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Theorem dchrisum0fmul 24908

Description: The function 𝐹, the divisor sum of a Dirichlet character, is a multiplicative function (but not completely multiplicative). Equation 9.4.27 of [Shapiro], p. 382. (Contributed by Mario Carneiro, 5-May-2016.)

**Hypotheses**
Ref	Expression
rpvmasum.z	⊢ 𝑍 = (ℤ/nℤ‘𝑁)
rpvmasum.l	⊢ 𝐿 = (ℤRHom‘𝑍)
rpvmasum.a	⊢ (𝜑 → 𝑁 ∈ ℕ)
rpvmasum2.g	⊢ 𝐺 = (DChr‘𝑁)
rpvmasum2.d	⊢ 𝐷 = (Base‘𝐺)
rpvmasum2.1	⊢ 1 = (0_g‘𝐺)
dchrisum0f.f	⊢ 𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} (𝑋‘(𝐿‘𝑣)))
dchrisum0f.x	⊢ (𝜑 → 𝑋 ∈ 𝐷)
dchrisum0fmul.a	⊢ (𝜑 → 𝐴 ∈ ℕ)
dchrisum0fmul.b	⊢ (𝜑 → 𝐵 ∈ ℕ)
dchrisum0fmul.m	⊢ (𝜑 → (𝐴 gcd 𝐵) = 1)

**Assertion**
Ref	Expression
dchrisum0fmul	⊢ (𝜑 → (𝐹‘(𝐴 · 𝐵)) = ((𝐹‘𝐴) · (𝐹‘𝐵)))

Distinct variable groups: 𝑞,𝑏,𝑣,𝐴 𝑁,𝑞 𝐵,𝑏,𝑞,𝑣 𝐿,𝑏,𝑣 𝑋,𝑏,𝑣 Allowed substitution hints: 𝜑(𝑣,𝑞,𝑏) 𝐷(𝑣,𝑞,𝑏) 1 (𝑣,𝑞,𝑏) 𝐹(𝑣,𝑞,𝑏) 𝐺(𝑣,𝑞,𝑏) 𝐿(𝑞) 𝑁(𝑣,𝑏) 𝑋(𝑞) 𝑍(𝑣,𝑞,𝑏)

Proof of Theorem dchrisum0fmul Dummy variables 𝑘 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dchrisum0fmul.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℕ)
2		dchrisum0fmul.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℕ)
3		dchrisum0fmul.m	. . 3 ⊢ (𝜑 → (𝐴 gcd 𝐵) = 1)
4		eqid 2605	. . 3 ⊢ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} = {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴}
5		eqid 2605	. . 3 ⊢ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵} = {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵}
6		eqid 2605	. . 3 ⊢ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝐴 · 𝐵)} = {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝐴 · 𝐵)}
7		rpvmasum2.g	. . . 4 ⊢ 𝐺 = (DChr‘𝑁)
8		rpvmasum.z	. . . 4 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
9		rpvmasum2.d	. . . 4 ⊢ 𝐷 = (Base‘𝐺)
10		rpvmasum.l	. . . 4 ⊢ 𝐿 = (ℤRHom‘𝑍)
11		dchrisum0f.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
12	11	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴}) → 𝑋 ∈ 𝐷)
13		elrabi 3323	. . . . . 6 ⊢ (𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} → 𝑗 ∈ ℕ)
14	13	nnzd 11309	. . . . 5 ⊢ (𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} → 𝑗 ∈ ℤ)
15	14	adantl 480	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴}) → 𝑗 ∈ ℤ)
16	7, 8, 9, 10, 12, 15	dchrzrhcl 24683	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴}) → (𝑋‘(𝐿‘𝑗)) ∈ ℂ)
17	11	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵}) → 𝑋 ∈ 𝐷)
18		elrabi 3323	. . . . . 6 ⊢ (𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵} → 𝑘 ∈ ℕ)
19	18	nnzd 11309	. . . . 5 ⊢ (𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵} → 𝑘 ∈ ℤ)
20	19	adantl 480	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵}) → 𝑘 ∈ ℤ)
21	7, 8, 9, 10, 17, 20	dchrzrhcl 24683	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵}) → (𝑋‘(𝐿‘𝑘)) ∈ ℂ)
22	14, 19	anim12i 587	. . . 4 ⊢ ((𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵}) → (𝑗 ∈ ℤ ∧ 𝑘 ∈ ℤ))
23	11	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → 𝑋 ∈ 𝐷)
24		simprl 789	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → 𝑗 ∈ ℤ)
25		simprr 791	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → 𝑘 ∈ ℤ)
26	7, 8, 9, 10, 23, 24, 25	dchrzrhmul 24684	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → (𝑋‘(𝐿‘(𝑗 · 𝑘))) = ((𝑋‘(𝐿‘𝑗)) · (𝑋‘(𝐿‘𝑘))))
27	26	eqcomd 2611	. . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → ((𝑋‘(𝐿‘𝑗)) · (𝑋‘(𝐿‘𝑘))) = (𝑋‘(𝐿‘(𝑗 · 𝑘))))
28	22, 27	sylan2 489	. . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵})) → ((𝑋‘(𝐿‘𝑗)) · (𝑋‘(𝐿‘𝑘))) = (𝑋‘(𝐿‘(𝑗 · 𝑘))))
29		fveq2 6084	. . . 4 ⊢ (𝑖 = (𝑗 · 𝑘) → (𝐿‘𝑖) = (𝐿‘(𝑗 · 𝑘)))
30	29	fveq2d 6088	. . 3 ⊢ (𝑖 = (𝑗 · 𝑘) → (𝑋‘(𝐿‘𝑖)) = (𝑋‘(𝐿‘(𝑗 · 𝑘))))
31	1, 2, 3, 4, 5, 6, 16, 21, 28, 30	fsumdvdsmul 24634	. 2 ⊢ (𝜑 → (Σ𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} (𝑋‘(𝐿‘𝑗)) · Σ𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵} (𝑋‘(𝐿‘𝑘))) = Σ𝑖 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝐴 · 𝐵)} (𝑋‘(𝐿‘𝑖)))
32		rpvmasum.a	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
33		rpvmasum2.1	. . . . 5 ⊢ 1 = (0_g‘𝐺)
34		dchrisum0f.f	. . . . 5 ⊢ 𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} (𝑋‘(𝐿‘𝑣)))
35	8, 10, 32, 7, 9, 33, 34	dchrisum0fval 24907	. . . 4 ⊢ (𝐴 ∈ ℕ → (𝐹‘𝐴) = Σ𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} (𝑋‘(𝐿‘𝑗)))
36	1, 35	syl 17	. . 3 ⊢ (𝜑 → (𝐹‘𝐴) = Σ𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} (𝑋‘(𝐿‘𝑗)))
37	8, 10, 32, 7, 9, 33, 34	dchrisum0fval 24907	. . . 4 ⊢ (𝐵 ∈ ℕ → (𝐹‘𝐵) = Σ𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵} (𝑋‘(𝐿‘𝑘)))
38	2, 37	syl 17	. . 3 ⊢ (𝜑 → (𝐹‘𝐵) = Σ𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵} (𝑋‘(𝐿‘𝑘)))
39	36, 38	oveq12d 6541	. 2 ⊢ (𝜑 → ((𝐹‘𝐴) · (𝐹‘𝐵)) = (Σ𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} (𝑋‘(𝐿‘𝑗)) · Σ𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵} (𝑋‘(𝐿‘𝑘))))
40	1, 2	nnmulcld 10911	. . 3 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℕ)
41	8, 10, 32, 7, 9, 33, 34	dchrisum0fval 24907	. . 3 ⊢ ((𝐴 · 𝐵) ∈ ℕ → (𝐹‘(𝐴 · 𝐵)) = Σ𝑖 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝐴 · 𝐵)} (𝑋‘(𝐿‘𝑖)))
42	40, 41	syl 17	. 2 ⊢ (𝜑 → (𝐹‘(𝐴 · 𝐵)) = Σ𝑖 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝐴 · 𝐵)} (𝑋‘(𝐿‘𝑖)))
43	31, 39, 42	3eqtr4rd 2650	1 ⊢ (𝜑 → (𝐹‘(𝐴 · 𝐵)) = ((𝐹‘𝐴) · (𝐹‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 382 = wceq 1474 ∈ wcel 1975 {crab 2895 class class class wbr 4573 ↦ cmpt 4633 ‘cfv 5786 (class class class)co 6523 1c1 9789 · cmul 9793 ℕcn 10863 ℤcz 11206 Σcsu 14206 ∥ cdvds 14763 gcd cgcd 14996 Basecbs 15637 0_gc0g 15865 ℤRHomczrh 19608 ℤ/nℤczn 19611 DChrcdchr 24670

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1711 ax-4 1726 ax-5 1825 ax-6 1873 ax-7 1920 ax-8 1977 ax-9 1984 ax-10 2004 ax-11 2019 ax-12 2031 ax-13 2228 ax-ext 2585 ax-rep 4689 ax-sep 4699 ax-nul 4708 ax-pow 4760 ax-pr 4824 ax-un 6820 ax-inf2 8394 ax-cnex 9844 ax-resscn 9845 ax-1cn 9846 ax-icn 9847 ax-addcl 9848 ax-addrcl 9849 ax-mulcl 9850 ax-mulrcl 9851 ax-mulcom 9852 ax-addass 9853 ax-mulass 9854 ax-distr 9855 ax-i2m1 9856 ax-1ne0 9857 ax-1rid 9858 ax-rnegex 9859 ax-rrecex 9860 ax-cnre 9861 ax-pre-lttri 9862 ax-pre-lttrn 9863 ax-pre-ltadd 9864 ax-pre-mulgt0 9865 ax-pre-sup 9866 ax-addf 9867 ax-mulf 9868

This theorem depends on definitions: df-bi 195 df-or 383 df-an 384 df-3or 1031 df-3an 1032 df-tru 1477 df-fal 1480 df-ex 1695 df-nf 1700 df-sb 1866 df-eu 2457 df-mo 2458 df-clab 2592 df-cleq 2598 df-clel 2601 df-nfc 2735 df-ne 2777 df-nel 2778 df-ral 2896 df-rex 2897 df-reu 2898 df-rmo 2899 df-rab 2900 df-v 3170 df-sbc 3398 df-csb 3495 df-dif 3538 df-un 3540 df-in 3542 df-ss 3549 df-pss 3551 df-nul 3870 df-if 4032 df-pw 4105 df-sn 4121 df-pr 4123 df-tp 4125 df-op 4127 df-uni 4363 df-int 4401 df-iun 4447 df-br 4574 df-opab 4634 df-mpt 4635 df-tr 4671 df-eprel 4935 df-id 4939 df-po 4945 df-so 4946 df-fr 4983 df-se 4984 df-we 4985 df-xp 5030 df-rel 5031 df-cnv 5032 df-co 5033 df-dm 5034 df-rn 5035 df-res 5036 df-ima 5037 df-pred 5579 df-ord 5625 df-on 5626 df-lim 5627 df-suc 5628 df-iota 5750 df-fun 5788 df-fn 5789 df-f 5790 df-f1 5791 df-fo 5792 df-f1o 5793 df-fv 5794 df-isom 5795 df-riota 6485 df-ov 6526 df-oprab 6527 df-mpt2 6528 df-om 6931 df-1st 7032 df-2nd 7033 df-tpos 7212 df-wrecs 7267 df-recs 7328 df-rdg 7366 df-1o 7420 df-oadd 7424 df-er 7602 df-ec 7604 df-qs 7608 df-map 7719 df-en 7815 df-dom 7816 df-sdom 7817 df-fin 7818 df-sup 8204 df-inf 8205 df-oi 8271 df-card 8621 df-pnf 9928 df-mnf 9929 df-xr 9930 df-ltxr 9931 df-le 9932 df-sub 10115 df-neg 10116 df-div 10530 df-nn 10864 df-2 10922 df-3 10923 df-4 10924 df-5 10925 df-6 10926 df-7 10927 df-8 10928 df-9 10929 df-n0 11136 df-z 11207 df-dec 11322 df-uz 11516 df-rp 11661 df-fz 12149 df-fzo 12286 df-fl 12406 df-mod 12482 df-seq 12615 df-exp 12674 df-hash 12931 df-cj 13629 df-re 13630 df-im 13631 df-sqrt 13765 df-abs 13766 df-clim 14009 df-sum 14207 df-dvds 14764 df-gcd 14997 df-struct 15639 df-ndx 15640 df-slot 15641 df-base 15642 df-sets 15643 df-ress 15644 df-plusg 15723 df-mulr 15724 df-starv 15725 df-sca 15726 df-vsca 15727 df-ip 15728 df-tset 15729 df-ple 15730 df-ds 15733 df-unif 15734 df-0g 15867 df-imas 15933 df-qus 15934 df-mgm 17007 df-sgrp 17049 df-mnd 17060 df-mhm 17100 df-grp 17190 df-minusg 17191 df-sbg 17192 df-mulg 17306 df-subg 17356 df-nsg 17357 df-eqg 17358 df-ghm 17423 df-cmn 17960 df-abl 17961 df-mgp 18255 df-ur 18267 df-ring 18314 df-cring 18315 df-oppr 18388 df-dvdsr 18406 df-unit 18407 df-rnghom 18480 df-subrg 18543 df-lmod 18630 df-lss 18696 df-lsp 18735 df-sra 18935 df-rgmod 18936 df-lidl 18937 df-rsp 18938 df-2idl 18995 df-cnfld 19510 df-zring 19580 df-zrh 19612 df-zn 19615 df-dchr 24671

This theorem is referenced by: dchrisum0flblem2 24911

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