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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 unity 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 2736 | . . . . . 6 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁) | |
2 | zrhpsgnevpm.s | . . . . . 6 ⊢ 𝑆 = (pmSgn‘𝑁) | |
3 | eqid 2736 | . . . . . 6 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
4 | 1, 2, 3 | psgnghm2 20970 | . . . . 5 ⊢ (𝑁 ∈ Fin → 𝑆 ∈ ((SymGrp‘𝑁) GrpHom ((mulGrp‘ℂfld) ↾s {1, -1}))) |
5 | eqid 2736 | . . . . . 6 ⊢ (Base‘(SymGrp‘𝑁)) = (Base‘(SymGrp‘𝑁)) | |
6 | eqid 2736 | . . . . . 6 ⊢ (Base‘((mulGrp‘ℂfld) ↾s {1, -1})) = (Base‘((mulGrp‘ℂfld) ↾s {1, -1})) | |
7 | 5, 6 | ghmf 19003 | . . . . 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 1134 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (pmEven‘𝑁)) → 𝑆:(Base‘(SymGrp‘𝑁))⟶(Base‘((mulGrp‘ℂfld) ↾s {1, -1}))) |
10 | 1, 5 | evpmss 20975 | . . . . 5 ⊢ (pmEven‘𝑁) ⊆ (Base‘(SymGrp‘𝑁)) |
11 | 10 | sseli 3938 | . . . 4 ⊢ (𝐹 ∈ (pmEven‘𝑁) → 𝐹 ∈ (Base‘(SymGrp‘𝑁))) |
12 | 11 | 3ad2ant3 1135 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (pmEven‘𝑁)) → 𝐹 ∈ (Base‘(SymGrp‘𝑁))) |
13 | fvco3 6937 | . . 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 20978 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝐹 ∈ (pmEven‘𝑁)) → (𝑆‘𝐹) = 1) |
16 | 15 | 3adant1 1130 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (pmEven‘𝑁)) → (𝑆‘𝐹) = 1) |
17 | 16 | fveq2d 6843 | . 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 20898 | . . 3 ⊢ (𝑅 ∈ Ring → (𝑌‘1) = 1 ) |
21 | 20 | 3ad2ant1 1133 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (pmEven‘𝑁)) → (𝑌‘1) = 1 ) |
22 | 14, 17, 21 | 3eqtrd 2780 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (pmEven‘𝑁)) → ((𝑌 ∘ 𝑆)‘𝐹) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 {cpr 4586 ∘ ccom 5635 ⟶wf 6489 ‘cfv 6493 (class class class)co 7353 Fincfn 8879 1c1 11048 -cneg 11382 Basecbs 17075 ↾s cress 17104 GrpHom cghm 18996 SymGrpcsymg 19139 pmSgncpsgn 19262 pmEvencevpm 19263 mulGrpcmgp 19887 1rcur 19904 Ringcrg 19950 ℂfldccnfld 20781 ℤRHomczrh 20885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 ax-addf 11126 ax-mulf 11127 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-xor 1510 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-ot 4593 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-1st 7917 df-2nd 7918 df-tpos 8153 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-1o 8408 df-2o 8409 df-er 8644 df-map 8763 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-card 9871 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-div 11809 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12410 df-xnn0 12482 df-z 12496 df-dec 12615 df-uz 12760 df-rp 12908 df-fz 13417 df-fzo 13560 df-seq 13899 df-exp 13960 df-hash 14223 df-word 14395 df-lsw 14443 df-concat 14451 df-s1 14476 df-substr 14521 df-pfx 14551 df-splice 14630 df-reverse 14639 df-s2 14729 df-struct 17011 df-sets 17028 df-slot 17046 df-ndx 17058 df-base 17076 df-ress 17105 df-plusg 17138 df-mulr 17139 df-starv 17140 df-tset 17144 df-ple 17145 df-ds 17147 df-unif 17148 df-0g 17315 df-gsum 17316 df-mre 17458 df-mrc 17459 df-acs 17461 df-mgm 18489 df-sgrp 18538 df-mnd 18549 df-mhm 18593 df-submnd 18594 df-efmnd 18671 df-grp 18743 df-minusg 18744 df-mulg 18864 df-subg 18916 df-ghm 18997 df-gim 19040 df-oppg 19115 df-symg 19140 df-pmtr 19215 df-psgn 19264 df-evpm 19265 df-cmn 19555 df-abl 19556 df-mgp 19888 df-ur 19905 df-ring 19952 df-cring 19953 df-oppr 20034 df-dvdsr 20055 df-unit 20056 df-invr 20086 df-dvr 20097 df-rnghom 20131 df-drng 20172 df-subrg 20205 df-cnfld 20782 df-zring 20855 df-zrh 20889 |
This theorem is referenced by: mdet0pr 21925 mdetralt 21941 |
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