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Mirrors > Home > MPE Home > Th. List > dchrisum0fmul | Structured version Visualization version GIF version |
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.) |
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 = (0g‘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) |
Ref | Expression |
---|---|
dchrisum0fmul | ⊢ (𝜑 → (𝐹‘(𝐴 · 𝐵)) = ((𝐹‘𝐴) · (𝐹‘𝐵))) |
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 2821 | . . 3 ⊢ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} = {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} | |
5 | eqid 2821 | . . 3 ⊢ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵} = {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵} | |
6 | eqid 2821 | . . 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 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴}) → 𝑋 ∈ 𝐷) |
13 | elrabi 3674 | . . . . . 6 ⊢ (𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} → 𝑗 ∈ ℕ) | |
14 | 13 | nnzd 12085 | . . . . 5 ⊢ (𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} → 𝑗 ∈ ℤ) |
15 | 14 | adantl 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴}) → 𝑗 ∈ ℤ) |
16 | 7, 8, 9, 10, 12, 15 | dchrzrhcl 25820 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴}) → (𝑋‘(𝐿‘𝑗)) ∈ ℂ) |
17 | 11 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵}) → 𝑋 ∈ 𝐷) |
18 | elrabi 3674 | . . . . . 6 ⊢ (𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵} → 𝑘 ∈ ℕ) | |
19 | 18 | nnzd 12085 | . . . . 5 ⊢ (𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵} → 𝑘 ∈ ℤ) |
20 | 19 | adantl 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵}) → 𝑘 ∈ ℤ) |
21 | 7, 8, 9, 10, 17, 20 | dchrzrhcl 25820 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵}) → (𝑋‘(𝐿‘𝑘)) ∈ ℂ) |
22 | 14, 19 | anim12i 614 | . . . 4 ⊢ ((𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵}) → (𝑗 ∈ ℤ ∧ 𝑘 ∈ ℤ)) |
23 | 11 | adantr 483 | . . . . . 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 25821 | . . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → (𝑋‘(𝐿‘(𝑗 · 𝑘))) = ((𝑋‘(𝐿‘𝑗)) · (𝑋‘(𝐿‘𝑘)))) |
27 | 26 | eqcomd 2827 | . . . 4 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → ((𝑋‘(𝐿‘𝑗)) · (𝑋‘(𝐿‘𝑘))) = (𝑋‘(𝐿‘(𝑗 · 𝑘)))) |
28 | 22, 27 | sylan2 594 | . . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐴} ∧ 𝑘 ∈ {𝑞 ∈ ℕ ∣ 𝑞 ∥ 𝐵})) → ((𝑋‘(𝐿‘𝑗)) · (𝑋‘(𝐿‘𝑘))) = (𝑋‘(𝐿‘(𝑗 · 𝑘)))) |
29 | 2fveq3 6674 | . . 3 ⊢ (𝑖 = (𝑗 · 𝑘) → (𝑋‘(𝐿‘𝑖)) = (𝑋‘(𝐿‘(𝑗 · 𝑘)))) | |
30 | 1, 2, 3, 4, 5, 6, 16, 21, 28, 29 | fsumdvdsmul 25771 | . 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 = (0g‘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 26080 | . . . 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 26080 | . . . 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 7173 | . 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 11689 | . . 3 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℕ) |
40 | 8, 10, 31, 7, 9, 32, 33 | dchrisum0fval 26080 | . . 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 2867 | 1 ⊢ (𝜑 → (𝐹‘(𝐴 · 𝐵)) = ((𝐹‘𝐴) · (𝐹‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {crab 3142 class class class wbr 5065 ↦ cmpt 5145 ‘cfv 6354 (class class class)co 7155 1c1 10537 · cmul 10541 ℕcn 11637 ℤcz 11980 Σcsu 15041 ∥ cdvds 15606 gcd cgcd 15842 Basecbs 16482 0gc0g 16712 ℤRHomczrh 20646 ℤ/nℤczn 20649 DChrcdchr 25807 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 ax-addf 10615 ax-mulf 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-tpos 7891 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-ec 8290 df-qs 8294 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-sup 8905 df-inf 8906 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-rp 12389 df-fz 12892 df-fzo 13033 df-fl 13161 df-mod 13237 df-seq 13369 df-exp 13429 df-hash 13690 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-clim 14844 df-sum 15042 df-dvds 15607 df-gcd 15843 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-starv 16579 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-unif 16587 df-0g 16714 df-imas 16780 df-qus 16781 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-mhm 17955 df-grp 18105 df-minusg 18106 df-sbg 18107 df-mulg 18224 df-subg 18275 df-nsg 18276 df-eqg 18277 df-ghm 18355 df-cmn 18907 df-abl 18908 df-mgp 19239 df-ur 19251 df-ring 19298 df-cring 19299 df-oppr 19372 df-dvdsr 19390 df-unit 19391 df-rnghom 19466 df-subrg 19532 df-lmod 19635 df-lss 19703 df-lsp 19743 df-sra 19943 df-rgmod 19944 df-lidl 19945 df-rsp 19946 df-2idl 20004 df-cnfld 20545 df-zring 20617 df-zrh 20650 df-zn 20653 df-dchr 25808 |
This theorem is referenced by: dchrisum0flblem2 26084 |
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