dchrzrhmul - Metamath Proof Explorer

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Theorem dchrzrhmul 27173

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 27167	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ)
5	1, 4	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
6	5	nnnn0d 12463	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
7		dchrmhm.z	. . . . . . . 8 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
8	7	zncrng 21469	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
9	6, 8	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ CRing)
10		crngring 20148	. . . . . 6 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
11	9, 10	syl 17	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ Ring)
12		dchrelbas4.l	. . . . . 6 ⊢ 𝐿 = (ℤRHom‘𝑍)
13	12	zrhrhm 21436	. . . . 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 21378	. . . . 5 ⊢ ℤ = (Base‘ℤ_ring)
18		zringmulr 21382	. . . . 5 ⊢ · = (._r‘ℤ_ring)
19		eqid 2729	. . . . 5 ⊢ (._r‘𝑍) = (._r‘𝑍)
20	17, 18, 19	rhmmul 20389	. . . 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 6830	. 2 ⊢ (𝜑 → (𝑋‘(𝐿‘(𝐴 · 𝐶))) = (𝑋‘((𝐿‘𝐴)(._r‘𝑍)(𝐿‘𝐶))))
23	2, 7, 3	dchrmhm 27168	. . . 4 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
24	23, 1	sselid 3935	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
25		eqid 2729	. . . . . 6 ⊢ (Base‘𝑍) = (Base‘𝑍)
26	17, 25	rhmf 20388	. . . . 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 7023	. . 3 ⊢ (𝜑 → (𝐿‘𝐴) ∈ (Base‘𝑍))
29	27, 16	ffvelcdmd 7023	. . 3 ⊢ (𝜑 → (𝐿‘𝐶) ∈ (Base‘𝑍))
30		eqid 2729	. . . . 5 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
31	30, 25	mgpbas 20048	. . . 4 ⊢ (Base‘𝑍) = (Base‘(mulGrp‘𝑍))
32	30, 19	mgpplusg 20047	. . . 4 ⊢ (._r‘𝑍) = (+_g‘(mulGrp‘𝑍))
33		eqid 2729	. . . . 5 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
34		cnfldmul 21287	. . . . 5 ⊢ · = (._r‘ℂ_fld)
35	33, 34	mgpplusg 20047	. . . 4 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
36	31, 32, 35	mhmlin 18685	. . 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 2764	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 2109 ⟶wf 6482 ‘cfv 6486 (class class class)co 7353 · cmul 11033 ℕcn 12146 ℕ₀cn0 12402 ℤcz 12489 Basecbs 17138 ._rcmulr 17180 MndHom cmhm 18673 mulGrpcmgp 20043 Ringcrg 20136 CRingccrg 20137 RingHom crh 20372 ℂ_fldccnfld 21279 ℤ_ringczring 21371 ℤRHomczrh 21424 ℤ/nℤczn 21427 DChrcdchr 27159

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-addf 11107 ax-mulf 11108

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 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-tpos 8166 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8632 df-ec 8634 df-qs 8638 df-map 8762 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-sup 9351 df-inf 9352 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12610 df-uz 12754 df-fz 13429 df-seq 13927 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-0g 17363 df-imas 17430 df-qus 17431 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-mhm 18675 df-grp 18833 df-minusg 18834 df-sbg 18835 df-mulg 18965 df-subg 19020 df-nsg 19021 df-eqg 19022 df-ghm 19110 df-cmn 19679 df-abl 19680 df-mgp 20044 df-rng 20056 df-ur 20085 df-ring 20138 df-cring 20139 df-oppr 20240 df-rhm 20375 df-subrng 20449 df-subrg 20473 df-lmod 20783 df-lss 20853 df-lsp 20893 df-sra 21095 df-rgmod 21096 df-lidl 21133 df-rsp 21134 df-2idl 21175 df-cnfld 21280 df-zring 21372 df-zrh 21428 df-zn 21431 df-dchr 27160

This theorem is referenced by: dchrmusum2 27421 dchrvmasumlem1 27422 dchrvmasum2lem 27423 dchrisum0fmul 27433

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