dchrzrhmul - Metamath Proof Explorer

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Theorem dchrzrhmul 26299

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 26293	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ)
5	1, 4	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
6	5	nnnn0d 12223	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
7		dchrmhm.z	. . . . . . . 8 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
8	7	zncrng 20664	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
9	6, 8	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ CRing)
10		crngring 19710	. . . . . 6 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
11	9, 10	syl 17	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ Ring)
12		dchrelbas4.l	. . . . . 6 ⊢ 𝐿 = (ℤRHom‘𝑍)
13	12	zrhrhm 20625	. . . . 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 20588	. . . . 5 ⊢ ℤ = (Base‘ℤ_ring)
18		zringmulr 20591	. . . . 5 ⊢ · = (._r‘ℤ_ring)
19		eqid 2738	. . . . 5 ⊢ (._r‘𝑍) = (._r‘𝑍)
20	17, 18, 19	rhmmul 19886	. . . 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 1369	. . 3 ⊢ (𝜑 → (𝐿‘(𝐴 · 𝐶)) = ((𝐿‘𝐴)(._r‘𝑍)(𝐿‘𝐶)))
22	21	fveq2d 6760	. 2 ⊢ (𝜑 → (𝑋‘(𝐿‘(𝐴 · 𝐶))) = (𝑋‘((𝐿‘𝐴)(._r‘𝑍)(𝐿‘𝐶))))
23	2, 7, 3	dchrmhm 26294	. . . 4 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
24	23, 1	sselid 3915	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
25		eqid 2738	. . . . . 6 ⊢ (Base‘𝑍) = (Base‘𝑍)
26	17, 25	rhmf 19885	. . . . 5 ⊢ (𝐿 ∈ (ℤ_ring RingHom 𝑍) → 𝐿:ℤ⟶(Base‘𝑍))
27	14, 26	syl 17	. . . 4 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑍))
28	27, 15	ffvelrnd 6944	. . 3 ⊢ (𝜑 → (𝐿‘𝐴) ∈ (Base‘𝑍))
29	27, 16	ffvelrnd 6944	. . 3 ⊢ (𝜑 → (𝐿‘𝐶) ∈ (Base‘𝑍))
30		eqid 2738	. . . . 5 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
31	30, 25	mgpbas 19641	. . . 4 ⊢ (Base‘𝑍) = (Base‘(mulGrp‘𝑍))
32	30, 19	mgpplusg 19639	. . . 4 ⊢ (._r‘𝑍) = (+_g‘(mulGrp‘𝑍))
33		eqid 2738	. . . . 5 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
34		cnfldmul 20516	. . . . 5 ⊢ · = (._r‘ℂ_fld)
35	33, 34	mgpplusg 19639	. . . 4 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
36	31, 32, 35	mhmlin 18352	. . 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 1369	. 2 ⊢ (𝜑 → (𝑋‘((𝐿‘𝐴)(._r‘𝑍)(𝐿‘𝐶))) = ((𝑋‘(𝐿‘𝐴)) · (𝑋‘(𝐿‘𝐶))))
38	22, 37	eqtrd 2778	1 ⊢ (𝜑 → (𝑋‘(𝐿‘(𝐴 · 𝐶))) = ((𝑋‘(𝐿‘𝐴)) · (𝑋‘(𝐿‘𝐶))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2108 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 · cmul 10807 ℕcn 11903 ℕ₀cn0 12163 ℤcz 12249 Basecbs 16840 ._rcmulr 16889 MndHom cmhm 18343 mulGrpcmgp 19635 Ringcrg 19698 CRingccrg 19699 RingHom crh 19871 ℂ_fldccnfld 20510 ℤ_ringzring 20582 ℤRHomczrh 20613 ℤ/nℤczn 20616 DChrcdchr 26285

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-tpos 8013 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-ec 8458 df-qs 8462 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-inf 9132 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-fz 13169 df-seq 13650 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-0g 17069 df-imas 17136 df-qus 17137 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-grp 18495 df-minusg 18496 df-sbg 18497 df-mulg 18616 df-subg 18667 df-nsg 18668 df-eqg 18669 df-ghm 18747 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-oppr 19777 df-rnghom 19874 df-subrg 19937 df-lmod 20040 df-lss 20109 df-lsp 20149 df-sra 20349 df-rgmod 20350 df-lidl 20351 df-rsp 20352 df-2idl 20416 df-cnfld 20511 df-zring 20583 df-zrh 20617 df-zn 20620 df-dchr 26286

This theorem is referenced by: dchrmusum2 26547 dchrvmasumlem1 26548 dchrvmasum2lem 26549 dchrisum0fmul 26559

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