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Theorem dchrisum0fmul 26854

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 2736	. . 3 ⊢ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} = {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴}
5		eqid 2736	. . 3 ⊢ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵} = {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵}
6		eqid 2736	. . 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 3639	. . . . . 6 ⊢ (𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} → 𝑗 ∈ ℕ)
14	13	nnzd 12526	. . . . 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 26593	. . 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 3639	. . . . . 6 ⊢ (𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵} → 𝑘 ∈ ℕ)
19	18	nnzd 12526	. . . . 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 26593	. . 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 769	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → 𝑗 ∈ ℤ)
25		simprr 771	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → 𝑘 ∈ ℤ)
26	7, 8, 9, 10, 23, 24, 25	dchrzrhmul 26594	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → (𝑋‘(𝐿‘(𝑗 · 𝑘))) = ((𝑋‘(𝐿‘𝑗)) · (𝑋‘(𝐿‘𝑘))))
27	26	eqcomd 2742	. . . 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 6847	. . 3 ⊢ (𝑖 = (𝑗 · 𝑘) → (𝑋‘(𝐿‘𝑖)) = (𝑋‘(𝐿‘(𝑗 · 𝑘))))
30	1, 2, 3, 4, 5, 6, 16, 21, 28, 29	fsumdvdsmul 26544	. 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 26853	. . . 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 26853	. . . 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 7375	. 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 12206	. . 3 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℕ)
40	8, 10, 31, 7, 9, 32, 33	dchrisum0fval 26853	. . 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 2787	1 ⊢ (𝜑 → (𝐹‘(𝐴 · 𝐵)) = ((𝐹‘𝐴) · (𝐹‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {crab 3407 class class class wbr 5105 ↦ cmpt 5188 ‘cfv 6496 (class class class)co 7357 1c1 11052 · cmul 11056 ℕcn 12153 ℤcz 12499 Σcsu 15570 ∥ cdvds 16136 gcd cgcd 16374 Basecbs 17083 0_gc0g 17321 ℤRHomczrh 20900 ℤ/nℤczn 20903 DChrcdchr 26580

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-tpos 8157 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-ec 8650 df-qs 8654 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-rp 12916 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-sum 15571 df-dvds 16137 df-gcd 16375 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-0g 17323 df-imas 17390 df-qus 17391 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-mhm 18601 df-grp 18751 df-minusg 18752 df-sbg 18753 df-mulg 18873 df-subg 18925 df-nsg 18926 df-eqg 18927 df-ghm 19006 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-cring 19967 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-rnghom 20146 df-subrg 20220 df-lmod 20324 df-lss 20393 df-lsp 20433 df-sra 20633 df-rgmod 20634 df-lidl 20635 df-rsp 20636 df-2idl 20702 df-cnfld 20797 df-zring 20870 df-zrh 20904 df-zn 20907 df-dchr 26581

This theorem is referenced by: dchrisum0flblem2 26857

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