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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 2820 | . . . . . 6 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁) | |
2 | zrhpsgnevpm.s | . . . . . 6 ⊢ 𝑆 = (pmSgn‘𝑁) | |
3 | eqid 2820 | . . . . . 6 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
4 | 1, 2, 3 | psgnghm2 20720 | . . . . 5 ⊢ (𝑁 ∈ Fin → 𝑆 ∈ ((SymGrp‘𝑁) GrpHom ((mulGrp‘ℂfld) ↾s {1, -1}))) |
5 | zrhpsgnodpm.p | . . . . . 6 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) | |
6 | eqid 2820 | . . . . . 6 ⊢ (Base‘((mulGrp‘ℂfld) ↾s {1, -1})) = (Base‘((mulGrp‘ℂfld) ↾s {1, -1})) | |
7 | 5, 6 | ghmf 18357 | . . . . 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 1129 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝑁))) → 𝑆:𝑃⟶(Base‘((mulGrp‘ℂfld) ↾s {1, -1}))) |
10 | eldifi 4096 | . . . 4 ⊢ (𝐹 ∈ (𝑃 ∖ (pmEven‘𝑁)) → 𝐹 ∈ 𝑃) | |
11 | 10 | 3ad2ant3 1130 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝑁))) → 𝐹 ∈ 𝑃) |
12 | fvco3 6753 | . . 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 20727 | . . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝑁))) → (𝑆‘𝐹) = -1) |
15 | 14 | 3adant1 1125 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝑁))) → (𝑆‘𝐹) = -1) |
16 | 15 | fveq2d 6667 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝑁))) → (𝑌‘(𝑆‘𝐹)) = (𝑌‘-1)) |
17 | zrhpsgnevpm.y | . . . . . . 7 ⊢ 𝑌 = (ℤRHom‘𝑅) | |
18 | 17 | zrhrhm 20654 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑌 ∈ (ℤring RingHom 𝑅)) |
19 | rhmghm 19472 | . . . . . 6 ⊢ (𝑌 ∈ (ℤring RingHom 𝑅) → 𝑌 ∈ (ℤring GrpHom 𝑅)) | |
20 | 18, 19 | syl 17 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑌 ∈ (ℤring GrpHom 𝑅)) |
21 | 1z 12006 | . . . . . 6 ⊢ 1 ∈ ℤ | |
22 | 21 | a1i 11 | . . . . 5 ⊢ (𝑅 ∈ Ring → 1 ∈ ℤ) |
23 | zringbas 20618 | . . . . . 6 ⊢ ℤ = (Base‘ℤring) | |
24 | eqid 2820 | . . . . . 6 ⊢ (invg‘ℤring) = (invg‘ℤring) | |
25 | zrhpsgnodpm.i | . . . . . 6 ⊢ 𝐼 = (invg‘𝑅) | |
26 | 23, 24, 25 | ghminv 18360 | . . . . 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 20629 | . . . . . . . 8 ⊢ (1 ∈ ℤ → -1 = ((invg‘ℤring)‘1)) | |
29 | 21, 28 | ax-mp 5 | . . . . . . 7 ⊢ -1 = ((invg‘ℤring)‘1) |
30 | 29 | eqcomi 2829 | . . . . . 6 ⊢ ((invg‘ℤring)‘1) = -1 |
31 | 30 | fveq2i 6666 | . . . . 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 20655 | . . . . 5 ⊢ (𝑅 ∈ Ring → (𝑌‘1) = 1 ) |
35 | 34 | fveq2d 6667 | . . . 4 ⊢ (𝑅 ∈ Ring → (𝐼‘(𝑌‘1)) = (𝐼‘ 1 )) |
36 | 27, 32, 35 | 3eqtr3d 2863 | . . 3 ⊢ (𝑅 ∈ Ring → (𝑌‘-1) = (𝐼‘ 1 )) |
37 | 36 | 3ad2ant1 1128 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ (𝑃 ∖ (pmEven‘𝑁))) → (𝑌‘-1) = (𝐼‘ 1 )) |
38 | 13, 16, 37 | 3eqtrd 2859 | 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 1082 = wceq 1536 ∈ wcel 2113 ∖ cdif 3926 {cpr 4562 ∘ ccom 5552 ⟶wf 6344 ‘cfv 6348 (class class class)co 7149 Fincfn 8502 1c1 10531 -cneg 10864 ℤcz 11975 Basecbs 16478 ↾s cress 16479 invgcminusg 18099 GrpHom cghm 18350 SymGrpcsymg 18490 pmSgncpsgn 18612 pmEvencevpm 18613 mulGrpcmgp 19234 1rcur 19246 Ringcrg 19292 RingHom crh 19459 ℂfldccnfld 20540 ℤringzring 20612 ℤRHomczrh 20642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-addf 10609 ax-mulf 10610 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-xor 1501 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-ot 4569 df-uni 4832 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 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 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-tpos 7885 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-2o 8096 df-oadd 8099 df-er 8282 df-map 8401 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-xnn0 11962 df-z 11976 df-dec 12093 df-uz 12238 df-rp 12384 df-fz 12890 df-fzo 13031 df-seq 13367 df-exp 13427 df-hash 13688 df-word 13859 df-lsw 13910 df-concat 13918 df-s1 13945 df-substr 13998 df-pfx 14028 df-splice 14107 df-reverse 14116 df-s2 14205 df-struct 16480 df-ndx 16481 df-slot 16482 df-base 16484 df-sets 16485 df-ress 16486 df-plusg 16573 df-mulr 16574 df-starv 16575 df-tset 16579 df-ple 16580 df-ds 16582 df-unif 16583 df-0g 16710 df-gsum 16711 df-mre 16852 df-mrc 16853 df-acs 16855 df-mgm 17847 df-sgrp 17896 df-mnd 17907 df-mhm 17951 df-submnd 17952 df-efmnd 18029 df-grp 18101 df-minusg 18102 df-mulg 18220 df-subg 18271 df-ghm 18351 df-gim 18394 df-oppg 18469 df-symg 18491 df-pmtr 18565 df-psgn 18614 df-evpm 18615 df-cmn 18903 df-abl 18904 df-mgp 19235 df-ur 19247 df-ring 19294 df-cring 19295 df-oppr 19368 df-dvdsr 19386 df-unit 19387 df-invr 19417 df-dvr 19428 df-rnghom 19462 df-drng 19499 df-subrg 19528 df-cnfld 20541 df-zring 20613 df-zrh 20646 |
This theorem is referenced by: mdetralt 21212 mdetunilem7 21222 |
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