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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 2798 | . . . 4 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅) | |
2 | 1 | zrhrhm 20205 | . . 3 ⊢ (𝑅 ∈ Ring → (ℤRHom‘𝑅) ∈ (ℤring RingHom 𝑅)) |
3 | eqid 2798 | . . . 4 ⊢ (mulGrp‘ℤring) = (mulGrp‘ℤring) | |
4 | eqid 2798 | . . . 4 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
5 | 3, 4 | rhmmhm 19470 | . . 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 2798 | . . . . 5 ⊢ (SymGrp‘𝐴) = (SymGrp‘𝐴) | |
8 | eqid 2798 | . . . . 5 ⊢ (pmSgn‘𝐴) = (pmSgn‘𝐴) | |
9 | eqid 2798 | . . . . 5 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
10 | 7, 8, 9 | psgnghm2 20270 | . . . 4 ⊢ (𝐴 ∈ Fin → (pmSgn‘𝐴) ∈ ((SymGrp‘𝐴) GrpHom ((mulGrp‘ℂfld) ↾s {1, -1}))) |
11 | ghmmhm 18360 | . . . 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 2798 | . . . . . . . 8 ⊢ ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) | |
14 | 13 | cnmsgnsubg 20266 | . . . . . . 7 ⊢ {1, -1} ∈ (SubGrp‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) |
15 | subgsubm 18293 | . . . . . . 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 20113 | . . . . . . 7 ⊢ ℂfld ∈ Ring | |
18 | cnfldbas 20095 | . . . . . . . . 9 ⊢ ℂ = (Base‘ℂfld) | |
19 | cnfld0 20115 | . . . . . . . . 9 ⊢ 0 = (0g‘ℂfld) | |
20 | cndrng 20120 | . . . . . . . . 9 ⊢ ℂfld ∈ DivRing | |
21 | 18, 19, 20 | drngui 19501 | . . . . . . . 8 ⊢ (ℂ ∖ {0}) = (Unit‘ℂfld) |
22 | eqid 2798 | . . . . . . . 8 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
23 | 21, 22 | unitsubm 19416 | . . . . . . 7 ⊢ (ℂfld ∈ Ring → (ℂ ∖ {0}) ∈ (SubMnd‘(mulGrp‘ℂfld))) |
24 | 13 | subsubm 17973 | . . . . . . 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 233 | . . . . 5 ⊢ ({1, -1} ∈ (SubMnd‘(mulGrp‘ℂfld)) ∧ {1, -1} ⊆ (ℂ ∖ {0})) |
27 | 26 | simpli 487 | . . . 4 ⊢ {1, -1} ∈ (SubMnd‘(mulGrp‘ℂfld)) |
28 | 1z 12000 | . . . . 5 ⊢ 1 ∈ ℤ | |
29 | neg1z 12006 | . . . . 5 ⊢ -1 ∈ ℤ | |
30 | prssi 4714 | . . . . 5 ⊢ ((1 ∈ ℤ ∧ -1 ∈ ℤ) → {1, -1} ⊆ ℤ) | |
31 | 28, 29, 30 | mp2an 691 | . . . 4 ⊢ {1, -1} ⊆ ℤ |
32 | zsubrg 20144 | . . . . 5 ⊢ ℤ ∈ (SubRing‘ℂfld) | |
33 | 22 | subrgsubm 19541 | . . . . 5 ⊢ (ℤ ∈ (SubRing‘ℂfld) → ℤ ∈ (SubMnd‘(mulGrp‘ℂfld))) |
34 | zringmpg 20185 | . . . . . . 7 ⊢ ((mulGrp‘ℂfld) ↾s ℤ) = (mulGrp‘ℤring) | |
35 | 34 | eqcomi 2807 | . . . . . 6 ⊢ (mulGrp‘ℤring) = ((mulGrp‘ℂfld) ↾s ℤ) |
36 | 35 | subsubm 17973 | . . . . 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 710 | . . 3 ⊢ {1, -1} ∈ (SubMnd‘(mulGrp‘ℤring)) |
39 | zex 11978 | . . . . . 6 ⊢ ℤ ∈ V | |
40 | ressabs 16555 | . . . . . 6 ⊢ ((ℤ ∈ V ∧ {1, -1} ⊆ ℤ) → (((mulGrp‘ℂfld) ↾s ℤ) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1})) | |
41 | 39, 31, 40 | mp2an 691 | . . . . 5 ⊢ (((mulGrp‘ℂfld) ↾s ℤ) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) |
42 | 34 | oveq1i 7145 | . . . . 5 ⊢ (((mulGrp‘ℂfld) ↾s ℤ) ↾s {1, -1}) = ((mulGrp‘ℤring) ↾s {1, -1}) |
43 | 41, 42 | eqtr3i 2823 | . . . 4 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℤring) ↾s {1, -1}) |
44 | 43 | resmhm2 17978 | . . 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 589 | . 2 ⊢ (𝐴 ∈ Fin → (pmSgn‘𝐴) ∈ ((SymGrp‘𝐴) MndHom (mulGrp‘ℤring))) |
46 | mhmco 17980 | . 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 598 | 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 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∖ cdif 3878 ⊆ wss 3881 {csn 4525 {cpr 4527 ∘ ccom 5523 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 ℂcc 10524 0cc0 10526 1c1 10527 -cneg 10860 ℤcz 11969 ↾s cress 16476 MndHom cmhm 17946 SubMndcsubmnd 17947 SubGrpcsubg 18265 GrpHom cghm 18347 SymGrpcsymg 18487 pmSgncpsgn 18609 mulGrpcmgp 19232 Ringcrg 19290 RingHom crh 19460 SubRingcsubrg 19524 ℂfldccnfld 20091 ℤringzring 20163 ℤRHomczrh 20193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-xor 1503 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-ot 4534 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 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 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 df-word 13858 df-lsw 13906 df-concat 13914 df-s1 13941 df-substr 13994 df-pfx 14024 df-splice 14103 df-reverse 14112 df-s2 14201 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-0g 16707 df-gsum 16708 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-efmnd 18026 df-grp 18098 df-minusg 18099 df-mulg 18217 df-subg 18268 df-ghm 18348 df-gim 18391 df-oppg 18466 df-symg 18488 df-pmtr 18562 df-psgn 18611 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-rnghom 19463 df-drng 19497 df-subrg 19526 df-cnfld 20092 df-zring 20164 df-zrh 20197 |
This theorem is referenced by: madetsumid 21066 mdetleib2 21193 mdetf 21200 mdetdiaglem 21203 mdetrlin 21207 mdetrsca 21208 mdetralt 21213 mdetunilem7 21223 mdetunilem8 21224 |
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