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Mirrors > Home > MPE Home > Th. List > psgnprfval1 | Structured version Visualization version GIF version |
Description: The permutation sign of the identity for a pair. (Contributed by AV, 11-Dec-2018.) |
Ref | Expression |
---|---|
psgnprfval.0 | ⊢ 𝐷 = {1, 2} |
psgnprfval.g | ⊢ 𝐺 = (SymGrp‘𝐷) |
psgnprfval.b | ⊢ 𝐵 = (Base‘𝐺) |
psgnprfval.t | ⊢ 𝑇 = ran (pmTrsp‘𝐷) |
psgnprfval.n | ⊢ 𝑁 = (pmSgn‘𝐷) |
Ref | Expression |
---|---|
psgnprfval1 | ⊢ (𝑁‘{〈1, 1〉, 〈2, 2〉}) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psgnprfval.0 | . . . . . . 7 ⊢ 𝐷 = {1, 2} | |
2 | prex 5323 | . . . . . . 7 ⊢ {1, 2} ∈ V | |
3 | 1, 2 | eqeltri 2906 | . . . . . 6 ⊢ 𝐷 ∈ V |
4 | psgnprfval.g | . . . . . . 7 ⊢ 𝐺 = (SymGrp‘𝐷) | |
5 | 4 | symgid 18459 | . . . . . 6 ⊢ (𝐷 ∈ V → ( I ↾ 𝐷) = (0g‘𝐺)) |
6 | 3, 5 | ax-mp 5 | . . . . 5 ⊢ ( I ↾ 𝐷) = (0g‘𝐺) |
7 | 6 | gsum0 17882 | . . . 4 ⊢ (𝐺 Σg ∅) = ( I ↾ 𝐷) |
8 | reseq2 5841 | . . . . . 6 ⊢ (𝐷 = {1, 2} → ( I ↾ 𝐷) = ( I ↾ {1, 2})) | |
9 | 1ex 10625 | . . . . . . 7 ⊢ 1 ∈ V | |
10 | 2nn 11698 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
11 | residpr 6897 | . . . . . . 7 ⊢ ((1 ∈ V ∧ 2 ∈ ℕ) → ( I ↾ {1, 2}) = {〈1, 1〉, 〈2, 2〉}) | |
12 | 9, 10, 11 | mp2an 688 | . . . . . 6 ⊢ ( I ↾ {1, 2}) = {〈1, 1〉, 〈2, 2〉} |
13 | 8, 12 | syl6eq 2869 | . . . . 5 ⊢ (𝐷 = {1, 2} → ( I ↾ 𝐷) = {〈1, 1〉, 〈2, 2〉}) |
14 | 1, 13 | ax-mp 5 | . . . 4 ⊢ ( I ↾ 𝐷) = {〈1, 1〉, 〈2, 2〉} |
15 | 7, 14 | eqtr2i 2842 | . . 3 ⊢ {〈1, 1〉, 〈2, 2〉} = (𝐺 Σg ∅) |
16 | 15 | fveq2i 6666 | . 2 ⊢ (𝑁‘{〈1, 1〉, 〈2, 2〉}) = (𝑁‘(𝐺 Σg ∅)) |
17 | wrd0 13877 | . . 3 ⊢ ∅ ∈ Word 𝑇 | |
18 | psgnprfval.t | . . . 4 ⊢ 𝑇 = ran (pmTrsp‘𝐷) | |
19 | psgnprfval.n | . . . 4 ⊢ 𝑁 = (pmSgn‘𝐷) | |
20 | 4, 18, 19 | psgnvalii 18566 | . . 3 ⊢ ((𝐷 ∈ V ∧ ∅ ∈ Word 𝑇) → (𝑁‘(𝐺 Σg ∅)) = (-1↑(♯‘∅))) |
21 | 3, 17, 20 | mp2an 688 | . 2 ⊢ (𝑁‘(𝐺 Σg ∅)) = (-1↑(♯‘∅)) |
22 | hash0 13716 | . . . 4 ⊢ (♯‘∅) = 0 | |
23 | 22 | oveq2i 7156 | . . 3 ⊢ (-1↑(♯‘∅)) = (-1↑0) |
24 | neg1cn 11739 | . . . 4 ⊢ -1 ∈ ℂ | |
25 | exp0 13421 | . . . 4 ⊢ (-1 ∈ ℂ → (-1↑0) = 1) | |
26 | 24, 25 | ax-mp 5 | . . 3 ⊢ (-1↑0) = 1 |
27 | 23, 26 | eqtri 2841 | . 2 ⊢ (-1↑(♯‘∅)) = 1 |
28 | 16, 21, 27 | 3eqtri 2845 | 1 ⊢ (𝑁‘{〈1, 1〉, 〈2, 2〉}) = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∅c0 4288 {cpr 4559 〈cop 4563 I cid 5452 ran crn 5549 ↾ cres 5550 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 0cc0 10525 1c1 10526 -cneg 10859 ℕcn 11626 2c2 11680 ↑cexp 13417 ♯chash 13678 Word cword 13849 Basecbs 16471 0gc0g 16701 Σg cgsu 16702 SymGrpcsymg 18433 pmTrspcpmtr 18498 pmSgncpsgn 18546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-xor 1496 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-ot 4566 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12881 df-fzo 13022 df-seq 13358 df-exp 13418 df-hash 13679 df-word 13850 df-lsw 13903 df-concat 13911 df-s1 13938 df-substr 13991 df-pfx 14021 df-splice 14100 df-reverse 14109 df-s2 14198 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-tset 16572 df-0g 16703 df-gsum 16704 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-submnd 17945 df-grp 18044 df-minusg 18045 df-subg 18214 df-ghm 18294 df-gim 18337 df-oppg 18412 df-symg 18434 df-pmtr 18499 df-psgn 18548 |
This theorem is referenced by: m2detleiblem1 21161 m2detleiblem5 21162 |
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