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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 2610 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
2 | 1 | gsum0 17101 | . . . 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 17639 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → 𝐵 = {{〈𝐴, 𝐴〉}}) |
7 | 6 | eleq2d 2673 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ {{〈𝐴, 𝐴〉}})) |
8 | 7 | biimpa 500 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ {{〈𝐴, 𝐴〉}}) |
9 | elsni 4142 | . . . . . 6 ⊢ (𝑋 ∈ {{〈𝐴, 𝐴〉}} → 𝑋 = {〈𝐴, 𝐴〉}) | |
10 | 5 | reseq2i 5314 | . . . . . . . . . 10 ⊢ ( I ↾ 𝐷) = ( I ↾ {𝐴}) |
11 | snex 4835 | . . . . . . . . . . . . 13 ⊢ {𝐴} ∈ V | |
12 | 11 | snid 4155 | . . . . . . . . . . . 12 ⊢ {𝐴} ∈ {{𝐴}} |
13 | 5, 12 | eqeltri 2684 | . . . . . . . . . . 11 ⊢ 𝐷 ∈ {{𝐴}} |
14 | 3 | symgid 17644 | . . . . . . . . . . 11 ⊢ (𝐷 ∈ {{𝐴}} → ( I ↾ 𝐷) = (0g‘𝐺)) |
15 | 13, 14 | mp1i 13 | . . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐷) = (0g‘𝐺)) |
16 | restidsing 5377 | . . . . . . . . . . 11 ⊢ ( I ↾ {𝐴}) = ({𝐴} × {𝐴}) | |
17 | xpsng 6312 | . . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ({𝐴} × {𝐴}) = {〈𝐴, 𝐴〉}) | |
18 | 17 | anidms 675 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} × {𝐴}) = {〈𝐴, 𝐴〉}) |
19 | 16, 18 | syl5eq 2656 | . . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ {𝐴}) = {〈𝐴, 𝐴〉}) |
20 | 10, 15, 19 | 3eqtr3a 2668 | . . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (0g‘𝐺) = {〈𝐴, 𝐴〉}) |
21 | 20 | adantr 480 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (0g‘𝐺) = {〈𝐴, 𝐴〉}) |
22 | id 22 | . . . . . . . . 9 ⊢ ({〈𝐴, 𝐴〉} = 𝑋 → {〈𝐴, 𝐴〉} = 𝑋) | |
23 | 22 | eqcoms 2618 | . . . . . . . 8 ⊢ (𝑋 = {〈𝐴, 𝐴〉} → {〈𝐴, 𝐴〉} = 𝑋) |
24 | 21, 23 | sylan9eqr 2666 | . . . . . . 7 ⊢ ((𝑋 = {〈𝐴, 𝐴〉} ∧ (𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵)) → (0g‘𝐺) = 𝑋) |
25 | 24 | ex 449 | . . . . . 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 37 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (0g‘𝐺) = 𝑋) |
28 | 2, 27 | syl5req 2657 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → 𝑋 = (𝐺 Σg ∅)) |
29 | 28 | fveq2d 6107 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = (𝑁‘(𝐺 Σg ∅))) |
30 | 5, 11 | eqeltri 2684 | . . . 4 ⊢ 𝐷 ∈ V |
31 | wrd0 13185 | . . . 4 ⊢ ∅ ∈ Word ∅ | |
32 | 30, 31 | pm3.2i 470 | . . 3 ⊢ (𝐷 ∈ V ∧ ∅ ∈ Word ∅) |
33 | 5 | fveq2i 6106 | . . . . . . 7 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘{𝐴}) |
34 | pmtrsn 17762 | . . . . . . 7 ⊢ (pmTrsp‘{𝐴}) = ∅ | |
35 | 33, 34 | eqtri 2632 | . . . . . 6 ⊢ (pmTrsp‘𝐷) = ∅ |
36 | 35 | rneqi 5273 | . . . . 5 ⊢ ran (pmTrsp‘𝐷) = ran ∅ |
37 | rn0 5298 | . . . . 5 ⊢ ran ∅ = ∅ | |
38 | 36, 37 | eqtr2i 2633 | . . . 4 ⊢ ∅ = ran (pmTrsp‘𝐷) |
39 | psgnsn.n | . . . 4 ⊢ 𝑁 = (pmSgn‘𝐷) | |
40 | 3, 38, 39 | psgnvalii 17752 | . . 3 ⊢ ((𝐷 ∈ V ∧ ∅ ∈ Word ∅) → (𝑁‘(𝐺 Σg ∅)) = (-1↑(#‘∅))) |
41 | 32, 40 | mp1i 13 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝐺 Σg ∅)) = (-1↑(#‘∅))) |
42 | hash0 13019 | . . . . 5 ⊢ (#‘∅) = 0 | |
43 | 42 | oveq2i 6560 | . . . 4 ⊢ (-1↑(#‘∅)) = (-1↑0) |
44 | neg1cn 11001 | . . . . 5 ⊢ -1 ∈ ℂ | |
45 | exp0 12726 | . . . . 5 ⊢ (-1 ∈ ℂ → (-1↑0) = 1) | |
46 | 44, 45 | ax-mp 5 | . . . 4 ⊢ (-1↑0) = 1 |
47 | 43, 46 | eqtri 2632 | . . 3 ⊢ (-1↑(#‘∅)) = 1 |
48 | 47 | a1i 11 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (-1↑(#‘∅)) = 1) |
49 | 29, 41, 48 | 3eqtrd 2648 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 {csn 4125 〈cop 4131 I cid 4948 × cxp 5036 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 Basecbs 15695 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: m1detdiag 20222 |
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