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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 5203 | . 2 ⊢ ∅ ∈ V | |
2 | wrd0 13883 | . 2 ⊢ ∅ ∈ Word ran (pmTrsp‘∅) | |
3 | eqid 2821 | . . . . . 6 ⊢ (0g‘(SymGrp‘∅)) = (0g‘(SymGrp‘∅)) | |
4 | 3 | gsum0 17888 | . . . . 5 ⊢ ((SymGrp‘∅) Σg ∅) = (0g‘(SymGrp‘∅)) |
5 | eqid 2821 | . . . . . . . . 9 ⊢ (SymGrp‘∅) = (SymGrp‘∅) | |
6 | 5 | symgid 18523 | . . . . . . . 8 ⊢ (∅ ∈ V → ( I ↾ ∅) = (0g‘(SymGrp‘∅))) |
7 | 1, 6 | ax-mp 5 | . . . . . . 7 ⊢ ( I ↾ ∅) = (0g‘(SymGrp‘∅)) |
8 | res0 5851 | . . . . . . 7 ⊢ ( I ↾ ∅) = ∅ | |
9 | 7, 8 | eqtr3i 2846 | . . . . . 6 ⊢ (0g‘(SymGrp‘∅)) = ∅ |
10 | 9 | a1i 11 | . . . . 5 ⊢ ((∅ ∈ V ∧ ∅ ∈ Word ran (pmTrsp‘∅)) → (0g‘(SymGrp‘∅)) = ∅) |
11 | 4, 10 | syl5req 2869 | . . . 4 ⊢ ((∅ ∈ V ∧ ∅ ∈ Word ran (pmTrsp‘∅)) → ∅ = ((SymGrp‘∅) Σg ∅)) |
12 | 11 | fveq2d 6668 | . . 3 ⊢ ((∅ ∈ V ∧ ∅ ∈ Word ran (pmTrsp‘∅)) → ((pmSgn‘∅)‘∅) = ((pmSgn‘∅)‘((SymGrp‘∅) Σg ∅))) |
13 | eqid 2821 | . . . 4 ⊢ ran (pmTrsp‘∅) = ran (pmTrsp‘∅) | |
14 | eqid 2821 | . . . 4 ⊢ (pmSgn‘∅) = (pmSgn‘∅) | |
15 | 5, 13, 14 | psgnvalii 18631 | . . 3 ⊢ ((∅ ∈ V ∧ ∅ ∈ Word ran (pmTrsp‘∅)) → ((pmSgn‘∅)‘((SymGrp‘∅) Σg ∅)) = (-1↑(♯‘∅))) |
16 | hash0 13722 | . . . . . 6 ⊢ (♯‘∅) = 0 | |
17 | 16 | oveq2i 7161 | . . . . 5 ⊢ (-1↑(♯‘∅)) = (-1↑0) |
18 | neg1cn 11745 | . . . . . 6 ⊢ -1 ∈ ℂ | |
19 | exp0 13427 | . . . . . 6 ⊢ (-1 ∈ ℂ → (-1↑0) = 1) | |
20 | 18, 19 | ax-mp 5 | . . . . 5 ⊢ (-1↑0) = 1 |
21 | 17, 20 | eqtri 2844 | . . . 4 ⊢ (-1↑(♯‘∅)) = 1 |
22 | 21 | a1i 11 | . . 3 ⊢ ((∅ ∈ V ∧ ∅ ∈ Word ran (pmTrsp‘∅)) → (-1↑(♯‘∅)) = 1) |
23 | 12, 15, 22 | 3eqtrd 2860 | . 2 ⊢ ((∅ ∈ V ∧ ∅ ∈ Word ran (pmTrsp‘∅)) → ((pmSgn‘∅)‘∅) = 1) |
24 | 1, 2, 23 | mp2an 690 | 1 ⊢ ((pmSgn‘∅)‘∅) = 1 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∅c0 4290 I cid 5453 ran crn 5550 ↾ cres 5551 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 0cc0 10531 1c1 10532 -cneg 10865 ↑cexp 13423 ♯chash 13684 Word cword 13855 0gc0g 16707 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-xor 1501 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-ot 4569 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 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-se 5509 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-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-tpos 7886 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-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 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-rp 12384 df-fz 12887 df-fzo 13028 df-seq 13364 df-exp 13424 df-hash 13685 df-word 13856 df-lsw 13909 df-concat 13917 df-s1 13944 df-substr 13997 df-pfx 14027 df-splice 14106 df-reverse 14115 df-s2 14204 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-0g 16709 df-gsum 16710 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-submnd 17951 df-efmnd 18028 df-grp 18100 df-minusg 18101 df-subg 18270 df-ghm 18350 df-gim 18393 df-oppg 18468 df-symg 18490 df-pmtr 18564 df-psgn 18613 |
This theorem is referenced by: mdet0pr 21195 |
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