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Mirrors > Home > MPE Home > Th. List > psgnprfval | Structured version Visualization version GIF version |
Description: The permutation sign function for a pair. (Contributed by AV, 10-Dec-2018.) |
Ref | Expression |
---|---|
psgnprfval.0 | ⊢ 𝐷 = {1, 2} |
psgnprfval.g | ⊢ 𝐺 = (SymGrp‘𝐷) |
psgnprfval.b | ⊢ 𝐵 = (Base‘𝐺) |
psgnprfval.t | ⊢ 𝑇 = ran (pmTrsp‘𝐷) |
psgnprfval.n | ⊢ 𝑁 = (pmSgn‘𝐷) |
Ref | Expression |
---|---|
psgnprfval | ⊢ (𝑋 ∈ 𝐵 → (𝑁‘𝑋) = (℩𝑠∃𝑤 ∈ Word 𝑇(𝑋 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ 𝐵) | |
2 | elpri 4535 | . . . . . 6 ⊢ (𝑋 ∈ {{〈1, 1〉, 〈2, 2〉}, {〈1, 2〉, 〈2, 1〉}} → (𝑋 = {〈1, 1〉, 〈2, 2〉} ∨ 𝑋 = {〈1, 2〉, 〈2, 1〉})) | |
3 | prfi 8860 | . . . . . . . . 9 ⊢ {〈1, 1〉, 〈2, 2〉} ∈ Fin | |
4 | eleq1 2820 | . . . . . . . . 9 ⊢ (𝑋 = {〈1, 1〉, 〈2, 2〉} → (𝑋 ∈ Fin ↔ {〈1, 1〉, 〈2, 2〉} ∈ Fin)) | |
5 | 3, 4 | mpbiri 261 | . . . . . . . 8 ⊢ (𝑋 = {〈1, 1〉, 〈2, 2〉} → 𝑋 ∈ Fin) |
6 | prfi 8860 | . . . . . . . . 9 ⊢ {〈1, 2〉, 〈2, 1〉} ∈ Fin | |
7 | eleq1 2820 | . . . . . . . . 9 ⊢ (𝑋 = {〈1, 2〉, 〈2, 1〉} → (𝑋 ∈ Fin ↔ {〈1, 2〉, 〈2, 1〉} ∈ Fin)) | |
8 | 6, 7 | mpbiri 261 | . . . . . . . 8 ⊢ (𝑋 = {〈1, 2〉, 〈2, 1〉} → 𝑋 ∈ Fin) |
9 | 5, 8 | jaoi 856 | . . . . . . 7 ⊢ ((𝑋 = {〈1, 1〉, 〈2, 2〉} ∨ 𝑋 = {〈1, 2〉, 〈2, 1〉}) → 𝑋 ∈ Fin) |
10 | diffi 8820 | . . . . . . 7 ⊢ (𝑋 ∈ Fin → (𝑋 ∖ I ) ∈ Fin) | |
11 | 9, 10 | syl 17 | . . . . . 6 ⊢ ((𝑋 = {〈1, 1〉, 〈2, 2〉} ∨ 𝑋 = {〈1, 2〉, 〈2, 1〉}) → (𝑋 ∖ I ) ∈ Fin) |
12 | dmfi 8868 | . . . . . 6 ⊢ ((𝑋 ∖ I ) ∈ Fin → dom (𝑋 ∖ I ) ∈ Fin) | |
13 | 2, 11, 12 | 3syl 18 | . . . . 5 ⊢ (𝑋 ∈ {{〈1, 1〉, 〈2, 2〉}, {〈1, 2〉, 〈2, 1〉}} → dom (𝑋 ∖ I ) ∈ Fin) |
14 | 1ex 10708 | . . . . . 6 ⊢ 1 ∈ V | |
15 | 2nn 11782 | . . . . . 6 ⊢ 2 ∈ ℕ | |
16 | psgnprfval.g | . . . . . . 7 ⊢ 𝐺 = (SymGrp‘𝐷) | |
17 | psgnprfval.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐺) | |
18 | psgnprfval.0 | . . . . . . 7 ⊢ 𝐷 = {1, 2} | |
19 | 16, 17, 18 | symg2bas 18632 | . . . . . 6 ⊢ ((1 ∈ V ∧ 2 ∈ ℕ) → 𝐵 = {{〈1, 1〉, 〈2, 2〉}, {〈1, 2〉, 〈2, 1〉}}) |
20 | 14, 15, 19 | mp2an 692 | . . . . 5 ⊢ 𝐵 = {{〈1, 1〉, 〈2, 2〉}, {〈1, 2〉, 〈2, 1〉}} |
21 | 13, 20 | eleq2s 2851 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → dom (𝑋 ∖ I ) ∈ Fin) |
22 | psgnprfval.n | . . . . 5 ⊢ 𝑁 = (pmSgn‘𝐷) | |
23 | 16, 22, 17 | psgneldm 18742 | . . . 4 ⊢ (𝑋 ∈ dom 𝑁 ↔ (𝑋 ∈ 𝐵 ∧ dom (𝑋 ∖ I ) ∈ Fin)) |
24 | 1, 21, 23 | sylanbrc 586 | . . 3 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ dom 𝑁) |
25 | psgnprfval.t | . . . 4 ⊢ 𝑇 = ran (pmTrsp‘𝐷) | |
26 | 16, 25, 22 | psgnval 18746 | . . 3 ⊢ (𝑋 ∈ dom 𝑁 → (𝑁‘𝑋) = (℩𝑠∃𝑤 ∈ Word 𝑇(𝑋 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))) |
27 | 24, 26 | syl 17 | . 2 ⊢ (𝑋 ∈ 𝐵 → (𝑁‘𝑋) = (℩𝑠∃𝑤 ∈ Word 𝑇(𝑋 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))) |
28 | 1, 27 | syl 17 | 1 ⊢ (𝑋 ∈ 𝐵 → (𝑁‘𝑋) = (℩𝑠∃𝑤 ∈ Word 𝑇(𝑋 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 846 = wceq 1542 ∈ wcel 2113 ∃wrex 3054 Vcvv 3397 ∖ cdif 3838 {cpr 4515 〈cop 4519 I cid 5424 dom cdm 5519 ran crn 5520 ℩cio 6289 ‘cfv 6333 (class class class)co 7164 Fincfn 8548 1c1 10609 -cneg 10942 ℕcn 11709 2c2 11764 ↑cexp 13514 ♯chash 13775 Word cword 13948 Basecbs 16579 Σg cgsu 16810 SymGrpcsymg 18606 pmTrspcpmtr 18680 pmSgncpsgn 18728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-2o 8125 df-oadd 8128 df-er 8313 df-map 8432 df-pm 8433 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-dju 9396 df-card 9434 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-div 11369 df-nn 11710 df-2 11772 df-3 11773 df-4 11774 df-5 11775 df-6 11776 df-7 11777 df-8 11778 df-9 11779 df-n0 11970 df-xnn0 12042 df-z 12056 df-uz 12318 df-fz 12975 df-fzo 13118 df-seq 13454 df-fac 13719 df-bc 13748 df-hash 13776 df-word 13949 df-struct 16581 df-ndx 16582 df-slot 16583 df-base 16585 df-sets 16586 df-ress 16587 df-plusg 16674 df-tset 16680 df-efmnd 18143 df-symg 18607 df-psgn 18730 |
This theorem is referenced by: (None) |
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