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Mirrors > Home > MPE Home > Th. List > zrhpsgnmhm | Structured version Visualization version GIF version |
Description: Embedding of permutation signs into an arbitrary ring is a homomorphism. (Contributed by SO, 9-Jul-2018.) |
Ref | Expression |
---|---|
zrhpsgnmhm | ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ Fin) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝐴)) ∈ ((SymGrp‘𝐴) MndHom (mulGrp‘𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . . 4 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅) | |
2 | 1 | zrhrhm 20587 | . . 3 ⊢ (𝑅 ∈ Ring → (ℤRHom‘𝑅) ∈ (ℤring RingHom 𝑅)) |
3 | eqid 2818 | . . . 4 ⊢ (mulGrp‘ℤring) = (mulGrp‘ℤring) | |
4 | eqid 2818 | . . . 4 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
5 | 3, 4 | rhmmhm 19403 | . . 3 ⊢ ((ℤRHom‘𝑅) ∈ (ℤring RingHom 𝑅) → (ℤRHom‘𝑅) ∈ ((mulGrp‘ℤring) MndHom (mulGrp‘𝑅))) |
6 | 2, 5 | syl 17 | . 2 ⊢ (𝑅 ∈ Ring → (ℤRHom‘𝑅) ∈ ((mulGrp‘ℤring) MndHom (mulGrp‘𝑅))) |
7 | eqid 2818 | . . . . 5 ⊢ (SymGrp‘𝐴) = (SymGrp‘𝐴) | |
8 | eqid 2818 | . . . . 5 ⊢ (pmSgn‘𝐴) = (pmSgn‘𝐴) | |
9 | eqid 2818 | . . . . 5 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
10 | 7, 8, 9 | psgnghm2 20653 | . . . 4 ⊢ (𝐴 ∈ Fin → (pmSgn‘𝐴) ∈ ((SymGrp‘𝐴) GrpHom ((mulGrp‘ℂfld) ↾s {1, -1}))) |
11 | ghmmhm 18306 | . . . 4 ⊢ ((pmSgn‘𝐴) ∈ ((SymGrp‘𝐴) GrpHom ((mulGrp‘ℂfld) ↾s {1, -1})) → (pmSgn‘𝐴) ∈ ((SymGrp‘𝐴) MndHom ((mulGrp‘ℂfld) ↾s {1, -1}))) | |
12 | 10, 11 | syl 17 | . . 3 ⊢ (𝐴 ∈ Fin → (pmSgn‘𝐴) ∈ ((SymGrp‘𝐴) MndHom ((mulGrp‘ℂfld) ↾s {1, -1}))) |
13 | eqid 2818 | . . . . . . . 8 ⊢ ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) | |
14 | 13 | cnmsgnsubg 20649 | . . . . . . 7 ⊢ {1, -1} ∈ (SubGrp‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) |
15 | subgsubm 18239 | . . . . . . 7 ⊢ ({1, -1} ∈ (SubGrp‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) → {1, -1} ∈ (SubMnd‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})))) | |
16 | 14, 15 | ax-mp 5 | . . . . . 6 ⊢ {1, -1} ∈ (SubMnd‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) |
17 | cnring 20495 | . . . . . . 7 ⊢ ℂfld ∈ Ring | |
18 | cnfldbas 20477 | . . . . . . . . 9 ⊢ ℂ = (Base‘ℂfld) | |
19 | cnfld0 20497 | . . . . . . . . 9 ⊢ 0 = (0g‘ℂfld) | |
20 | cndrng 20502 | . . . . . . . . 9 ⊢ ℂfld ∈ DivRing | |
21 | 18, 19, 20 | drngui 19437 | . . . . . . . 8 ⊢ (ℂ ∖ {0}) = (Unit‘ℂfld) |
22 | eqid 2818 | . . . . . . . 8 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
23 | 21, 22 | unitsubm 19349 | . . . . . . 7 ⊢ (ℂfld ∈ Ring → (ℂ ∖ {0}) ∈ (SubMnd‘(mulGrp‘ℂfld))) |
24 | 13 | subsubm 17969 | . . . . . . 7 ⊢ ((ℂ ∖ {0}) ∈ (SubMnd‘(mulGrp‘ℂfld)) → ({1, -1} ∈ (SubMnd‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) ↔ ({1, -1} ∈ (SubMnd‘(mulGrp‘ℂfld)) ∧ {1, -1} ⊆ (ℂ ∖ {0})))) |
25 | 17, 23, 24 | mp2b 10 | . . . . . 6 ⊢ ({1, -1} ∈ (SubMnd‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) ↔ ({1, -1} ∈ (SubMnd‘(mulGrp‘ℂfld)) ∧ {1, -1} ⊆ (ℂ ∖ {0}))) |
26 | 16, 25 | mpbi 231 | . . . . 5 ⊢ ({1, -1} ∈ (SubMnd‘(mulGrp‘ℂfld)) ∧ {1, -1} ⊆ (ℂ ∖ {0})) |
27 | 26 | simpli 484 | . . . 4 ⊢ {1, -1} ∈ (SubMnd‘(mulGrp‘ℂfld)) |
28 | 1z 12000 | . . . . 5 ⊢ 1 ∈ ℤ | |
29 | neg1z 12006 | . . . . 5 ⊢ -1 ∈ ℤ | |
30 | prssi 4746 | . . . . 5 ⊢ ((1 ∈ ℤ ∧ -1 ∈ ℤ) → {1, -1} ⊆ ℤ) | |
31 | 28, 29, 30 | mp2an 688 | . . . 4 ⊢ {1, -1} ⊆ ℤ |
32 | zsubrg 20526 | . . . . 5 ⊢ ℤ ∈ (SubRing‘ℂfld) | |
33 | 22 | subrgsubm 19477 | . . . . 5 ⊢ (ℤ ∈ (SubRing‘ℂfld) → ℤ ∈ (SubMnd‘(mulGrp‘ℂfld))) |
34 | zringmpg 20567 | . . . . . . 7 ⊢ ((mulGrp‘ℂfld) ↾s ℤ) = (mulGrp‘ℤring) | |
35 | 34 | eqcomi 2827 | . . . . . 6 ⊢ (mulGrp‘ℤring) = ((mulGrp‘ℂfld) ↾s ℤ) |
36 | 35 | subsubm 17969 | . . . . 5 ⊢ (ℤ ∈ (SubMnd‘(mulGrp‘ℂfld)) → ({1, -1} ∈ (SubMnd‘(mulGrp‘ℤring)) ↔ ({1, -1} ∈ (SubMnd‘(mulGrp‘ℂfld)) ∧ {1, -1} ⊆ ℤ))) |
37 | 32, 33, 36 | mp2b 10 | . . . 4 ⊢ ({1, -1} ∈ (SubMnd‘(mulGrp‘ℤring)) ↔ ({1, -1} ∈ (SubMnd‘(mulGrp‘ℂfld)) ∧ {1, -1} ⊆ ℤ)) |
38 | 27, 31, 37 | mpbir2an 707 | . . 3 ⊢ {1, -1} ∈ (SubMnd‘(mulGrp‘ℤring)) |
39 | zex 11978 | . . . . . 6 ⊢ ℤ ∈ V | |
40 | ressabs 16551 | . . . . . 6 ⊢ ((ℤ ∈ V ∧ {1, -1} ⊆ ℤ) → (((mulGrp‘ℂfld) ↾s ℤ) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1})) | |
41 | 39, 31, 40 | mp2an 688 | . . . . 5 ⊢ (((mulGrp‘ℂfld) ↾s ℤ) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) |
42 | 34 | oveq1i 7155 | . . . . 5 ⊢ (((mulGrp‘ℂfld) ↾s ℤ) ↾s {1, -1}) = ((mulGrp‘ℤring) ↾s {1, -1}) |
43 | 41, 42 | eqtr3i 2843 | . . . 4 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℤring) ↾s {1, -1}) |
44 | 43 | resmhm2 17974 | . . 3 ⊢ (((pmSgn‘𝐴) ∈ ((SymGrp‘𝐴) MndHom ((mulGrp‘ℂfld) ↾s {1, -1})) ∧ {1, -1} ∈ (SubMnd‘(mulGrp‘ℤring))) → (pmSgn‘𝐴) ∈ ((SymGrp‘𝐴) MndHom (mulGrp‘ℤring))) |
45 | 12, 38, 44 | sylancl 586 | . 2 ⊢ (𝐴 ∈ Fin → (pmSgn‘𝐴) ∈ ((SymGrp‘𝐴) MndHom (mulGrp‘ℤring))) |
46 | mhmco 17976 | . 2 ⊢ (((ℤRHom‘𝑅) ∈ ((mulGrp‘ℤring) MndHom (mulGrp‘𝑅)) ∧ (pmSgn‘𝐴) ∈ ((SymGrp‘𝐴) MndHom (mulGrp‘ℤring))) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝐴)) ∈ ((SymGrp‘𝐴) MndHom (mulGrp‘𝑅))) | |
47 | 6, 45, 46 | syl2an 595 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ Fin) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝐴)) ∈ ((SymGrp‘𝐴) MndHom (mulGrp‘𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∖ cdif 3930 ⊆ wss 3933 {csn 4557 {cpr 4559 ∘ ccom 5552 ‘cfv 6348 (class class class)co 7145 Fincfn 8497 ℂcc 10523 0cc0 10525 1c1 10526 -cneg 10859 ℤcz 11969 ↾s cress 16472 MndHom cmhm 17942 SubMndcsubmnd 17943 SubGrpcsubg 18211 GrpHom cghm 18293 SymGrpcsymg 18433 pmSgncpsgn 18546 mulGrpcmgp 19168 Ringcrg 19226 RingHom crh 19393 SubRingcsubrg 19460 ℂfldccnfld 20473 ℤringzring 20545 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-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: madetsumid 20998 mdetleib2 21125 mdetf 21132 mdetdiaglem 21135 mdetrlin 21139 mdetrsca 21140 mdetralt 21145 mdetunilem7 21155 mdetunilem8 21156 |
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