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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 2725 | . . . 4 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅) | |
2 | 1 | zrhrhm 21441 | . . 3 ⊢ (𝑅 ∈ Ring → (ℤRHom‘𝑅) ∈ (ℤring RingHom 𝑅)) |
3 | eqid 2725 | . . . 4 ⊢ (mulGrp‘ℤring) = (mulGrp‘ℤring) | |
4 | eqid 2725 | . . . 4 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
5 | 3, 4 | rhmmhm 20422 | . . 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 2725 | . . . . 5 ⊢ (SymGrp‘𝐴) = (SymGrp‘𝐴) | |
8 | eqid 2725 | . . . . 5 ⊢ (pmSgn‘𝐴) = (pmSgn‘𝐴) | |
9 | eqid 2725 | . . . . 5 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
10 | 7, 8, 9 | psgnghm2 21517 | . . . 4 ⊢ (𝐴 ∈ Fin → (pmSgn‘𝐴) ∈ ((SymGrp‘𝐴) GrpHom ((mulGrp‘ℂfld) ↾s {1, -1}))) |
11 | ghmmhm 19184 | . . . 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 2725 | . . . . . . . 8 ⊢ ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) | |
14 | 13 | cnmsgnsubg 21513 | . . . . . . 7 ⊢ {1, -1} ∈ (SubGrp‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) |
15 | subgsubm 19107 | . . . . . . 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 21322 | . . . . . . 7 ⊢ ℂfld ∈ Ring | |
18 | cnfldbas 21287 | . . . . . . . . 9 ⊢ ℂ = (Base‘ℂfld) | |
19 | cnfld0 21324 | . . . . . . . . 9 ⊢ 0 = (0g‘ℂfld) | |
20 | cndrng 21330 | . . . . . . . . 9 ⊢ ℂfld ∈ DivRing | |
21 | 18, 19, 20 | drngui 20634 | . . . . . . . 8 ⊢ (ℂ ∖ {0}) = (Unit‘ℂfld) |
22 | eqid 2725 | . . . . . . . 8 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
23 | 21, 22 | unitsubm 20329 | . . . . . . 7 ⊢ (ℂfld ∈ Ring → (ℂ ∖ {0}) ∈ (SubMnd‘(mulGrp‘ℂfld))) |
24 | 13 | subsubm 18772 | . . . . . . 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 229 | . . . . 5 ⊢ ({1, -1} ∈ (SubMnd‘(mulGrp‘ℂfld)) ∧ {1, -1} ⊆ (ℂ ∖ {0})) |
27 | 26 | simpli 482 | . . . 4 ⊢ {1, -1} ∈ (SubMnd‘(mulGrp‘ℂfld)) |
28 | 1z 12622 | . . . . 5 ⊢ 1 ∈ ℤ | |
29 | neg1z 12628 | . . . . 5 ⊢ -1 ∈ ℤ | |
30 | prssi 4820 | . . . . 5 ⊢ ((1 ∈ ℤ ∧ -1 ∈ ℤ) → {1, -1} ⊆ ℤ) | |
31 | 28, 29, 30 | mp2an 690 | . . . 4 ⊢ {1, -1} ⊆ ℤ |
32 | zsubrg 21357 | . . . . 5 ⊢ ℤ ∈ (SubRing‘ℂfld) | |
33 | 22 | subrgsubm 20528 | . . . . 5 ⊢ (ℤ ∈ (SubRing‘ℂfld) → ℤ ∈ (SubMnd‘(mulGrp‘ℂfld))) |
34 | zringmpg 21401 | . . . . . . 7 ⊢ ((mulGrp‘ℂfld) ↾s ℤ) = (mulGrp‘ℤring) | |
35 | 34 | eqcomi 2734 | . . . . . 6 ⊢ (mulGrp‘ℤring) = ((mulGrp‘ℂfld) ↾s ℤ) |
36 | 35 | subsubm 18772 | . . . . 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 709 | . . 3 ⊢ {1, -1} ∈ (SubMnd‘(mulGrp‘ℤring)) |
39 | zex 12597 | . . . . . 6 ⊢ ℤ ∈ V | |
40 | ressabs 17229 | . . . . . 6 ⊢ ((ℤ ∈ V ∧ {1, -1} ⊆ ℤ) → (((mulGrp‘ℂfld) ↾s ℤ) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1})) | |
41 | 39, 31, 40 | mp2an 690 | . . . . 5 ⊢ (((mulGrp‘ℂfld) ↾s ℤ) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) |
42 | 34 | oveq1i 7426 | . . . . 5 ⊢ (((mulGrp‘ℂfld) ↾s ℤ) ↾s {1, -1}) = ((mulGrp‘ℤring) ↾s {1, -1}) |
43 | 41, 42 | eqtr3i 2755 | . . . 4 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℤring) ↾s {1, -1}) |
44 | 43 | resmhm2 18777 | . . 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 584 | . 2 ⊢ (𝐴 ∈ Fin → (pmSgn‘𝐴) ∈ ((SymGrp‘𝐴) MndHom (mulGrp‘ℤring))) |
46 | mhmco 18779 | . 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 594 | 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 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ∖ cdif 3936 ⊆ wss 3939 {csn 4624 {cpr 4626 ∘ ccom 5676 ‘cfv 6543 (class class class)co 7416 Fincfn 8962 ℂcc 11136 0cc0 11138 1c1 11139 -cneg 11475 ℤcz 12588 ↾s cress 17208 MndHom cmhm 18737 SubMndcsubmnd 18738 SubGrpcsubg 19079 GrpHom cghm 19171 SymGrpcsymg 19325 pmSgncpsgn 19448 mulGrpcmgp 20078 Ringcrg 20177 RingHom crh 20412 SubRingcsubrg 20510 ℂfldccnfld 21283 ℤringczring 21376 ℤRHomczrh 21429 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-addf 11217 ax-mulf 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-xor 1505 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-xnn0 12575 df-z 12589 df-dec 12708 df-uz 12853 df-rp 13007 df-fz 13517 df-fzo 13660 df-seq 13999 df-exp 14059 df-hash 14322 df-word 14497 df-lsw 14545 df-concat 14553 df-s1 14578 df-substr 14623 df-pfx 14653 df-splice 14732 df-reverse 14741 df-s2 14831 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-0g 17422 df-gsum 17423 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-submnd 18740 df-efmnd 18825 df-grp 18897 df-minusg 18898 df-mulg 19028 df-subg 19082 df-ghm 19172 df-gim 19217 df-oppg 19301 df-symg 19326 df-pmtr 19401 df-psgn 19450 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-ring 20179 df-cring 20180 df-oppr 20277 df-dvdsr 20300 df-unit 20301 df-invr 20331 df-dvr 20344 df-rhm 20415 df-subrng 20487 df-subrg 20512 df-drng 20630 df-cnfld 21284 df-zring 21377 df-zrh 21433 |
This theorem is referenced by: madetsumid 22381 mdetleib2 22508 mdetf 22515 mdetdiaglem 22518 mdetrlin 22522 mdetrsca 22523 mdetralt 22528 mdetunilem7 22538 mdetunilem8 22539 |
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