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Theorem dchrisum0fmul 27568

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 2740	. . 3 ⊢ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} = {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴}
5		eqid 2740	. . 3 ⊢ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵} = {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵}
6		eqid 2740	. . 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 480	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴}) → 𝑋 ∈ 𝐷)
13		elrabi 3703	. . . . . 6 ⊢ (𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} → 𝑗 ∈ ℕ)
14	13	nnzd 12666	. . . . 5 ⊢ (𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} → 𝑗 ∈ ℤ)
15	14	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴}) → 𝑗 ∈ ℤ)
16	7, 8, 9, 10, 12, 15	dchrzrhcl 27307	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴}) → (𝑋‘(𝐿‘𝑗)) ∈ ℂ)
17	11	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵}) → 𝑋 ∈ 𝐷)
18		elrabi 3703	. . . . . 6 ⊢ (𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵} → 𝑘 ∈ ℕ)
19	18	nnzd 12666	. . . . 5 ⊢ (𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵} → 𝑘 ∈ ℤ)
20	19	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵}) → 𝑘 ∈ ℤ)
21	7, 8, 9, 10, 17, 20	dchrzrhcl 27307	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵}) → (𝑋‘(𝐿‘𝑘)) ∈ ℂ)
22	14, 19	anim12i 612	. . . 4 ⊢ ((𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵}) → (𝑗 ∈ ℤ ∧ 𝑘 ∈ ℤ))
23	11	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → 𝑋 ∈ 𝐷)
24		simprl 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → 𝑗 ∈ ℤ)
25		simprr 772	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → 𝑘 ∈ ℤ)
26	7, 8, 9, 10, 23, 24, 25	dchrzrhmul 27308	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → (𝑋‘(𝐿‘(𝑗 · 𝑘))) = ((𝑋‘(𝐿‘𝑗)) · (𝑋‘(𝐿‘𝑘))))
27	26	eqcomd 2746	. . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → ((𝑋‘(𝐿‘𝑗)) · (𝑋‘(𝐿‘𝑘))) = (𝑋‘(𝐿‘(𝑗 · 𝑘))))
28	22, 27	sylan2 592	. . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵})) → ((𝑋‘(𝐿‘𝑗)) · (𝑋‘(𝐿‘𝑘))) = (𝑋‘(𝐿‘(𝑗 · 𝑘))))
29		2fveq3 6925	. . 3 ⊢ (𝑖 = (𝑗 · 𝑘) → (𝑋‘(𝐿‘𝑖)) = (𝑋‘(𝐿‘(𝑗 · 𝑘))))
30	1, 2, 3, 4, 5, 6, 16, 21, 28, 29	fsumdvdsmul 27256	. 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 27567	. . . 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 27567	. . . 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 7466	. 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 12346	. . 3 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℕ)
40	8, 10, 31, 7, 9, 32, 33	dchrisum0fval 27567	. . 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 2791	1 ⊢ (𝜑 → (𝐹‘(𝐴 · 𝐵)) = ((𝐹‘𝐴) · (𝐹‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 {crab 3443 class class class wbr 5166 ↦ cmpt 5249 ‘cfv 6573 (class class class)co 7448 1c1 11185 · cmul 11189 ℕcn 12293 ℤcz 12639 Σcsu 15734 ∥ cdvds 16302 gcd cgcd 16540 Basecbs 17258 0_gc0g 17499 ℤRHomczrh 21533 ℤ/nℤczn 21536 DChrcdchr 27294

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263 ax-mulf 11264

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-tpos 8267 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-er 8763 df-ec 8765 df-qs 8769 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-rp 13058 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-clim 15534 df-sum 15735 df-dvds 16303 df-gcd 16541 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-0g 17501 df-imas 17568 df-qus 17569 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-grp 18976 df-minusg 18977 df-sbg 18978 df-mulg 19108 df-subg 19163 df-nsg 19164 df-eqg 19165 df-ghm 19253 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-cring 20263 df-oppr 20360 df-dvdsr 20383 df-unit 20384 df-rhm 20498 df-subrng 20572 df-subrg 20597 df-lmod 20882 df-lss 20953 df-lsp 20993 df-sra 21195 df-rgmod 21196 df-lidl 21241 df-rsp 21242 df-2idl 21283 df-cnfld 21388 df-zring 21481 df-zrh 21537 df-zn 21540 df-dchr 27295

This theorem is referenced by: dchrisum0flblem2 27571

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