dchrzrhmul - Metamath Proof Explorer

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Theorem dchrzrhmul 27218

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 27212	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ)
5	1, 4	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
6	5	nnnn0d 12467	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
7		dchrmhm.z	. . . . . . . 8 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
8	7	zncrng 21504	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
9	6, 8	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ CRing)
10		crngring 20185	. . . . . 6 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
11	9, 10	syl 17	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ Ring)
12		dchrelbas4.l	. . . . . 6 ⊢ 𝐿 = (ℤRHom‘𝑍)
13	12	zrhrhm 21471	. . . . 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 21413	. . . . 5 ⊢ ℤ = (Base‘ℤ_ring)
18		zringmulr 21417	. . . . 5 ⊢ · = (._r‘ℤ_ring)
19		eqid 2737	. . . . 5 ⊢ (._r‘𝑍) = (._r‘𝑍)
20	17, 18, 19	rhmmul 20426	. . . 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 1374	. . 3 ⊢ (𝜑 → (𝐿‘(𝐴 · 𝐶)) = ((𝐿‘𝐴)(._r‘𝑍)(𝐿‘𝐶)))
22	21	fveq2d 6839	. 2 ⊢ (𝜑 → (𝑋‘(𝐿‘(𝐴 · 𝐶))) = (𝑋‘((𝐿‘𝐴)(._r‘𝑍)(𝐿‘𝐶))))
23	2, 7, 3	dchrmhm 27213	. . . 4 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
24	23, 1	sselid 3932	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
25		eqid 2737	. . . . . 6 ⊢ (Base‘𝑍) = (Base‘𝑍)
26	17, 25	rhmf 20425	. . . . 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 7032	. . 3 ⊢ (𝜑 → (𝐿‘𝐴) ∈ (Base‘𝑍))
29	27, 16	ffvelcdmd 7032	. . 3 ⊢ (𝜑 → (𝐿‘𝐶) ∈ (Base‘𝑍))
30		eqid 2737	. . . . 5 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
31	30, 25	mgpbas 20085	. . . 4 ⊢ (Base‘𝑍) = (Base‘(mulGrp‘𝑍))
32	30, 19	mgpplusg 20084	. . . 4 ⊢ (._r‘𝑍) = (+_g‘(mulGrp‘𝑍))
33		eqid 2737	. . . . 5 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
34		cnfldmul 21322	. . . . 5 ⊢ · = (._r‘ℂ_fld)
35	33, 34	mgpplusg 20084	. . . 4 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
36	31, 32, 35	mhmlin 18723	. . 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 1374	. 2 ⊢ (𝜑 → (𝑋‘((𝐿‘𝐴)(._r‘𝑍)(𝐿‘𝐶))) = ((𝑋‘(𝐿‘𝐴)) · (𝑋‘(𝐿‘𝐶))))
38	22, 37	eqtrd 2772	1 ⊢ (𝜑 → (𝑋‘(𝐿‘(𝐴 · 𝐶))) = ((𝑋‘(𝐿‘𝐴)) · (𝑋‘(𝐿‘𝐶))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ⟶wf 6489 ‘cfv 6493 (class class class)co 7361 · cmul 11036 ℕcn 12150 ℕ₀cn0 12406 ℤcz 12493 Basecbs 17141 ._rcmulr 17183 MndHom cmhm 18711 mulGrpcmgp 20080 Ringcrg 20173 CRingccrg 20174 RingHom crh 20410 ℂ_fldccnfld 21314 ℤ_ringczring 21406 ℤRHomczrh 21459 ℤ/nℤczn 21462 DChrcdchr 27204

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7683 ax-cnex 11087 ax-resscn 11088 ax-1cn 11089 ax-icn 11090 ax-addcl 11091 ax-addrcl 11092 ax-mulcl 11093 ax-mulrcl 11094 ax-mulcom 11095 ax-addass 11096 ax-mulass 11097 ax-distr 11098 ax-i2m1 11099 ax-1ne0 11100 ax-1rid 11101 ax-rnegex 11102 ax-rrecex 11103 ax-cnre 11104 ax-pre-lttri 11105 ax-pre-lttrn 11106 ax-pre-ltadd 11107 ax-pre-mulgt0 11108 ax-addf 11110 ax-mulf 11111

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7812 df-1st 7936 df-2nd 7937 df-tpos 8171 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-er 8638 df-ec 8640 df-qs 8644 df-map 8770 df-en 8889 df-dom 8890 df-sdom 8891 df-fin 8892 df-sup 9350 df-inf 9351 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-nn 12151 df-2 12213 df-3 12214 df-4 12215 df-5 12216 df-6 12217 df-7 12218 df-8 12219 df-9 12220 df-n0 12407 df-z 12494 df-dec 12613 df-uz 12757 df-fz 13429 df-seq 13930 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17142 df-ress 17163 df-plusg 17195 df-mulr 17196 df-starv 17197 df-sca 17198 df-vsca 17199 df-ip 17200 df-tset 17201 df-ple 17202 df-ds 17204 df-unif 17205 df-0g 17366 df-imas 17434 df-qus 17435 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-mhm 18713 df-grp 18871 df-minusg 18872 df-sbg 18873 df-mulg 19003 df-subg 19058 df-nsg 19059 df-eqg 19060 df-ghm 19147 df-cmn 19716 df-abl 19717 df-mgp 20081 df-rng 20093 df-ur 20122 df-ring 20175 df-cring 20176 df-oppr 20278 df-rhm 20413 df-subrng 20484 df-subrg 20508 df-lmod 20818 df-lss 20888 df-lsp 20928 df-sra 21130 df-rgmod 21131 df-lidl 21168 df-rsp 21169 df-2idl 21210 df-cnfld 21315 df-zring 21407 df-zrh 21463 df-zn 21466 df-dchr 27205

This theorem is referenced by: dchrmusum2 27466 dchrvmasumlem1 27467 dchrvmasum2lem 27468 dchrisum0fmul 27478

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