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Theorem dchrisum0fmul 26654

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 2738	. . 3 ⊢ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} = {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴}
5		eqid 2738	. . 3 ⊢ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵} = {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵}
6		eqid 2738	. . 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 481	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴}) → 𝑋 ∈ 𝐷)
13		elrabi 3618	. . . . . 6 ⊢ (𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} → 𝑗 ∈ ℕ)
14	13	nnzd 12425	. . . . 5 ⊢ (𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} → 𝑗 ∈ ℤ)
15	14	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴}) → 𝑗 ∈ ℤ)
16	7, 8, 9, 10, 12, 15	dchrzrhcl 26393	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴}) → (𝑋‘(𝐿‘𝑗)) ∈ ℂ)
17	11	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵}) → 𝑋 ∈ 𝐷)
18		elrabi 3618	. . . . . 6 ⊢ (𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵} → 𝑘 ∈ ℕ)
19	18	nnzd 12425	. . . . 5 ⊢ (𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵} → 𝑘 ∈ ℤ)
20	19	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵}) → 𝑘 ∈ ℤ)
21	7, 8, 9, 10, 17, 20	dchrzrhcl 26393	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵}) → (𝑋‘(𝐿‘𝑘)) ∈ ℂ)
22	14, 19	anim12i 613	. . . 4 ⊢ ((𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵}) → (𝑗 ∈ ℤ ∧ 𝑘 ∈ ℤ))
23	11	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → 𝑋 ∈ 𝐷)
24		simprl 768	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → 𝑗 ∈ ℤ)
25		simprr 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → 𝑘 ∈ ℤ)
26	7, 8, 9, 10, 23, 24, 25	dchrzrhmul 26394	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → (𝑋‘(𝐿‘(𝑗 · 𝑘))) = ((𝑋‘(𝐿‘𝑗)) · (𝑋‘(𝐿‘𝑘))))
27	26	eqcomd 2744	. . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → ((𝑋‘(𝐿‘𝑗)) · (𝑋‘(𝐿‘𝑘))) = (𝑋‘(𝐿‘(𝑗 · 𝑘))))
28	22, 27	sylan2 593	. . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵})) → ((𝑋‘(𝐿‘𝑗)) · (𝑋‘(𝐿‘𝑘))) = (𝑋‘(𝐿‘(𝑗 · 𝑘))))
29		2fveq3 6779	. . 3 ⊢ (𝑖 = (𝑗 · 𝑘) → (𝑋‘(𝐿‘𝑖)) = (𝑋‘(𝐿‘(𝑗 · 𝑘))))
30	1, 2, 3, 4, 5, 6, 16, 21, 28, 29	fsumdvdsmul 26344	. 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)))
31		rpvmasum.a	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
32		rpvmasum2.1	. . . . 5 ⊢ 1 = (0_g‘𝐺)
33		dchrisum0f.f	. . . . 5 ⊢ 𝐹 = (𝑏 ∈ ℕ ↦ Σ𝑣 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝑏} (𝑋‘(𝐿‘𝑣)))
34	8, 10, 31, 7, 9, 32, 33	dchrisum0fval 26653	. . . 4 ⊢ (𝐴 ∈ ℕ → (𝐹‘𝐴) = Σ𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} (𝑋‘(𝐿‘𝑗)))
35	1, 34	syl 17	. . 3 ⊢ (𝜑 → (𝐹‘𝐴) = Σ𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} (𝑋‘(𝐿‘𝑗)))
36	8, 10, 31, 7, 9, 32, 33	dchrisum0fval 26653	. . . 4 ⊢ (𝐵 ∈ ℕ → (𝐹‘𝐵) = Σ𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵} (𝑋‘(𝐿‘𝑘)))
37	2, 36	syl 17	. . 3 ⊢ (𝜑 → (𝐹‘𝐵) = Σ𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵} (𝑋‘(𝐿‘𝑘)))
38	35, 37	oveq12d 7293	. 2 ⊢ (𝜑 → ((𝐹‘𝐴) · (𝐹‘𝐵)) = (Σ𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} (𝑋‘(𝐿‘𝑗)) · Σ𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵} (𝑋‘(𝐿‘𝑘))))
39	1, 2	nnmulcld 12026	. . 3 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℕ)
40	8, 10, 31, 7, 9, 32, 33	dchrisum0fval 26653	. . 3 ⊢ ((𝐴 · 𝐵) ∈ ℕ → (𝐹‘(𝐴 · 𝐵)) = Σ𝑖 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝐴 · 𝐵)} (𝑋‘(𝐿‘𝑖)))
41	39, 40	syl 17	. 2 ⊢ (𝜑 → (𝐹‘(𝐴 · 𝐵)) = Σ𝑖 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ (𝐴 · 𝐵)} (𝑋‘(𝐿‘𝑖)))
42	30, 38, 41	3eqtr4rd 2789	1 ⊢ (𝜑 → (𝐹‘(𝐴 · 𝐵)) = ((𝐹‘𝐴) · (𝐹‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 {crab 3068 class class class wbr 5074 ↦ cmpt 5157 ‘cfv 6433 (class class class)co 7275 1c1 10872 · cmul 10876 ℕcn 11973 ℤcz 12319 Σcsu 15397 ∥ cdvds 15963 gcd cgcd 16201 Basecbs 16912 0_gc0g 17150 ℤRHomczrh 20701 ℤ/nℤczn 20704 DChrcdchr 26380

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8042 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-ec 8500 df-qs 8504 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-sum 15398 df-dvds 15964 df-gcd 16202 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-0g 17152 df-imas 17219 df-qus 17220 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-grp 18580 df-minusg 18581 df-sbg 18582 df-mulg 18701 df-subg 18752 df-nsg 18753 df-eqg 18754 df-ghm 18832 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-cring 19786 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-rnghom 19959 df-subrg 20022 df-lmod 20125 df-lss 20194 df-lsp 20234 df-sra 20434 df-rgmod 20435 df-lidl 20436 df-rsp 20437 df-2idl 20503 df-cnfld 20598 df-zring 20671 df-zrh 20705 df-zn 20708 df-dchr 26381

This theorem is referenced by: dchrisum0flblem2 26657

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