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Mirrors > Home > MPE Home > Th. List > zrhpsgnevpm | Structured version Visualization version GIF version |
Description: The sign of an even permutation embedded into a ring is the multiplicative neutral element of the ring. (Contributed by SO, 9-Jul-2018.) |
Ref | Expression |
---|---|
zrhpsgnevpm.y | ⊢ 𝑌 = (ℤRHom‘𝑅) |
zrhpsgnevpm.s | ⊢ 𝑆 = (pmSgn‘𝑁) |
zrhpsgnevpm.o | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
zrhpsgnevpm | ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (pmEven‘𝑁)) → ((𝑌 ∘ 𝑆)‘𝐹) = 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . . . . 6 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁) | |
2 | zrhpsgnevpm.s | . . . . . 6 ⊢ 𝑆 = (pmSgn‘𝑁) | |
3 | eqid 2818 | . . . . . 6 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
4 | 1, 2, 3 | psgnghm2 20653 | . . . . 5 ⊢ (𝑁 ∈ Fin → 𝑆 ∈ ((SymGrp‘𝑁) GrpHom ((mulGrp‘ℂfld) ↾s {1, -1}))) |
5 | eqid 2818 | . . . . . 6 ⊢ (Base‘(SymGrp‘𝑁)) = (Base‘(SymGrp‘𝑁)) | |
6 | eqid 2818 | . . . . . 6 ⊢ (Base‘((mulGrp‘ℂfld) ↾s {1, -1})) = (Base‘((mulGrp‘ℂfld) ↾s {1, -1})) | |
7 | 5, 6 | ghmf 18300 | . . . . 5 ⊢ (𝑆 ∈ ((SymGrp‘𝑁) GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})) → 𝑆:(Base‘(SymGrp‘𝑁))⟶(Base‘((mulGrp‘ℂfld) ↾s {1, -1}))) |
8 | 4, 7 | syl 17 | . . . 4 ⊢ (𝑁 ∈ Fin → 𝑆:(Base‘(SymGrp‘𝑁))⟶(Base‘((mulGrp‘ℂfld) ↾s {1, -1}))) |
9 | 8 | 3ad2ant2 1126 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (pmEven‘𝑁)) → 𝑆:(Base‘(SymGrp‘𝑁))⟶(Base‘((mulGrp‘ℂfld) ↾s {1, -1}))) |
10 | 1, 5 | evpmss 20658 | . . . . 5 ⊢ (pmEven‘𝑁) ⊆ (Base‘(SymGrp‘𝑁)) |
11 | 10 | sseli 3960 | . . . 4 ⊢ (𝐹 ∈ (pmEven‘𝑁) → 𝐹 ∈ (Base‘(SymGrp‘𝑁))) |
12 | 11 | 3ad2ant3 1127 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (pmEven‘𝑁)) → 𝐹 ∈ (Base‘(SymGrp‘𝑁))) |
13 | fvco3 6753 | . . 3 ⊢ ((𝑆:(Base‘(SymGrp‘𝑁))⟶(Base‘((mulGrp‘ℂfld) ↾s {1, -1})) ∧ 𝐹 ∈ (Base‘(SymGrp‘𝑁))) → ((𝑌 ∘ 𝑆)‘𝐹) = (𝑌‘(𝑆‘𝐹))) | |
14 | 9, 12, 13 | syl2anc 584 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (pmEven‘𝑁)) → ((𝑌 ∘ 𝑆)‘𝐹) = (𝑌‘(𝑆‘𝐹))) |
15 | 1, 5, 2 | psgnevpm 20661 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝐹 ∈ (pmEven‘𝑁)) → (𝑆‘𝐹) = 1) |
16 | 15 | 3adant1 1122 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (pmEven‘𝑁)) → (𝑆‘𝐹) = 1) |
17 | 16 | fveq2d 6667 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (pmEven‘𝑁)) → (𝑌‘(𝑆‘𝐹)) = (𝑌‘1)) |
18 | zrhpsgnevpm.y | . . . 4 ⊢ 𝑌 = (ℤRHom‘𝑅) | |
19 | zrhpsgnevpm.o | . . . 4 ⊢ 1 = (1r‘𝑅) | |
20 | 18, 19 | zrh1 20588 | . . 3 ⊢ (𝑅 ∈ Ring → (𝑌‘1) = 1 ) |
21 | 20 | 3ad2ant1 1125 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (pmEven‘𝑁)) → (𝑌‘1) = 1 ) |
22 | 14, 17, 21 | 3eqtrd 2857 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (pmEven‘𝑁)) → ((𝑌 ∘ 𝑆)‘𝐹) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 {cpr 4559 ∘ ccom 5552 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 Fincfn 8497 1c1 10526 -cneg 10859 Basecbs 16471 ↾s cress 16472 GrpHom cghm 18293 SymGrpcsymg 18433 pmSgncpsgn 18546 pmEvencevpm 18547 mulGrpcmgp 19168 1rcur 19180 Ringcrg 19226 ℂfldccnfld 20473 ℤRHomczrh 20575 |
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-mulg 18163 df-subg 18214 df-ghm 18294 df-gim 18337 df-oppg 18412 df-symg 18434 df-pmtr 18499 df-psgn 18548 df-evpm 18549 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-rnghom 19396 df-drng 19433 df-subrg 19462 df-cnfld 20474 df-zring 20546 df-zrh 20579 |
This theorem is referenced by: mdet0pr 21129 mdetralt 21145 |
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