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Mirrors > Home > MPE Home > Th. List > zrhpsgnodpm | Structured version Visualization version GIF version |
Description: The sign of an odd permutation embedded into a ring is the additive inverse of 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) |
zrhpsgnodpm.p | ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) |
zrhpsgnodpm.i | ⊢ 𝐼 = (invg‘𝑅) |
Ref | Expression |
---|---|
zrhpsgnodpm | ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝑁))) → ((𝑌 ∘ 𝑆)‘𝐹) = (𝐼‘ 1 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . . . . 6 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁) | |
2 | zrhpsgnevpm.s | . . . . . 6 ⊢ 𝑆 = (pmSgn‘𝑁) | |
3 | eqid 2737 | . . . . . 6 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
4 | 1, 2, 3 | psgnghm2 20543 | . . . . 5 ⊢ (𝑁 ∈ Fin → 𝑆 ∈ ((SymGrp‘𝑁) GrpHom ((mulGrp‘ℂfld) ↾s {1, -1}))) |
5 | zrhpsgnodpm.p | . . . . . 6 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) | |
6 | eqid 2737 | . . . . . 6 ⊢ (Base‘((mulGrp‘ℂfld) ↾s {1, -1})) = (Base‘((mulGrp‘ℂfld) ↾s {1, -1})) | |
7 | 5, 6 | ghmf 18626 | . . . . 5 ⊢ (𝑆 ∈ ((SymGrp‘𝑁) GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})) → 𝑆:𝑃⟶(Base‘((mulGrp‘ℂfld) ↾s {1, -1}))) |
8 | 4, 7 | syl 17 | . . . 4 ⊢ (𝑁 ∈ Fin → 𝑆:𝑃⟶(Base‘((mulGrp‘ℂfld) ↾s {1, -1}))) |
9 | 8 | 3ad2ant2 1136 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝑁))) → 𝑆:𝑃⟶(Base‘((mulGrp‘ℂfld) ↾s {1, -1}))) |
10 | eldifi 4041 | . . . 4 ⊢ (𝐹 ∈ (𝑃 ∖ (pmEven‘𝑁)) → 𝐹 ∈ 𝑃) | |
11 | 10 | 3ad2ant3 1137 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝑁))) → 𝐹 ∈ 𝑃) |
12 | fvco3 6810 | . . 3 ⊢ ((𝑆:𝑃⟶(Base‘((mulGrp‘ℂfld) ↾s {1, -1})) ∧ 𝐹 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘𝐹) = (𝑌‘(𝑆‘𝐹))) | |
13 | 9, 11, 12 | syl2anc 587 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝑁))) → ((𝑌 ∘ 𝑆)‘𝐹) = (𝑌‘(𝑆‘𝐹))) |
14 | 1, 5, 2 | psgnodpm 20550 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝑁))) → (𝑆‘𝐹) = -1) |
15 | 14 | 3adant1 1132 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝑁))) → (𝑆‘𝐹) = -1) |
16 | 15 | fveq2d 6721 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝑁))) → (𝑌‘(𝑆‘𝐹)) = (𝑌‘-1)) |
17 | zrhpsgnevpm.y | . . . . . . 7 ⊢ 𝑌 = (ℤRHom‘𝑅) | |
18 | 17 | zrhrhm 20478 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑌 ∈ (ℤring RingHom 𝑅)) |
19 | rhmghm 19745 | . . . . . 6 ⊢ (𝑌 ∈ (ℤring RingHom 𝑅) → 𝑌 ∈ (ℤring GrpHom 𝑅)) | |
20 | 18, 19 | syl 17 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑌 ∈ (ℤring GrpHom 𝑅)) |
21 | 1z 12207 | . . . . . 6 ⊢ 1 ∈ ℤ | |
22 | 21 | a1i 11 | . . . . 5 ⊢ (𝑅 ∈ Ring → 1 ∈ ℤ) |
23 | zringbas 20441 | . . . . . 6 ⊢ ℤ = (Base‘ℤring) | |
24 | eqid 2737 | . . . . . 6 ⊢ (invg‘ℤring) = (invg‘ℤring) | |
25 | zrhpsgnodpm.i | . . . . . 6 ⊢ 𝐼 = (invg‘𝑅) | |
26 | 23, 24, 25 | ghminv 18629 | . . . . 5 ⊢ ((𝑌 ∈ (ℤring GrpHom 𝑅) ∧ 1 ∈ ℤ) → (𝑌‘((invg‘ℤring)‘1)) = (𝐼‘(𝑌‘1))) |
27 | 20, 22, 26 | syl2anc 587 | . . . 4 ⊢ (𝑅 ∈ Ring → (𝑌‘((invg‘ℤring)‘1)) = (𝐼‘(𝑌‘1))) |
28 | zringinvg 20452 | . . . . . . . 8 ⊢ (1 ∈ ℤ → -1 = ((invg‘ℤring)‘1)) | |
29 | 21, 28 | ax-mp 5 | . . . . . . 7 ⊢ -1 = ((invg‘ℤring)‘1) |
30 | 29 | eqcomi 2746 | . . . . . 6 ⊢ ((invg‘ℤring)‘1) = -1 |
31 | 30 | fveq2i 6720 | . . . . 5 ⊢ (𝑌‘((invg‘ℤring)‘1)) = (𝑌‘-1) |
32 | 31 | a1i 11 | . . . 4 ⊢ (𝑅 ∈ Ring → (𝑌‘((invg‘ℤring)‘1)) = (𝑌‘-1)) |
33 | zrhpsgnevpm.o | . . . . . 6 ⊢ 1 = (1r‘𝑅) | |
34 | 17, 33 | zrh1 20479 | . . . . 5 ⊢ (𝑅 ∈ Ring → (𝑌‘1) = 1 ) |
35 | 34 | fveq2d 6721 | . . . 4 ⊢ (𝑅 ∈ Ring → (𝐼‘(𝑌‘1)) = (𝐼‘ 1 )) |
36 | 27, 32, 35 | 3eqtr3d 2785 | . . 3 ⊢ (𝑅 ∈ Ring → (𝑌‘-1) = (𝐼‘ 1 )) |
37 | 36 | 3ad2ant1 1135 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝑁))) → (𝑌‘-1) = (𝐼‘ 1 )) |
38 | 13, 16, 37 | 3eqtrd 2781 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝑁))) → ((𝑌 ∘ 𝑆)‘𝐹) = (𝐼‘ 1 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ∖ cdif 3863 {cpr 4543 ∘ ccom 5555 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 Fincfn 8626 1c1 10730 -cneg 11063 ℤcz 12176 Basecbs 16760 ↾s cress 16784 invgcminusg 18366 GrpHom cghm 18619 SymGrpcsymg 18759 pmSgncpsgn 18881 pmEvencevpm 18882 mulGrpcmgp 19504 1rcur 19516 Ringcrg 19562 RingHom crh 19732 ℂfldccnfld 20363 ℤringzring 20435 ℤRHomczrh 20466 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-addf 10808 ax-mulf 10809 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-xor 1508 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-ot 4550 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-tpos 7968 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-xnn0 12163 df-z 12177 df-dec 12294 df-uz 12439 df-rp 12587 df-fz 13096 df-fzo 13239 df-seq 13575 df-exp 13636 df-hash 13897 df-word 14070 df-lsw 14118 df-concat 14126 df-s1 14153 df-substr 14206 df-pfx 14236 df-splice 14315 df-reverse 14324 df-s2 14413 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-starv 16817 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-0g 16946 df-gsum 16947 df-mre 17089 df-mrc 17090 df-acs 17092 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-mhm 18218 df-submnd 18219 df-efmnd 18296 df-grp 18368 df-minusg 18369 df-mulg 18489 df-subg 18540 df-ghm 18620 df-gim 18663 df-oppg 18738 df-symg 18760 df-pmtr 18834 df-psgn 18883 df-evpm 18884 df-cmn 19172 df-abl 19173 df-mgp 19505 df-ur 19517 df-ring 19564 df-cring 19565 df-oppr 19641 df-dvdsr 19659 df-unit 19660 df-invr 19690 df-dvr 19701 df-rnghom 19735 df-drng 19769 df-subrg 19798 df-cnfld 20364 df-zring 20436 df-zrh 20470 |
This theorem is referenced by: mdetralt 21505 mdetunilem7 21515 |
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