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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 2823 | . . . . . 6 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁) | |
2 | zrhpsgnevpm.s | . . . . . 6 ⊢ 𝑆 = (pmSgn‘𝑁) | |
3 | eqid 2823 | . . . . . 6 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
4 | 1, 2, 3 | psgnghm2 20727 | . . . . 5 ⊢ (𝑁 ∈ Fin → 𝑆 ∈ ((SymGrp‘𝑁) GrpHom ((mulGrp‘ℂfld) ↾s {1, -1}))) |
5 | zrhpsgnodpm.p | . . . . . 6 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) | |
6 | eqid 2823 | . . . . . 6 ⊢ (Base‘((mulGrp‘ℂfld) ↾s {1, -1})) = (Base‘((mulGrp‘ℂfld) ↾s {1, -1})) | |
7 | 5, 6 | ghmf 18364 | . . . . 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 1130 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝑁))) → 𝑆:𝑃⟶(Base‘((mulGrp‘ℂfld) ↾s {1, -1}))) |
10 | eldifi 4105 | . . . 4 ⊢ (𝐹 ∈ (𝑃 ∖ (pmEven‘𝑁)) → 𝐹 ∈ 𝑃) | |
11 | 10 | 3ad2ant3 1131 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝑁))) → 𝐹 ∈ 𝑃) |
12 | fvco3 6762 | . . 3 ⊢ ((𝑆:𝑃⟶(Base‘((mulGrp‘ℂfld) ↾s {1, -1})) ∧ 𝐹 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘𝐹) = (𝑌‘(𝑆‘𝐹))) | |
13 | 9, 11, 12 | syl2anc 586 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝑁))) → ((𝑌 ∘ 𝑆)‘𝐹) = (𝑌‘(𝑆‘𝐹))) |
14 | 1, 5, 2 | psgnodpm 20734 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝑁))) → (𝑆‘𝐹) = -1) |
15 | 14 | 3adant1 1126 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝑁))) → (𝑆‘𝐹) = -1) |
16 | 15 | fveq2d 6676 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝑁))) → (𝑌‘(𝑆‘𝐹)) = (𝑌‘-1)) |
17 | zrhpsgnevpm.y | . . . . . . 7 ⊢ 𝑌 = (ℤRHom‘𝑅) | |
18 | 17 | zrhrhm 20661 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑌 ∈ (ℤring RingHom 𝑅)) |
19 | rhmghm 19479 | . . . . . 6 ⊢ (𝑌 ∈ (ℤring RingHom 𝑅) → 𝑌 ∈ (ℤring GrpHom 𝑅)) | |
20 | 18, 19 | syl 17 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑌 ∈ (ℤring GrpHom 𝑅)) |
21 | 1z 12015 | . . . . . 6 ⊢ 1 ∈ ℤ | |
22 | 21 | a1i 11 | . . . . 5 ⊢ (𝑅 ∈ Ring → 1 ∈ ℤ) |
23 | zringbas 20625 | . . . . . 6 ⊢ ℤ = (Base‘ℤring) | |
24 | eqid 2823 | . . . . . 6 ⊢ (invg‘ℤring) = (invg‘ℤring) | |
25 | zrhpsgnodpm.i | . . . . . 6 ⊢ 𝐼 = (invg‘𝑅) | |
26 | 23, 24, 25 | ghminv 18367 | . . . . 5 ⊢ ((𝑌 ∈ (ℤring GrpHom 𝑅) ∧ 1 ∈ ℤ) → (𝑌‘((invg‘ℤring)‘1)) = (𝐼‘(𝑌‘1))) |
27 | 20, 22, 26 | syl2anc 586 | . . . 4 ⊢ (𝑅 ∈ Ring → (𝑌‘((invg‘ℤring)‘1)) = (𝐼‘(𝑌‘1))) |
28 | zringinvg 20636 | . . . . . . . 8 ⊢ (1 ∈ ℤ → -1 = ((invg‘ℤring)‘1)) | |
29 | 21, 28 | ax-mp 5 | . . . . . . 7 ⊢ -1 = ((invg‘ℤring)‘1) |
30 | 29 | eqcomi 2832 | . . . . . 6 ⊢ ((invg‘ℤring)‘1) = -1 |
31 | 30 | fveq2i 6675 | . . . . 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 20662 | . . . . 5 ⊢ (𝑅 ∈ Ring → (𝑌‘1) = 1 ) |
35 | 34 | fveq2d 6676 | . . . 4 ⊢ (𝑅 ∈ Ring → (𝐼‘(𝑌‘1)) = (𝐼‘ 1 )) |
36 | 27, 32, 35 | 3eqtr3d 2866 | . . 3 ⊢ (𝑅 ∈ Ring → (𝑌‘-1) = (𝐼‘ 1 )) |
37 | 36 | 3ad2ant1 1129 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝑁))) → (𝑌‘-1) = (𝐼‘ 1 )) |
38 | 13, 16, 37 | 3eqtrd 2862 | 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 1083 = wceq 1537 ∈ wcel 2114 ∖ cdif 3935 {cpr 4571 ∘ ccom 5561 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 Fincfn 8511 1c1 10540 -cneg 10873 ℤcz 11984 Basecbs 16485 ↾s cress 16486 invgcminusg 18106 GrpHom cghm 18357 SymGrpcsymg 18497 pmSgncpsgn 18619 pmEvencevpm 18620 mulGrpcmgp 19241 1rcur 19253 Ringcrg 19299 RingHom crh 19466 ℂfldccnfld 20547 ℤringzring 20619 ℤRHomczrh 20649 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-xor 1502 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-ot 4578 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-xnn0 11971 df-z 11985 df-dec 12102 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-word 13865 df-lsw 13917 df-concat 13925 df-s1 13952 df-substr 14005 df-pfx 14035 df-splice 14114 df-reverse 14123 df-s2 14212 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-0g 16717 df-gsum 16718 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-efmnd 18036 df-grp 18108 df-minusg 18109 df-mulg 18227 df-subg 18278 df-ghm 18358 df-gim 18401 df-oppg 18476 df-symg 18498 df-pmtr 18572 df-psgn 18621 df-evpm 18622 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-dvr 19435 df-rnghom 19469 df-drng 19506 df-subrg 19535 df-cnfld 20548 df-zring 20620 df-zrh 20653 |
This theorem is referenced by: mdetralt 21219 mdetunilem7 21229 |
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