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Mirrors > Home > MPE Home > Th. List > psgn0fv0 | Structured version Visualization version GIF version |
Description: The permutation sign function for an empty set at an empty set is 1. (Contributed by AV, 27-Feb-2019.) |
Ref | Expression |
---|---|
psgn0fv0 | ⊢ ((pmSgn‘∅)‘∅) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5263 | . 2 ⊢ ∅ ∈ V | |
2 | wrd0 14381 | . 2 ⊢ ∅ ∈ Word ran (pmTrsp‘∅) | |
3 | eqid 2738 | . . . . . 6 ⊢ (0g‘(SymGrp‘∅)) = (0g‘(SymGrp‘∅)) | |
4 | 3 | gsum0 18499 | . . . . 5 ⊢ ((SymGrp‘∅) Σg ∅) = (0g‘(SymGrp‘∅)) |
5 | eqid 2738 | . . . . . . . . 9 ⊢ (SymGrp‘∅) = (SymGrp‘∅) | |
6 | 5 | symgid 19142 | . . . . . . . 8 ⊢ (∅ ∈ V → ( I ↾ ∅) = (0g‘(SymGrp‘∅))) |
7 | 1, 6 | ax-mp 5 | . . . . . . 7 ⊢ ( I ↾ ∅) = (0g‘(SymGrp‘∅)) |
8 | res0 5940 | . . . . . . 7 ⊢ ( I ↾ ∅) = ∅ | |
9 | 7, 8 | eqtr3i 2768 | . . . . . 6 ⊢ (0g‘(SymGrp‘∅)) = ∅ |
10 | 9 | a1i 11 | . . . . 5 ⊢ ((∅ ∈ V ∧ ∅ ∈ Word ran (pmTrsp‘∅)) → (0g‘(SymGrp‘∅)) = ∅) |
11 | 4, 10 | eqtr2id 2791 | . . . 4 ⊢ ((∅ ∈ V ∧ ∅ ∈ Word ran (pmTrsp‘∅)) → ∅ = ((SymGrp‘∅) Σg ∅)) |
12 | 11 | fveq2d 6844 | . . 3 ⊢ ((∅ ∈ V ∧ ∅ ∈ Word ran (pmTrsp‘∅)) → ((pmSgn‘∅)‘∅) = ((pmSgn‘∅)‘((SymGrp‘∅) Σg ∅))) |
13 | eqid 2738 | . . . 4 ⊢ ran (pmTrsp‘∅) = ran (pmTrsp‘∅) | |
14 | eqid 2738 | . . . 4 ⊢ (pmSgn‘∅) = (pmSgn‘∅) | |
15 | 5, 13, 14 | psgnvalii 19250 | . . 3 ⊢ ((∅ ∈ V ∧ ∅ ∈ Word ran (pmTrsp‘∅)) → ((pmSgn‘∅)‘((SymGrp‘∅) Σg ∅)) = (-1↑(♯‘∅))) |
16 | hash0 14221 | . . . . . 6 ⊢ (♯‘∅) = 0 | |
17 | 16 | oveq2i 7363 | . . . . 5 ⊢ (-1↑(♯‘∅)) = (-1↑0) |
18 | neg1cn 12226 | . . . . . 6 ⊢ -1 ∈ ℂ | |
19 | exp0 13926 | . . . . . 6 ⊢ (-1 ∈ ℂ → (-1↑0) = 1) | |
20 | 18, 19 | ax-mp 5 | . . . . 5 ⊢ (-1↑0) = 1 |
21 | 17, 20 | eqtri 2766 | . . . 4 ⊢ (-1↑(♯‘∅)) = 1 |
22 | 21 | a1i 11 | . . 3 ⊢ ((∅ ∈ V ∧ ∅ ∈ Word ran (pmTrsp‘∅)) → (-1↑(♯‘∅)) = 1) |
23 | 12, 15, 22 | 3eqtrd 2782 | . 2 ⊢ ((∅ ∈ V ∧ ∅ ∈ Word ran (pmTrsp‘∅)) → ((pmSgn‘∅)‘∅) = 1) |
24 | 1, 2, 23 | mp2an 691 | 1 ⊢ ((pmSgn‘∅)‘∅) = 1 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ∅c0 4281 I cid 5529 ran crn 5633 ↾ cres 5634 ‘cfv 6494 (class class class)co 7352 ℂcc 11008 0cc0 11010 1c1 11011 -cneg 11345 ↑cexp 13922 ♯chash 14184 Word cword 14356 0gc0g 17281 Σg cgsu 17282 SymGrpcsymg 19107 pmTrspcpmtr 19182 pmSgncpsgn 19230 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-cnex 11066 ax-resscn 11067 ax-1cn 11068 ax-icn 11069 ax-addcl 11070 ax-addrcl 11071 ax-mulcl 11072 ax-mulrcl 11073 ax-mulcom 11074 ax-addass 11075 ax-mulass 11076 ax-distr 11077 ax-i2m1 11078 ax-1ne0 11079 ax-1rid 11080 ax-rnegex 11081 ax-rrecex 11082 ax-cnre 11083 ax-pre-lttri 11084 ax-pre-lttrn 11085 ax-pre-ltadd 11086 ax-pre-mulgt0 11087 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-xor 1511 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-ot 4594 df-uni 4865 df-int 4907 df-iun 4955 df-iin 4956 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-se 5588 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-isom 6503 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7796 df-1st 7914 df-2nd 7915 df-tpos 8150 df-frecs 8205 df-wrecs 8236 df-recs 8310 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8607 df-map 8726 df-en 8843 df-dom 8844 df-sdom 8845 df-fin 8846 df-card 9834 df-pnf 11150 df-mnf 11151 df-xr 11152 df-ltxr 11153 df-le 11154 df-sub 11346 df-neg 11347 df-div 11772 df-nn 12113 df-2 12175 df-3 12176 df-4 12177 df-5 12178 df-6 12179 df-7 12180 df-8 12181 df-9 12182 df-n0 12373 df-xnn0 12445 df-z 12459 df-uz 12723 df-rp 12871 df-fz 13380 df-fzo 13523 df-seq 13862 df-exp 13923 df-hash 14185 df-word 14357 df-lsw 14405 df-concat 14413 df-s1 14438 df-substr 14487 df-pfx 14517 df-splice 14596 df-reverse 14605 df-s2 14695 df-struct 16979 df-sets 16996 df-slot 17014 df-ndx 17026 df-base 17044 df-ress 17073 df-plusg 17106 df-tset 17112 df-0g 17283 df-gsum 17284 df-mre 17426 df-mrc 17427 df-acs 17429 df-mgm 18457 df-sgrp 18506 df-mnd 18517 df-mhm 18561 df-submnd 18562 df-efmnd 18639 df-grp 18711 df-minusg 18712 df-subg 18884 df-ghm 18965 df-gim 19008 df-oppg 19083 df-symg 19108 df-pmtr 19183 df-psgn 19232 |
This theorem is referenced by: mdet0pr 21893 |
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