dchrzrhmul - Metamath Proof Explorer

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Theorem dchrzrhmul 27209

Description: A Dirichlet character is completely multiplicative. (Contributed by Mario Carneiro, 4-May-2016.)

**Hypotheses**
Ref	Expression
dchrmhm.g	⊢ 𝐺 = (DChr‘𝑁)
dchrmhm.z	⊢ 𝑍 = (ℤ/nℤ‘𝑁)
dchrmhm.b	⊢ 𝐷 = (Base‘𝐺)
dchrelbas4.l	⊢ 𝐿 = (ℤRHom‘𝑍)
dchrzrh1.x	⊢ (𝜑 → 𝑋 ∈ 𝐷)
dchrzrh1.a	⊢ (𝜑 → 𝐴 ∈ ℤ)
dchrzrh1.c	⊢ (𝜑 → 𝐶 ∈ ℤ)

**Assertion**
Ref	Expression
dchrzrhmul	⊢ (𝜑 → (𝑋‘(𝐿‘(𝐴 · 𝐶))) = ((𝑋‘(𝐿‘𝐴)) · (𝑋‘(𝐿‘𝐶))))

**Proof of Theorem dchrzrhmul**
Step	Hyp	Ref	Expression
1		dchrzrh1.x	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
2		dchrmhm.g	. . . . . . . . . 10 ⊢ 𝐺 = (DChr‘𝑁)
3		dchrmhm.b	. . . . . . . . . 10 ⊢ 𝐷 = (Base‘𝐺)
4	2, 3	dchrrcl 27203	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ)
5	1, 4	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
6	5	nnnn0d 12562	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
7		dchrmhm.z	. . . . . . . 8 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
8	7	zncrng 21505	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
9	6, 8	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ CRing)
10		crngring 20205	. . . . . 6 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
11	9, 10	syl 17	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ Ring)
12		dchrelbas4.l	. . . . . 6 ⊢ 𝐿 = (ℤRHom‘𝑍)
13	12	zrhrhm 21472	. . . . 5 ⊢ (𝑍 ∈ Ring → 𝐿 ∈ (ℤ_ring RingHom 𝑍))
14	11, 13	syl 17	. . . 4 ⊢ (𝜑 → 𝐿 ∈ (ℤ_ring RingHom 𝑍))
15		dchrzrh1.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℤ)
16		dchrzrh1.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℤ)
17		zringbas 21414	. . . . 5 ⊢ ℤ = (Base‘ℤ_ring)
18		zringmulr 21418	. . . . 5 ⊢ · = (._r‘ℤ_ring)
19		eqid 2735	. . . . 5 ⊢ (._r‘𝑍) = (._r‘𝑍)
20	17, 18, 19	rhmmul 20446	. . . 4 ⊢ ((𝐿 ∈ (ℤ_ring RingHom 𝑍) ∧ 𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐿‘(𝐴 · 𝐶)) = ((𝐿‘𝐴)(._r‘𝑍)(𝐿‘𝐶)))
21	14, 15, 16, 20	syl3anc 1373	. . 3 ⊢ (𝜑 → (𝐿‘(𝐴 · 𝐶)) = ((𝐿‘𝐴)(._r‘𝑍)(𝐿‘𝐶)))
22	21	fveq2d 6880	. 2 ⊢ (𝜑 → (𝑋‘(𝐿‘(𝐴 · 𝐶))) = (𝑋‘((𝐿‘𝐴)(._r‘𝑍)(𝐿‘𝐶))))
23	2, 7, 3	dchrmhm 27204	. . . 4 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
24	23, 1	sselid 3956	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
25		eqid 2735	. . . . . 6 ⊢ (Base‘𝑍) = (Base‘𝑍)
26	17, 25	rhmf 20445	. . . . 5 ⊢ (𝐿 ∈ (ℤ_ring RingHom 𝑍) → 𝐿:ℤ⟶(Base‘𝑍))
27	14, 26	syl 17	. . . 4 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑍))
28	27, 15	ffvelcdmd 7075	. . 3 ⊢ (𝜑 → (𝐿‘𝐴) ∈ (Base‘𝑍))
29	27, 16	ffvelcdmd 7075	. . 3 ⊢ (𝜑 → (𝐿‘𝐶) ∈ (Base‘𝑍))
30		eqid 2735	. . . . 5 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
31	30, 25	mgpbas 20105	. . . 4 ⊢ (Base‘𝑍) = (Base‘(mulGrp‘𝑍))
32	30, 19	mgpplusg 20104	. . . 4 ⊢ (._r‘𝑍) = (+_g‘(mulGrp‘𝑍))
33		eqid 2735	. . . . 5 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
34		cnfldmul 21323	. . . . 5 ⊢ · = (._r‘ℂ_fld)
35	33, 34	mgpplusg 20104	. . . 4 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
36	31, 32, 35	mhmlin 18771	. . 3 ⊢ ((𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) ∧ (𝐿‘𝐴) ∈ (Base‘𝑍) ∧ (𝐿‘𝐶) ∈ (Base‘𝑍)) → (𝑋‘((𝐿‘𝐴)(._r‘𝑍)(𝐿‘𝐶))) = ((𝑋‘(𝐿‘𝐴)) · (𝑋‘(𝐿‘𝐶))))
37	24, 28, 29, 36	syl3anc 1373	. 2 ⊢ (𝜑 → (𝑋‘((𝐿‘𝐴)(._r‘𝑍)(𝐿‘𝐶))) = ((𝑋‘(𝐿‘𝐴)) · (𝑋‘(𝐿‘𝐶))))
38	22, 37	eqtrd 2770	1 ⊢ (𝜑 → (𝑋‘(𝐿‘(𝐴 · 𝐶))) = ((𝑋‘(𝐿‘𝐴)) · (𝑋‘(𝐿‘𝐶))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ⟶wf 6527 ‘cfv 6531 (class class class)co 7405 · cmul 11134 ℕcn 12240 ℕ₀cn0 12501 ℤcz 12588 Basecbs 17228 ._rcmulr 17272 MndHom cmhm 18759 mulGrpcmgp 20100 Ringcrg 20193 CRingccrg 20194 RingHom crh 20429 ℂ_fldccnfld 21315 ℤ_ringczring 21407 ℤRHomczrh 21460 ℤ/nℤczn 21463 DChrcdchr 27195

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-addf 11208 ax-mulf 11209

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8719 df-ec 8721 df-qs 8725 df-map 8842 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-sup 9454 df-inf 9455 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 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 12502 df-z 12589 df-dec 12709 df-uz 12853 df-fz 13525 df-seq 14020 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-starv 17286 df-sca 17287 df-vsca 17288 df-ip 17289 df-tset 17290 df-ple 17291 df-ds 17293 df-unif 17294 df-0g 17455 df-imas 17522 df-qus 17523 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-mhm 18761 df-grp 18919 df-minusg 18920 df-sbg 18921 df-mulg 19051 df-subg 19106 df-nsg 19107 df-eqg 19108 df-ghm 19196 df-cmn 19763 df-abl 19764 df-mgp 20101 df-rng 20113 df-ur 20142 df-ring 20195 df-cring 20196 df-oppr 20297 df-rhm 20432 df-subrng 20506 df-subrg 20530 df-lmod 20819 df-lss 20889 df-lsp 20929 df-sra 21131 df-rgmod 21132 df-lidl 21169 df-rsp 21170 df-2idl 21211 df-cnfld 21316 df-zring 21408 df-zrh 21464 df-zn 21467 df-dchr 27196

This theorem is referenced by: dchrmusum2 27457 dchrvmasumlem1 27458 dchrvmasum2lem 27459 dchrisum0fmul 27469

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