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Mirrors > Home > MPE Home > Th. List > psgnvalfi | Structured version Visualization version GIF version |
Description: Value of the permutation sign function for permutations of finite sets. (Contributed by AV, 13-Jan-2019.) |
Ref | Expression |
---|---|
psgnfvalfi.g | ⊢ 𝐺 = (SymGrp‘𝐷) |
psgnfvalfi.b | ⊢ 𝐵 = (Base‘𝐺) |
psgnfvalfi.t | ⊢ 𝑇 = ran (pmTrsp‘𝐷) |
psgnfvalfi.n | ⊢ 𝑁 = (pmSgn‘𝐷) |
Ref | Expression |
---|---|
psgnvalfi | ⊢ ((𝐷 ∈ Fin ∧ 𝑃 ∈ 𝐵) → (𝑁‘𝑃) = (℩𝑠∃𝑤 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 485 | . . 3 ⊢ ((𝐷 ∈ Fin ∧ 𝑃 ∈ 𝐵) → 𝑃 ∈ 𝐵) | |
2 | psgnfvalfi.g | . . . 4 ⊢ 𝐺 = (SymGrp‘𝐷) | |
3 | psgnfvalfi.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
4 | 2, 3 | sygbasnfpfi 18376 | . . 3 ⊢ ((𝐷 ∈ Fin ∧ 𝑃 ∈ 𝐵) → dom (𝑃 ∖ I ) ∈ Fin) |
5 | psgnfvalfi.n | . . . 4 ⊢ 𝑁 = (pmSgn‘𝐷) | |
6 | 2, 5, 3 | psgneldm 18367 | . . 3 ⊢ (𝑃 ∈ dom 𝑁 ↔ (𝑃 ∈ 𝐵 ∧ dom (𝑃 ∖ I ) ∈ Fin)) |
7 | 1, 4, 6 | sylanbrc 583 | . 2 ⊢ ((𝐷 ∈ Fin ∧ 𝑃 ∈ 𝐵) → 𝑃 ∈ dom 𝑁) |
8 | psgnfvalfi.t | . . 3 ⊢ 𝑇 = ran (pmTrsp‘𝐷) | |
9 | 2, 8, 5 | psgnval 18371 | . 2 ⊢ (𝑃 ∈ dom 𝑁 → (𝑁‘𝑃) = (℩𝑠∃𝑤 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))) |
10 | 7, 9 | syl 17 | 1 ⊢ ((𝐷 ∈ Fin ∧ 𝑃 ∈ 𝐵) → (𝑁‘𝑃) = (℩𝑠∃𝑤 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ∃wrex 3106 ∖ cdif 3860 I cid 5352 dom cdm 5448 ran crn 5449 ℩cio 6192 ‘cfv 6230 (class class class)co 7021 Fincfn 8362 1c1 10389 -cneg 10723 ↑cexp 13284 ♯chash 13545 Word cword 13712 Basecbs 16317 Σg cgsu 16548 SymGrpcsymg 18241 pmTrspcpmtr 18305 pmSgncpsgn 18353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5086 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-cnex 10444 ax-resscn 10445 ax-1cn 10446 ax-icn 10447 ax-addcl 10448 ax-addrcl 10449 ax-mulcl 10450 ax-mulrcl 10451 ax-mulcom 10452 ax-addass 10453 ax-mulass 10454 ax-distr 10455 ax-i2m1 10456 ax-1ne0 10457 ax-1rid 10458 ax-rnegex 10459 ax-rrecex 10460 ax-cnre 10461 ax-pre-lttri 10462 ax-pre-lttrn 10463 ax-pre-ltadd 10464 ax-pre-mulgt0 10465 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-int 4787 df-iun 4831 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-om 7442 df-1st 7550 df-2nd 7551 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-1o 7958 df-oadd 7962 df-er 8144 df-map 8263 df-en 8363 df-dom 8364 df-sdom 8365 df-fin 8366 df-card 9219 df-pnf 10528 df-mnf 10529 df-xr 10530 df-ltxr 10531 df-le 10532 df-sub 10724 df-neg 10725 df-nn 11492 df-2 11553 df-3 11554 df-4 11555 df-5 11556 df-6 11557 df-7 11558 df-8 11559 df-9 11560 df-n0 11751 df-z 11835 df-uz 12099 df-fz 12748 df-fzo 12889 df-hash 13546 df-word 13713 df-struct 16319 df-ndx 16320 df-slot 16321 df-base 16323 df-plusg 16412 df-tset 16418 df-symg 18242 df-psgn 18355 |
This theorem is referenced by: psgndif 20433 |
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