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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 2732 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
2 | 1 | gsum0 18609 | . . . 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 19299 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → 𝐵 = {{⟨𝐴, 𝐴⟩}}) |
7 | 6 | eleq2d 2819 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ {{⟨𝐴, 𝐴⟩}})) |
8 | 7 | biimpa 477 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ {{⟨𝐴, 𝐴⟩}}) |
9 | elsni 4645 | . . . . . 6 ⊢ (𝑋 ∈ {{⟨𝐴, 𝐴⟩}} → 𝑋 = {⟨𝐴, 𝐴⟩}) | |
10 | 5 | reseq2i 5978 | . . . . . . . . . 10 ⊢ ( I ↾ 𝐷) = ( I ↾ {𝐴}) |
11 | snex 5431 | . . . . . . . . . . . . 13 ⊢ {𝐴} ∈ V | |
12 | 11 | snid 4664 | . . . . . . . . . . . 12 ⊢ {𝐴} ∈ {{𝐴}} |
13 | 5, 12 | eqeltri 2829 | . . . . . . . . . . 11 ⊢ 𝐷 ∈ {{𝐴}} |
14 | 3 | symgid 19310 | . . . . . . . . . . 11 ⊢ (𝐷 ∈ {{𝐴}} → ( I ↾ 𝐷) = (0g‘𝐺)) |
15 | 13, 14 | mp1i 13 | . . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐷) = (0g‘𝐺)) |
16 | restidsing 6052 | . . . . . . . . . . 11 ⊢ ( I ↾ {𝐴}) = ({𝐴} × {𝐴}) | |
17 | xpsng 7139 | . . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ({𝐴} × {𝐴}) = {⟨𝐴, 𝐴⟩}) | |
18 | 17 | anidms 567 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} × {𝐴}) = {⟨𝐴, 𝐴⟩}) |
19 | 16, 18 | eqtrid 2784 | . . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ {𝐴}) = {⟨𝐴, 𝐴⟩}) |
20 | 10, 15, 19 | 3eqtr3a 2796 | . . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (0g‘𝐺) = {⟨𝐴, 𝐴⟩}) |
21 | 20 | adantr 481 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (0g‘𝐺) = {⟨𝐴, 𝐴⟩}) |
22 | id 22 | . . . . . . . . 9 ⊢ ({⟨𝐴, 𝐴⟩} = 𝑋 → {⟨𝐴, 𝐴⟩} = 𝑋) | |
23 | 22 | eqcoms 2740 | . . . . . . . 8 ⊢ (𝑋 = {⟨𝐴, 𝐴⟩} → {⟨𝐴, 𝐴⟩} = 𝑋) |
24 | 21, 23 | sylan9eqr 2794 | . . . . . . 7 ⊢ ((𝑋 = {⟨𝐴, 𝐴⟩} ∧ (𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵)) → (0g‘𝐺) = 𝑋) |
25 | 24 | ex 413 | . . . . . 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 | eqtr2id 2785 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → 𝑋 = (𝐺 Σg ∅)) |
29 | 28 | fveq2d 6895 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = (𝑁‘(𝐺 Σg ∅))) |
30 | 5, 11 | eqeltri 2829 | . . . 4 ⊢ 𝐷 ∈ V |
31 | wrd0 14493 | . . . 4 ⊢ ∅ ∈ Word ∅ | |
32 | 30, 31 | pm3.2i 471 | . . 3 ⊢ (𝐷 ∈ V ∧ ∅ ∈ Word ∅) |
33 | 5 | fveq2i 6894 | . . . . . . 7 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘{𝐴}) |
34 | pmtrsn 19428 | . . . . . . 7 ⊢ (pmTrsp‘{𝐴}) = ∅ | |
35 | 33, 34 | eqtri 2760 | . . . . . 6 ⊢ (pmTrsp‘𝐷) = ∅ |
36 | 35 | rneqi 5936 | . . . . 5 ⊢ ran (pmTrsp‘𝐷) = ran ∅ |
37 | rn0 5925 | . . . . 5 ⊢ ran ∅ = ∅ | |
38 | 36, 37 | eqtr2i 2761 | . . . 4 ⊢ ∅ = ran (pmTrsp‘𝐷) |
39 | psgnsn.n | . . . 4 ⊢ 𝑁 = (pmSgn‘𝐷) | |
40 | 3, 38, 39 | psgnvalii 19418 | . . 3 ⊢ ((𝐷 ∈ V ∧ ∅ ∈ Word ∅) → (𝑁‘(𝐺 Σg ∅)) = (-1↑(♯‘∅))) |
41 | 32, 40 | mp1i 13 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝐺 Σg ∅)) = (-1↑(♯‘∅))) |
42 | hash0 14331 | . . . . 5 ⊢ (♯‘∅) = 0 | |
43 | 42 | oveq2i 7422 | . . . 4 ⊢ (-1↑(♯‘∅)) = (-1↑0) |
44 | neg1cn 12330 | . . . . 5 ⊢ -1 ∈ ℂ | |
45 | exp0 14035 | . . . . 5 ⊢ (-1 ∈ ℂ → (-1↑0) = 1) | |
46 | 44, 45 | ax-mp 5 | . . . 4 ⊢ (-1↑0) = 1 |
47 | 43, 46 | eqtri 2760 | . . 3 ⊢ (-1↑(♯‘∅)) = 1 |
48 | 47 | a1i 11 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (-1↑(♯‘∅)) = 1) |
49 | 29, 41, 48 | 3eqtrd 2776 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ∅c0 4322 {csn 4628 ⟨cop 4634 I cid 5573 × cxp 5674 ran crn 5677 ↾ cres 5678 ‘cfv 6543 (class class class)co 7411 ℂcc 11110 0cc0 11112 1c1 11113 -cneg 11449 ↑cexp 14031 ♯chash 14294 Word cword 14468 Basecbs 17148 0gc0g 17389 Σg cgsu 17390 SymGrpcsymg 19275 pmTrspcpmtr 19350 pmSgncpsgn 19398 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-xor 1510 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-ot 4637 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 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 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-tpos 8213 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-xnn0 12549 df-z 12563 df-uz 12827 df-rp 12979 df-fz 13489 df-fzo 13632 df-seq 13971 df-exp 14032 df-hash 14295 df-word 14469 df-lsw 14517 df-concat 14525 df-s1 14550 df-substr 14595 df-pfx 14625 df-splice 14704 df-reverse 14713 df-s2 14803 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-tset 17220 df-0g 17391 df-gsum 17392 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18705 df-submnd 18706 df-efmnd 18786 df-grp 18858 df-minusg 18859 df-subg 19039 df-ghm 19128 df-gim 19173 df-oppg 19251 df-symg 19276 df-pmtr 19351 df-psgn 19400 |
This theorem is referenced by: m1detdiag 22319 |
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