Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4582 | . . . . . 6 ⊢ (𝑋 ∈ {{〈1, 1〉, 〈2, 2〉}, {〈1, 2〉, 〈2, 1〉}} → (𝑋 = {〈1, 1〉, 〈2, 2〉} ∨ 𝑋 = {〈1, 2〉, 〈2, 1〉})) | |
3 | prfi 8787 | . . . . . . . . 9 ⊢ {〈1, 1〉, 〈2, 2〉} ∈ Fin | |
4 | eleq1 2900 | . . . . . . . . 9 ⊢ (𝑋 = {〈1, 1〉, 〈2, 2〉} → (𝑋 ∈ Fin ↔ {〈1, 1〉, 〈2, 2〉} ∈ Fin)) | |
5 | 3, 4 | mpbiri 260 | . . . . . . . 8 ⊢ (𝑋 = {〈1, 1〉, 〈2, 2〉} → 𝑋 ∈ Fin) |
6 | prfi 8787 | . . . . . . . . 9 ⊢ {〈1, 2〉, 〈2, 1〉} ∈ Fin | |
7 | eleq1 2900 | . . . . . . . . 9 ⊢ (𝑋 = {〈1, 2〉, 〈2, 1〉} → (𝑋 ∈ Fin ↔ {〈1, 2〉, 〈2, 1〉} ∈ Fin)) | |
8 | 6, 7 | mpbiri 260 | . . . . . . . 8 ⊢ (𝑋 = {〈1, 2〉, 〈2, 1〉} → 𝑋 ∈ Fin) |
9 | 5, 8 | jaoi 853 | . . . . . . 7 ⊢ ((𝑋 = {〈1, 1〉, 〈2, 2〉} ∨ 𝑋 = {〈1, 2〉, 〈2, 1〉}) → 𝑋 ∈ Fin) |
10 | diffi 8744 | . . . . . . 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 8796 | . . . . . 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 10631 | . . . . . 6 ⊢ 1 ∈ V | |
15 | 2nn 11704 | . . . . . 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 18515 | . . . . . 6 ⊢ ((1 ∈ V ∧ 2 ∈ ℕ) → 𝐵 = {{〈1, 1〉, 〈2, 2〉}, {〈1, 2〉, 〈2, 1〉}}) |
20 | 14, 15, 19 | mp2an 690 | . . . . 5 ⊢ 𝐵 = {{〈1, 1〉, 〈2, 2〉}, {〈1, 2〉, 〈2, 1〉}} |
21 | 13, 20 | eleq2s 2931 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → dom (𝑋 ∖ I ) ∈ Fin) |
22 | psgnprfval.n | . . . . 5 ⊢ 𝑁 = (pmSgn‘𝐷) | |
23 | 16, 22, 17 | psgneldm 18625 | . . . 4 ⊢ (𝑋 ∈ dom 𝑁 ↔ (𝑋 ∈ 𝐵 ∧ dom (𝑋 ∖ I ) ∈ Fin)) |
24 | 1, 21, 23 | sylanbrc 585 | . . 3 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ dom 𝑁) |
25 | psgnprfval.t | . . . 4 ⊢ 𝑇 = ran (pmTrsp‘𝐷) | |
26 | 16, 25, 22 | psgnval 18629 | . . 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 398 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 Vcvv 3494 ∖ cdif 3932 {cpr 4562 〈cop 4566 I cid 5453 dom cdm 5549 ran crn 5550 ℩cio 6306 ‘cfv 6349 (class class class)co 7150 Fincfn 8503 1c1 10532 -cneg 10865 ℕcn 11632 2c2 11686 ↑cexp 13423 ♯chash 13684 Word cword 13855 Basecbs 16477 Σg cgsu 16708 SymGrpcsymg 18489 pmTrspcpmtr 18563 pmSgncpsgn 18611 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-xnn0 11962 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-seq 13364 df-fac 13628 df-bc 13657 df-hash 13685 df-word 13856 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-tset 16578 df-efmnd 18028 df-symg 18490 df-psgn 18613 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |