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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 2797 | . . . 4 ⊢ (SymGrp‘𝐷) = (SymGrp‘𝐷) | |
2 | 1 | symgid 18264 | . . 3 ⊢ (𝐷 ∈ Fin → ( I ↾ 𝐷) = (0g‘(SymGrp‘𝐷))) |
3 | 2 | fveq2d 6549 | . 2 ⊢ (𝐷 ∈ Fin → (𝑆‘( I ↾ 𝐷)) = (𝑆‘(0g‘(SymGrp‘𝐷)))) |
4 | psgnid.s | . . . 4 ⊢ 𝑆 = (pmSgn‘𝐷) | |
5 | eqid 2797 | . . . 4 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
6 | 1, 4, 5 | psgnghm2 20411 | . . 3 ⊢ (𝐷 ∈ Fin → 𝑆 ∈ ((SymGrp‘𝐷) GrpHom ((mulGrp‘ℂfld) ↾s {1, -1}))) |
7 | eqid 2797 | . . . 4 ⊢ (0g‘(SymGrp‘𝐷)) = (0g‘(SymGrp‘𝐷)) | |
8 | cnring 20253 | . . . . . 6 ⊢ ℂfld ∈ Ring | |
9 | eqid 2797 | . . . . . . 7 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
10 | 9 | ringmgp 18997 | . . . . . 6 ⊢ (ℂfld ∈ Ring → (mulGrp‘ℂfld) ∈ Mnd) |
11 | 8, 10 | ax-mp 5 | . . . . 5 ⊢ (mulGrp‘ℂfld) ∈ Mnd |
12 | 1ex 10490 | . . . . . 6 ⊢ 1 ∈ V | |
13 | 12 | prid1 4611 | . . . . 5 ⊢ 1 ∈ {1, -1} |
14 | ax-1cn 10448 | . . . . . 6 ⊢ 1 ∈ ℂ | |
15 | neg1cn 11605 | . . . . . 6 ⊢ -1 ∈ ℂ | |
16 | prssi 4667 | . . . . . 6 ⊢ ((1 ∈ ℂ ∧ -1 ∈ ℂ) → {1, -1} ⊆ ℂ) | |
17 | 14, 15, 16 | mp2an 688 | . . . . 5 ⊢ {1, -1} ⊆ ℂ |
18 | cnfldbas 20235 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
19 | 9, 18 | mgpbas 18939 | . . . . . 6 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
20 | cnfld1 20256 | . . . . . . 7 ⊢ 1 = (1r‘ℂfld) | |
21 | 9, 20 | ringidval 18947 | . . . . . 6 ⊢ 1 = (0g‘(mulGrp‘ℂfld)) |
22 | 5, 19, 21 | ress0g 17762 | . . . . 5 ⊢ (((mulGrp‘ℂfld) ∈ Mnd ∧ 1 ∈ {1, -1} ∧ {1, -1} ⊆ ℂ) → 1 = (0g‘((mulGrp‘ℂfld) ↾s {1, -1}))) |
23 | 11, 13, 17, 22 | mp3an 1453 | . . . 4 ⊢ 1 = (0g‘((mulGrp‘ℂfld) ↾s {1, -1})) |
24 | 7, 23 | ghmid 18109 | . . 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 2833 | 1 ⊢ (𝐷 ∈ Fin → (𝑆‘( I ↾ 𝐷)) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1525 ∈ wcel 2083 ⊆ wss 3865 {cpr 4480 I cid 5354 ↾ cres 5452 ‘cfv 6232 (class class class)co 7023 Fincfn 8364 ℂcc 10388 1c1 10391 -cneg 10724 ↾s cress 16317 0gc0g 16546 Mndcmnd 17737 GrpHom cghm 18100 SymGrpcsymg 18240 pmSgncpsgn 18352 mulGrpcmgp 18933 Ringcrg 18991 ℂfldccnfld 20231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 ax-addf 10469 ax-mulf 10470 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-xor 1497 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-ot 4487 df-uni 4752 df-int 4789 df-iun 4833 df-iin 4834 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-se 5410 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-isom 6241 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-1st 7552 df-2nd 7553 df-tpos 7750 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-2o 7961 df-oadd 7964 df-er 8146 df-map 8265 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-card 9221 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-div 11152 df-nn 11493 df-2 11554 df-3 11555 df-4 11556 df-5 11557 df-6 11558 df-7 11559 df-8 11560 df-9 11561 df-n0 11752 df-xnn0 11822 df-z 11836 df-dec 11953 df-uz 12098 df-rp 12244 df-fz 12747 df-fzo 12888 df-seq 13224 df-exp 13284 df-hash 13545 df-word 13712 df-lsw 13765 df-concat 13773 df-s1 13798 df-substr 13843 df-pfx 13873 df-splice 13952 df-reverse 13961 df-s2 14050 df-struct 16318 df-ndx 16319 df-slot 16320 df-base 16322 df-sets 16323 df-ress 16324 df-plusg 16411 df-mulr 16412 df-starv 16413 df-tset 16417 df-ple 16418 df-ds 16420 df-unif 16421 df-0g 16548 df-gsum 16549 df-mre 16690 df-mrc 16691 df-acs 16693 df-mgm 17685 df-sgrp 17727 df-mnd 17738 df-mhm 17778 df-submnd 17779 df-grp 17868 df-minusg 17869 df-subg 18034 df-ghm 18101 df-gim 18144 df-oppg 18219 df-symg 18241 df-pmtr 18305 df-psgn 18354 df-cmn 18639 df-abl 18640 df-mgp 18934 df-ur 18946 df-ring 18993 df-cring 18994 df-oppr 19067 df-dvdsr 19085 df-unit 19086 df-invr 19116 df-dvr 19127 df-drng 19198 df-cnfld 20232 |
This theorem is referenced by: evpmid 30424 psgnfzto1st 30665 |
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