↑(♯‘𝑊)) = (-1↑(♯‘𝑤))))) |
25 | 18, 20, 24 | iota2d 6531 | . . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → (∃𝑤 ∈ Word 𝑇((𝐺 Σg 𝑊) = (𝐺 Σg 𝑤) ∧ (-1↑(♯‘𝑊)) = (-1↑(♯‘𝑤))) ↔ (℩𝑠∃𝑤 ∈ Word 𝑇((𝐺 Σg 𝑊) = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))) = (-1↑(♯‘𝑊)))) |
26 | 17, 25 | mpbid 231 | . 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → (℩𝑠∃𝑤 ∈ Word 𝑇((𝐺 Σg 𝑊) = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))) = (-1↑(♯‘𝑊))) |
27 | 6, 26 | eqtrd 2765 | 1 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → (𝑁‘(𝐺 Σg 𝑊)) = (-1↑(♯‘𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∃!weu 2556 ∃wrex 3060 Vcvv 3463 dom cdm 5672 ran crn 5673 ℩cio 6493 ‘cfv 6543 (class class class)co 7416 1c1 11139 -cneg 11475 ↑cexp 14058 ♯chash 14321 Word cword 14496 Σg cgsu 17421 SymGrpcsymg 19325 pmTrspcpmtr 19400 pmSgncpsgn 19448 |
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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-xor 1505 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-xnn0 12575 df-z 12589 df-uz 12853 df-rp 13007 df-fz 13517 df-fzo 13660 df-seq 13999 df-exp 14059 df-hash 14322 df-word 14497 df-lsw 14545 df-concat 14553 df-s1 14578 df-substr 14623 df-pfx 14653 df-splice 14732 df-reverse 14741 df-s2 14831 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-tset 17251 df-0g 17422 df-gsum 17423 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-submnd 18740 df-efmnd 18825 df-grp 18897 df-minusg 18898 df-subg 19082 df-ghm 19172 df-gim 19217 df-oppg 19301 df-symg 19326 df-pmtr 19401 df-psgn 19450 |
This theorem is referenced by: psgnpmtr 19469 psgn0fv0 19470 psgnsn 19479 psgnprfval1 19481 psgnghm 21516 cyc3genpm 32918 |
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