dchrzrhmul - Metamath Proof Explorer

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Theorem dchrzrhmul 27190

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 27184	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ)
5	1, 4	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
6	5	nnnn0d 12448	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
7		dchrmhm.z	. . . . . . . 8 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
8	7	zncrng 21487	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
9	6, 8	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ CRing)
10		crngring 20169	. . . . . 6 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
11	9, 10	syl 17	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ Ring)
12		dchrelbas4.l	. . . . . 6 ⊢ 𝐿 = (ℤRHom‘𝑍)
13	12	zrhrhm 21454	. . . . 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 21396	. . . . 5 ⊢ ℤ = (Base‘ℤ_ring)
18		zringmulr 21400	. . . . 5 ⊢ · = (._r‘ℤ_ring)
19		eqid 2731	. . . . 5 ⊢ (._r‘𝑍) = (._r‘𝑍)
20	17, 18, 19	rhmmul 20409	. . . 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 6832	. 2 ⊢ (𝜑 → (𝑋‘(𝐿‘(𝐴 · 𝐶))) = (𝑋‘((𝐿‘𝐴)(._r‘𝑍)(𝐿‘𝐶))))
23	2, 7, 3	dchrmhm 27185	. . . 4 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
24	23, 1	sselid 3927	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
25		eqid 2731	. . . . . 6 ⊢ (Base‘𝑍) = (Base‘𝑍)
26	17, 25	rhmf 20408	. . . . 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 7024	. . 3 ⊢ (𝜑 → (𝐿‘𝐴) ∈ (Base‘𝑍))
29	27, 16	ffvelcdmd 7024	. . 3 ⊢ (𝜑 → (𝐿‘𝐶) ∈ (Base‘𝑍))
30		eqid 2731	. . . . 5 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
31	30, 25	mgpbas 20069	. . . 4 ⊢ (Base‘𝑍) = (Base‘(mulGrp‘𝑍))
32	30, 19	mgpplusg 20068	. . . 4 ⊢ (._r‘𝑍) = (+_g‘(mulGrp‘𝑍))
33		eqid 2731	. . . . 5 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
34		cnfldmul 21305	. . . . 5 ⊢ · = (._r‘ℂ_fld)
35	33, 34	mgpplusg 20068	. . . 4 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
36	31, 32, 35	mhmlin 18707	. . 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 2766	1 ⊢ (𝜑 → (𝑋‘(𝐿‘(𝐴 · 𝐶))) = ((𝑋‘(𝐿‘𝐴)) · (𝑋‘(𝐿‘𝐶))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2111 ⟶wf 6483 ‘cfv 6487 (class class class)co 7352 · cmul 11017 ℕcn 12131 ℕ₀cn0 12387 ℤcz 12474 Basecbs 17126 ._rcmulr 17168 MndHom cmhm 18695 mulGrpcmgp 20064 Ringcrg 20157 CRingccrg 20158 RingHom crh 20393 ℂ_fldccnfld 21297 ℤ_ringczring 21389 ℤRHomczrh 21442 ℤ/nℤczn 21445 DChrcdchr 27176

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11068 ax-resscn 11069 ax-1cn 11070 ax-icn 11071 ax-addcl 11072 ax-addrcl 11073 ax-mulcl 11074 ax-mulrcl 11075 ax-mulcom 11076 ax-addass 11077 ax-mulass 11078 ax-distr 11079 ax-i2m1 11080 ax-1ne0 11081 ax-1rid 11082 ax-rnegex 11083 ax-rrecex 11084 ax-cnre 11085 ax-pre-lttri 11086 ax-pre-lttrn 11087 ax-pre-ltadd 11088 ax-pre-mulgt0 11089 ax-addf 11091 ax-mulf 11092

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7803 df-1st 7927 df-2nd 7928 df-tpos 8162 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-er 8628 df-ec 8630 df-qs 8634 df-map 8758 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-sup 9332 df-inf 9333 df-pnf 11154 df-mnf 11155 df-xr 11156 df-ltxr 11157 df-le 11158 df-sub 11352 df-neg 11353 df-nn 12132 df-2 12194 df-3 12195 df-4 12196 df-5 12197 df-6 12198 df-7 12199 df-8 12200 df-9 12201 df-n0 12388 df-z 12475 df-dec 12595 df-uz 12739 df-fz 13414 df-seq 13915 df-struct 17064 df-sets 17081 df-slot 17099 df-ndx 17111 df-base 17127 df-ress 17148 df-plusg 17180 df-mulr 17181 df-starv 17182 df-sca 17183 df-vsca 17184 df-ip 17185 df-tset 17186 df-ple 17187 df-ds 17189 df-unif 17190 df-0g 17351 df-imas 17418 df-qus 17419 df-mgm 18554 df-sgrp 18633 df-mnd 18649 df-mhm 18697 df-grp 18855 df-minusg 18856 df-sbg 18857 df-mulg 18987 df-subg 19042 df-nsg 19043 df-eqg 19044 df-ghm 19131 df-cmn 19700 df-abl 19701 df-mgp 20065 df-rng 20077 df-ur 20106 df-ring 20159 df-cring 20160 df-oppr 20261 df-rhm 20396 df-subrng 20467 df-subrg 20491 df-lmod 20801 df-lss 20871 df-lsp 20911 df-sra 21113 df-rgmod 21114 df-lidl 21151 df-rsp 21152 df-2idl 21193 df-cnfld 21298 df-zring 21390 df-zrh 21446 df-zn 21449 df-dchr 27177

This theorem is referenced by: dchrmusum2 27438 dchrvmasumlem1 27439 dchrvmasum2lem 27440 dchrisum0fmul 27450

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