Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > psgnid | Structured version Visualization version GIF version |
Description: Permutation sign of the identity. (Contributed by Thierry Arnoux, 21-Aug-2020.) |
Ref | Expression |
---|---|
psgnid.s | ⊢ 𝑆 = (pmSgn‘𝐷) |
Ref | Expression |
---|---|
psgnid | ⊢ (𝐷 ∈ Fin → (𝑆‘( I ↾ 𝐷)) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . . 4 ⊢ (SymGrp‘𝐷) = (SymGrp‘𝐷) | |
2 | 1 | symgid 18459 | . . 3 ⊢ (𝐷 ∈ Fin → ( I ↾ 𝐷) = (0g‘(SymGrp‘𝐷))) |
3 | 2 | fveq2d 6667 | . 2 ⊢ (𝐷 ∈ Fin → (𝑆‘( I ↾ 𝐷)) = (𝑆‘(0g‘(SymGrp‘𝐷)))) |
4 | psgnid.s | . . . 4 ⊢ 𝑆 = (pmSgn‘𝐷) | |
5 | eqid 2818 | . . . 4 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
6 | 1, 4, 5 | psgnghm2 20653 | . . 3 ⊢ (𝐷 ∈ Fin → 𝑆 ∈ ((SymGrp‘𝐷) GrpHom ((mulGrp‘ℂfld) ↾s {1, -1}))) |
7 | eqid 2818 | . . . 4 ⊢ (0g‘(SymGrp‘𝐷)) = (0g‘(SymGrp‘𝐷)) | |
8 | cnring 20495 | . . . . . 6 ⊢ ℂfld ∈ Ring | |
9 | eqid 2818 | . . . . . . 7 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
10 | 9 | ringmgp 19232 | . . . . . 6 ⊢ (ℂfld ∈ Ring → (mulGrp‘ℂfld) ∈ Mnd) |
11 | 8, 10 | ax-mp 5 | . . . . 5 ⊢ (mulGrp‘ℂfld) ∈ Mnd |
12 | 1ex 10625 | . . . . . 6 ⊢ 1 ∈ V | |
13 | 12 | prid1 4690 | . . . . 5 ⊢ 1 ∈ {1, -1} |
14 | ax-1cn 10583 | . . . . . 6 ⊢ 1 ∈ ℂ | |
15 | neg1cn 11739 | . . . . . 6 ⊢ -1 ∈ ℂ | |
16 | prssi 4746 | . . . . . 6 ⊢ ((1 ∈ ℂ ∧ -1 ∈ ℂ) → {1, -1} ⊆ ℂ) | |
17 | 14, 15, 16 | mp2an 688 | . . . . 5 ⊢ {1, -1} ⊆ ℂ |
18 | cnfldbas 20477 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
19 | 9, 18 | mgpbas 19174 | . . . . . 6 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
20 | cnfld1 20498 | . . . . . . 7 ⊢ 1 = (1r‘ℂfld) | |
21 | 9, 20 | ringidval 19182 | . . . . . 6 ⊢ 1 = (0g‘(mulGrp‘ℂfld)) |
22 | 5, 19, 21 | ress0g 17927 | . . . . 5 ⊢ (((mulGrp‘ℂfld) ∈ Mnd ∧ 1 ∈ {1, -1} ∧ {1, -1} ⊆ ℂ) → 1 = (0g‘((mulGrp‘ℂfld) ↾s {1, -1}))) |
23 | 11, 13, 17, 22 | mp3an 1452 | . . . 4 ⊢ 1 = (0g‘((mulGrp‘ℂfld) ↾s {1, -1})) |
24 | 7, 23 | ghmid 18302 | . . 3 ⊢ (𝑆 ∈ ((SymGrp‘𝐷) GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})) → (𝑆‘(0g‘(SymGrp‘𝐷))) = 1) |
25 | 6, 24 | syl 17 | . 2 ⊢ (𝐷 ∈ Fin → (𝑆‘(0g‘(SymGrp‘𝐷))) = 1) |
26 | 3, 25 | eqtrd 2853 | 1 ⊢ (𝐷 ∈ Fin → (𝑆‘( I ↾ 𝐷)) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 {cpr 4559 I cid 5452 ↾ cres 5550 ‘cfv 6348 (class class class)co 7145 Fincfn 8497 ℂcc 10523 1c1 10526 -cneg 10859 ↾s cress 16472 0gc0g 16701 Mndcmnd 17899 GrpHom cghm 18293 SymGrpcsymg 18433 pmSgncpsgn 18546 mulGrpcmgp 19168 Ringcrg 19226 ℂfldccnfld 20473 |
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 ax-addf 10604 ax-mulf 10605 |
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-dec 12087 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-mulr 16567 df-starv 16568 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 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 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-drng 19433 df-cnfld 20474 |
This theorem is referenced by: psgnfzto1st 30674 evpmid 30717 |
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