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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 2823 | . . . 4 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅) | |
2 | 1 | zrhrhm 20661 | . . 3 ⊢ (𝑅 ∈ Ring → (ℤRHom‘𝑅) ∈ (ℤring RingHom 𝑅)) |
3 | eqid 2823 | . . . 4 ⊢ (mulGrp‘ℤring) = (mulGrp‘ℤring) | |
4 | eqid 2823 | . . . 4 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
5 | 3, 4 | rhmmhm 19476 | . . 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 2823 | . . . . 5 ⊢ (SymGrp‘𝐴) = (SymGrp‘𝐴) | |
8 | eqid 2823 | . . . . 5 ⊢ (pmSgn‘𝐴) = (pmSgn‘𝐴) | |
9 | eqid 2823 | . . . . 5 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
10 | 7, 8, 9 | psgnghm2 20727 | . . . 4 ⊢ (𝐴 ∈ Fin → (pmSgn‘𝐴) ∈ ((SymGrp‘𝐴) GrpHom ((mulGrp‘ℂfld) ↾s {1, -1}))) |
11 | ghmmhm 18370 | . . . 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 2823 | . . . . . . . 8 ⊢ ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) | |
14 | 13 | cnmsgnsubg 20723 | . . . . . . 7 ⊢ {1, -1} ∈ (SubGrp‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) |
15 | subgsubm 18303 | . . . . . . 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 20569 | . . . . . . 7 ⊢ ℂfld ∈ Ring | |
18 | cnfldbas 20551 | . . . . . . . . 9 ⊢ ℂ = (Base‘ℂfld) | |
19 | cnfld0 20571 | . . . . . . . . 9 ⊢ 0 = (0g‘ℂfld) | |
20 | cndrng 20576 | . . . . . . . . 9 ⊢ ℂfld ∈ DivRing | |
21 | 18, 19, 20 | drngui 19510 | . . . . . . . 8 ⊢ (ℂ ∖ {0}) = (Unit‘ℂfld) |
22 | eqid 2823 | . . . . . . . 8 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
23 | 21, 22 | unitsubm 19422 | . . . . . . 7 ⊢ (ℂfld ∈ Ring → (ℂ ∖ {0}) ∈ (SubMnd‘(mulGrp‘ℂfld))) |
24 | 13 | subsubm 17983 | . . . . . . 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 232 | . . . . 5 ⊢ ({1, -1} ∈ (SubMnd‘(mulGrp‘ℂfld)) ∧ {1, -1} ⊆ (ℂ ∖ {0})) |
27 | 26 | simpli 486 | . . . 4 ⊢ {1, -1} ∈ (SubMnd‘(mulGrp‘ℂfld)) |
28 | 1z 12015 | . . . . 5 ⊢ 1 ∈ ℤ | |
29 | neg1z 12021 | . . . . 5 ⊢ -1 ∈ ℤ | |
30 | prssi 4756 | . . . . 5 ⊢ ((1 ∈ ℤ ∧ -1 ∈ ℤ) → {1, -1} ⊆ ℤ) | |
31 | 28, 29, 30 | mp2an 690 | . . . 4 ⊢ {1, -1} ⊆ ℤ |
32 | zsubrg 20600 | . . . . 5 ⊢ ℤ ∈ (SubRing‘ℂfld) | |
33 | 22 | subrgsubm 19550 | . . . . 5 ⊢ (ℤ ∈ (SubRing‘ℂfld) → ℤ ∈ (SubMnd‘(mulGrp‘ℂfld))) |
34 | zringmpg 20641 | . . . . . . 7 ⊢ ((mulGrp‘ℂfld) ↾s ℤ) = (mulGrp‘ℤring) | |
35 | 34 | eqcomi 2832 | . . . . . 6 ⊢ (mulGrp‘ℤring) = ((mulGrp‘ℂfld) ↾s ℤ) |
36 | 35 | subsubm 17983 | . . . . 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 11993 | . . . . . 6 ⊢ ℤ ∈ V | |
40 | ressabs 16565 | . . . . . 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 7168 | . . . . 5 ⊢ (((mulGrp‘ℂfld) ↾s ℤ) ↾s {1, -1}) = ((mulGrp‘ℤring) ↾s {1, -1}) |
43 | 41, 42 | eqtr3i 2848 | . . . 4 ⊢ ((mulGrp‘ℂfld) ↾s {1, -1}) = ((mulGrp‘ℤring) ↾s {1, -1}) |
44 | 43 | resmhm2 17988 | . . 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 588 | . 2 ⊢ (𝐴 ∈ Fin → (pmSgn‘𝐴) ∈ ((SymGrp‘𝐴) MndHom (mulGrp‘ℤring))) |
46 | mhmco 17990 | . 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 597 | 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 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∖ cdif 3935 ⊆ wss 3938 {csn 4569 {cpr 4571 ∘ ccom 5561 ‘cfv 6357 (class class class)co 7158 Fincfn 8511 ℂcc 10537 0cc0 10539 1c1 10540 -cneg 10873 ℤcz 11984 ↾s cress 16486 MndHom cmhm 17956 SubMndcsubmnd 17957 SubGrpcsubg 18275 GrpHom cghm 18357 SymGrpcsymg 18497 pmSgncpsgn 18619 mulGrpcmgp 19241 Ringcrg 19299 RingHom crh 19466 SubRingcsubrg 19533 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-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: madetsumid 21072 mdetleib2 21199 mdetf 21206 mdetdiaglem 21209 mdetrlin 21213 mdetrsca 21214 mdetralt 21219 mdetunilem7 21229 mdetunilem8 21230 |
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