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Mirrors > Home > MPE Home > Th. List > psgnsn | Structured version Visualization version GIF version |
Description: The permutation sign function for a singleton. (Contributed by AV, 6-Aug-2019.) |
Ref | Expression |
---|---|
psgnsn.0 | ⊢ 𝐷 = {𝐴} |
psgnsn.g | ⊢ 𝐺 = (SymGrp‘𝐷) |
psgnsn.b | ⊢ 𝐵 = (Base‘𝐺) |
psgnsn.n | ⊢ 𝑁 = (pmSgn‘𝐷) |
Ref | Expression |
---|---|
psgnsn | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
2 | 1 | gsum0 17886 | . . . 4 ⊢ (𝐺 Σg ∅) = (0g‘𝐺) |
3 | psgnsn.g | . . . . . . . 8 ⊢ 𝐺 = (SymGrp‘𝐷) | |
4 | psgnsn.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝐺) | |
5 | psgnsn.0 | . . . . . . . 8 ⊢ 𝐷 = {𝐴} | |
6 | 3, 4, 5 | symg1bas 18511 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → 𝐵 = {{〈𝐴, 𝐴〉}}) |
7 | 6 | eleq2d 2875 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ {{〈𝐴, 𝐴〉}})) |
8 | 7 | biimpa 480 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ {{〈𝐴, 𝐴〉}}) |
9 | elsni 4542 | . . . . . 6 ⊢ (𝑋 ∈ {{〈𝐴, 𝐴〉}} → 𝑋 = {〈𝐴, 𝐴〉}) | |
10 | 5 | reseq2i 5815 | . . . . . . . . . 10 ⊢ ( I ↾ 𝐷) = ( I ↾ {𝐴}) |
11 | snex 5297 | . . . . . . . . . . . . 13 ⊢ {𝐴} ∈ V | |
12 | 11 | snid 4561 | . . . . . . . . . . . 12 ⊢ {𝐴} ∈ {{𝐴}} |
13 | 5, 12 | eqeltri 2886 | . . . . . . . . . . 11 ⊢ 𝐷 ∈ {{𝐴}} |
14 | 3 | symgid 18521 | . . . . . . . . . . 11 ⊢ (𝐷 ∈ {{𝐴}} → ( I ↾ 𝐷) = (0g‘𝐺)) |
15 | 13, 14 | mp1i 13 | . . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐷) = (0g‘𝐺)) |
16 | restidsing 5889 | . . . . . . . . . . 11 ⊢ ( I ↾ {𝐴}) = ({𝐴} × {𝐴}) | |
17 | xpsng 6878 | . . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ({𝐴} × {𝐴}) = {〈𝐴, 𝐴〉}) | |
18 | 17 | anidms 570 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} × {𝐴}) = {〈𝐴, 𝐴〉}) |
19 | 16, 18 | syl5eq 2845 | . . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ {𝐴}) = {〈𝐴, 𝐴〉}) |
20 | 10, 15, 19 | 3eqtr3a 2857 | . . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (0g‘𝐺) = {〈𝐴, 𝐴〉}) |
21 | 20 | adantr 484 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (0g‘𝐺) = {〈𝐴, 𝐴〉}) |
22 | id 22 | . . . . . . . . 9 ⊢ ({〈𝐴, 𝐴〉} = 𝑋 → {〈𝐴, 𝐴〉} = 𝑋) | |
23 | 22 | eqcoms 2806 | . . . . . . . 8 ⊢ (𝑋 = {〈𝐴, 𝐴〉} → {〈𝐴, 𝐴〉} = 𝑋) |
24 | 21, 23 | sylan9eqr 2855 | . . . . . . 7 ⊢ ((𝑋 = {〈𝐴, 𝐴〉} ∧ (𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵)) → (0g‘𝐺) = 𝑋) |
25 | 24 | ex 416 | . . . . . 6 ⊢ (𝑋 = {〈𝐴, 𝐴〉} → ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (0g‘𝐺) = 𝑋)) |
26 | 9, 25 | syl 17 | . . . . 5 ⊢ (𝑋 ∈ {{〈𝐴, 𝐴〉}} → ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (0g‘𝐺) = 𝑋)) |
27 | 8, 26 | mpcom 38 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (0g‘𝐺) = 𝑋) |
28 | 2, 27 | syl5req 2846 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → 𝑋 = (𝐺 Σg ∅)) |
29 | 28 | fveq2d 6649 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = (𝑁‘(𝐺 Σg ∅))) |
30 | 5, 11 | eqeltri 2886 | . . . 4 ⊢ 𝐷 ∈ V |
31 | wrd0 13882 | . . . 4 ⊢ ∅ ∈ Word ∅ | |
32 | 30, 31 | pm3.2i 474 | . . 3 ⊢ (𝐷 ∈ V ∧ ∅ ∈ Word ∅) |
33 | 5 | fveq2i 6648 | . . . . . . 7 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘{𝐴}) |
34 | pmtrsn 18639 | . . . . . . 7 ⊢ (pmTrsp‘{𝐴}) = ∅ | |
35 | 33, 34 | eqtri 2821 | . . . . . 6 ⊢ (pmTrsp‘𝐷) = ∅ |
36 | 35 | rneqi 5771 | . . . . 5 ⊢ ran (pmTrsp‘𝐷) = ran ∅ |
37 | rn0 5760 | . . . . 5 ⊢ ran ∅ = ∅ | |
38 | 36, 37 | eqtr2i 2822 | . . . 4 ⊢ ∅ = ran (pmTrsp‘𝐷) |
39 | psgnsn.n | . . . 4 ⊢ 𝑁 = (pmSgn‘𝐷) | |
40 | 3, 38, 39 | psgnvalii 18629 | . . 3 ⊢ ((𝐷 ∈ V ∧ ∅ ∈ Word ∅) → (𝑁‘(𝐺 Σg ∅)) = (-1↑(♯‘∅))) |
41 | 32, 40 | mp1i 13 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝐺 Σg ∅)) = (-1↑(♯‘∅))) |
42 | hash0 13724 | . . . . 5 ⊢ (♯‘∅) = 0 | |
43 | 42 | oveq2i 7146 | . . . 4 ⊢ (-1↑(♯‘∅)) = (-1↑0) |
44 | neg1cn 11739 | . . . . 5 ⊢ -1 ∈ ℂ | |
45 | exp0 13429 | . . . . 5 ⊢ (-1 ∈ ℂ → (-1↑0) = 1) | |
46 | 44, 45 | ax-mp 5 | . . . 4 ⊢ (-1↑0) = 1 |
47 | 43, 46 | eqtri 2821 | . . 3 ⊢ (-1↑(♯‘∅)) = 1 |
48 | 47 | a1i 11 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (-1↑(♯‘∅)) = 1) |
49 | 29, 41, 48 | 3eqtrd 2837 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 {csn 4525 〈cop 4531 I cid 5424 × cxp 5517 ran crn 5520 ↾ cres 5521 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 0cc0 10526 1c1 10527 -cneg 10860 ↑cexp 13425 ♯chash 13686 Word cword 13857 Basecbs 16475 0gc0g 16705 Σg cgsu 16706 SymGrpcsymg 18487 pmTrspcpmtr 18561 pmSgncpsgn 18609 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-xor 1503 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-ot 4534 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 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 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 df-word 13858 df-lsw 13906 df-concat 13914 df-s1 13941 df-substr 13994 df-pfx 14024 df-splice 14103 df-reverse 14112 df-s2 14201 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-tset 16576 df-0g 16707 df-gsum 16708 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-efmnd 18026 df-grp 18098 df-minusg 18099 df-subg 18268 df-ghm 18348 df-gim 18391 df-oppg 18466 df-symg 18488 df-pmtr 18562 df-psgn 18611 |
This theorem is referenced by: m1detdiag 21202 |
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