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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 4718 | . 2 ⊢ ∅ ∈ V | |
2 | wrd0 13185 | . 2 ⊢ ∅ ∈ Word ran (pmTrsp‘∅) | |
3 | eqid 2610 | . . . . . 6 ⊢ (0g‘(SymGrp‘∅)) = (0g‘(SymGrp‘∅)) | |
4 | 3 | gsum0 17101 | . . . . 5 ⊢ ((SymGrp‘∅) Σg ∅) = (0g‘(SymGrp‘∅)) |
5 | eqid 2610 | . . . . . . . . 9 ⊢ (SymGrp‘∅) = (SymGrp‘∅) | |
6 | 5 | symgid 17644 | . . . . . . . 8 ⊢ (∅ ∈ V → ( I ↾ ∅) = (0g‘(SymGrp‘∅))) |
7 | 1, 6 | ax-mp 5 | . . . . . . 7 ⊢ ( I ↾ ∅) = (0g‘(SymGrp‘∅)) |
8 | res0 5321 | . . . . . . 7 ⊢ ( I ↾ ∅) = ∅ | |
9 | 7, 8 | eqtr3i 2634 | . . . . . 6 ⊢ (0g‘(SymGrp‘∅)) = ∅ |
10 | 9 | a1i 11 | . . . . 5 ⊢ ((∅ ∈ V ∧ ∅ ∈ Word ran (pmTrsp‘∅)) → (0g‘(SymGrp‘∅)) = ∅) |
11 | 4, 10 | syl5req 2657 | . . . 4 ⊢ ((∅ ∈ V ∧ ∅ ∈ Word ran (pmTrsp‘∅)) → ∅ = ((SymGrp‘∅) Σg ∅)) |
12 | 11 | fveq2d 6107 | . . 3 ⊢ ((∅ ∈ V ∧ ∅ ∈ Word ran (pmTrsp‘∅)) → ((pmSgn‘∅)‘∅) = ((pmSgn‘∅)‘((SymGrp‘∅) Σg ∅))) |
13 | eqid 2610 | . . . 4 ⊢ ran (pmTrsp‘∅) = ran (pmTrsp‘∅) | |
14 | eqid 2610 | . . . 4 ⊢ (pmSgn‘∅) = (pmSgn‘∅) | |
15 | 5, 13, 14 | psgnvalii 17752 | . . 3 ⊢ ((∅ ∈ V ∧ ∅ ∈ Word ran (pmTrsp‘∅)) → ((pmSgn‘∅)‘((SymGrp‘∅) Σg ∅)) = (-1↑(#‘∅))) |
16 | hash0 13019 | . . . . . 6 ⊢ (#‘∅) = 0 | |
17 | 16 | oveq2i 6560 | . . . . 5 ⊢ (-1↑(#‘∅)) = (-1↑0) |
18 | neg1cn 11001 | . . . . . 6 ⊢ -1 ∈ ℂ | |
19 | exp0 12726 | . . . . . 6 ⊢ (-1 ∈ ℂ → (-1↑0) = 1) | |
20 | 18, 19 | ax-mp 5 | . . . . 5 ⊢ (-1↑0) = 1 |
21 | 17, 20 | eqtri 2632 | . . . 4 ⊢ (-1↑(#‘∅)) = 1 |
22 | 21 | a1i 11 | . . 3 ⊢ ((∅ ∈ V ∧ ∅ ∈ Word ran (pmTrsp‘∅)) → (-1↑(#‘∅)) = 1) |
23 | 12, 15, 22 | 3eqtrd 2648 | . 2 ⊢ ((∅ ∈ V ∧ ∅ ∈ Word ran (pmTrsp‘∅)) → ((pmSgn‘∅)‘∅) = 1) |
24 | 1, 2, 23 | mp2an 704 | 1 ⊢ ((pmSgn‘∅)‘∅) = 1 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 I cid 4948 ran crn 5039 ↾ cres 5040 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 -cneg 10146 ↑cexp 12722 #chash 12979 Word cword 13146 0gc0g 15923 Σg cgsu 15924 SymGrpcsymg 17620 pmTrspcpmtr 17684 pmSgncpsgn 17732 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-xor 1457 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-ot 4134 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-word 13154 df-lsw 13155 df-concat 13156 df-s1 13157 df-substr 13158 df-splice 13159 df-reverse 13160 df-s2 13444 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-tset 15787 df-0g 15925 df-gsum 15926 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-subg 17414 df-ghm 17481 df-gim 17524 df-oppg 17599 df-symg 17621 df-pmtr 17685 df-psgn 17734 |
This theorem is referenced by: mdet0pr 20217 |
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