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Mirrors > Home > MPE Home > Th. List > dchrzrhmul | Structured version Visualization version GIF version |
Description: A Dirichlet character is completely multiplicative. (Contributed by Mario Carneiro, 4-May-2016.) |
Ref | Expression |
---|---|
dchrmhm.g | β’ πΊ = (DChrβπ) |
dchrmhm.z | β’ π = (β€/nβ€βπ) |
dchrmhm.b | β’ π· = (BaseβπΊ) |
dchrelbas4.l | β’ πΏ = (β€RHomβπ) |
dchrzrh1.x | β’ (π β π β π·) |
dchrzrh1.a | β’ (π β π΄ β β€) |
dchrzrh1.c | β’ (π β πΆ β β€) |
Ref | Expression |
---|---|
dchrzrhmul | β’ (π β (πβ(πΏβ(π΄ Β· πΆ))) = ((πβ(πΏβπ΄)) Β· (πβ(πΏβπΆ)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dchrzrh1.x | . . . . . . . . 9 β’ (π β π β π·) | |
2 | dchrmhm.g | . . . . . . . . . 10 β’ πΊ = (DChrβπ) | |
3 | dchrmhm.b | . . . . . . . . . 10 β’ π· = (BaseβπΊ) | |
4 | 2, 3 | dchrrcl 27189 | . . . . . . . . 9 β’ (π β π· β π β β) |
5 | 1, 4 | syl 17 | . . . . . . . 8 β’ (π β π β β) |
6 | 5 | nnnn0d 12560 | . . . . . . 7 β’ (π β π β β0) |
7 | dchrmhm.z | . . . . . . . 8 β’ π = (β€/nβ€βπ) | |
8 | 7 | zncrng 21480 | . . . . . . 7 β’ (π β β0 β π β CRing) |
9 | 6, 8 | syl 17 | . . . . . 6 β’ (π β π β CRing) |
10 | crngring 20187 | . . . . . 6 β’ (π β CRing β π β Ring) | |
11 | 9, 10 | syl 17 | . . . . 5 β’ (π β π β Ring) |
12 | dchrelbas4.l | . . . . . 6 β’ πΏ = (β€RHomβπ) | |
13 | 12 | zrhrhm 21439 | . . . . 5 β’ (π β Ring β πΏ β (β€ring RingHom π)) |
14 | 11, 13 | syl 17 | . . . 4 β’ (π β πΏ β (β€ring RingHom π)) |
15 | dchrzrh1.a | . . . 4 β’ (π β π΄ β β€) | |
16 | dchrzrh1.c | . . . 4 β’ (π β πΆ β β€) | |
17 | zringbas 21381 | . . . . 5 β’ β€ = (Baseββ€ring) | |
18 | zringmulr 21385 | . . . . 5 β’ Β· = (.rββ€ring) | |
19 | eqid 2725 | . . . . 5 β’ (.rβπ) = (.rβπ) | |
20 | 17, 18, 19 | rhmmul 20427 | . . . 4 β’ ((πΏ β (β€ring RingHom π) β§ π΄ β β€ β§ πΆ β β€) β (πΏβ(π΄ Β· πΆ)) = ((πΏβπ΄)(.rβπ)(πΏβπΆ))) |
21 | 14, 15, 16, 20 | syl3anc 1368 | . . 3 β’ (π β (πΏβ(π΄ Β· πΆ)) = ((πΏβπ΄)(.rβπ)(πΏβπΆ))) |
22 | 21 | fveq2d 6895 | . 2 β’ (π β (πβ(πΏβ(π΄ Β· πΆ))) = (πβ((πΏβπ΄)(.rβπ)(πΏβπΆ)))) |
23 | 2, 7, 3 | dchrmhm 27190 | . . . 4 β’ π· β ((mulGrpβπ) MndHom (mulGrpββfld)) |
24 | 23, 1 | sselid 3970 | . . 3 β’ (π β π β ((mulGrpβπ) MndHom (mulGrpββfld))) |
25 | eqid 2725 | . . . . . 6 β’ (Baseβπ) = (Baseβπ) | |
26 | 17, 25 | rhmf 20426 | . . . . 5 β’ (πΏ β (β€ring RingHom π) β πΏ:β€βΆ(Baseβπ)) |
27 | 14, 26 | syl 17 | . . . 4 β’ (π β πΏ:β€βΆ(Baseβπ)) |
28 | 27, 15 | ffvelcdmd 7089 | . . 3 β’ (π β (πΏβπ΄) β (Baseβπ)) |
29 | 27, 16 | ffvelcdmd 7089 | . . 3 β’ (π β (πΏβπΆ) β (Baseβπ)) |
30 | eqid 2725 | . . . . 5 β’ (mulGrpβπ) = (mulGrpβπ) | |
31 | 30, 25 | mgpbas 20082 | . . . 4 β’ (Baseβπ) = (Baseβ(mulGrpβπ)) |
32 | 30, 19 | mgpplusg 20080 | . . . 4 β’ (.rβπ) = (+gβ(mulGrpβπ)) |
33 | eqid 2725 | . . . . 5 β’ (mulGrpββfld) = (mulGrpββfld) | |
34 | cnfldmul 21289 | . . . . 5 β’ Β· = (.rββfld) | |
35 | 33, 34 | mgpplusg 20080 | . . . 4 β’ Β· = (+gβ(mulGrpββfld)) |
36 | 31, 32, 35 | mhmlin 18747 | . . 3 β’ ((π β ((mulGrpβπ) MndHom (mulGrpββfld)) β§ (πΏβπ΄) β (Baseβπ) β§ (πΏβπΆ) β (Baseβπ)) β (πβ((πΏβπ΄)(.rβπ)(πΏβπΆ))) = ((πβ(πΏβπ΄)) Β· (πβ(πΏβπΆ)))) |
37 | 24, 28, 29, 36 | syl3anc 1368 | . 2 β’ (π β (πβ((πΏβπ΄)(.rβπ)(πΏβπΆ))) = ((πβ(πΏβπ΄)) Β· (πβ(πΏβπΆ)))) |
38 | 22, 37 | eqtrd 2765 | 1 β’ (π β (πβ(πΏβ(π΄ Β· πΆ))) = ((πβ(πΏβπ΄)) Β· (πβ(πΏβπΆ)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βΆwf 6538 βcfv 6542 (class class class)co 7415 Β· cmul 11141 βcn 12240 β0cn0 12500 β€cz 12586 Basecbs 17177 .rcmulr 17231 MndHom cmhm 18735 mulGrpcmgp 20076 Ringcrg 20175 CRingccrg 20176 RingHom crh 20410 βfldccnfld 21281 β€ringczring 21374 β€RHomczrh 21427 β€/nβ€czn 21430 DChrcdchr 27181 |
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 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-addf 11215 ax-mulf 11216 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 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 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 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-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 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-tpos 8228 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-ec 8723 df-qs 8727 df-map 8843 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-sup 9463 df-inf 9464 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-dec 12706 df-uz 12851 df-fz 13515 df-seq 13997 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-starv 17245 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-ds 17252 df-unif 17253 df-0g 17420 df-imas 17487 df-qus 17488 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18737 df-grp 18895 df-minusg 18896 df-sbg 18897 df-mulg 19026 df-subg 19080 df-nsg 19081 df-eqg 19082 df-ghm 19170 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-ur 20124 df-ring 20177 df-cring 20178 df-oppr 20275 df-rhm 20413 df-subrng 20485 df-subrg 20510 df-lmod 20747 df-lss 20818 df-lsp 20858 df-sra 21060 df-rgmod 21061 df-lidl 21106 df-rsp 21107 df-2idl 21146 df-cnfld 21282 df-zring 21375 df-zrh 21431 df-zn 21434 df-dchr 27182 |
This theorem is referenced by: dchrmusum2 27443 dchrvmasumlem1 27444 dchrvmasum2lem 27445 dchrisum0fmul 27455 |
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