dchrvmaeq0 - Metamath Proof Explorer

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Theorem dchrvmaeq0 27831

Description: The set 𝑊 is the collection of all non-principal Dirichlet characters such that the sum Σ𝑛 ∈ ℕ, 𝑋(𝑛) / 𝑛 is equal to zero. (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)
rpvmasum.g	⊢ 𝐺 = (DChr‘𝑁)
rpvmasum.d	⊢ 𝐷 = (Base‘𝐺)
rpvmasum.1	⊢ 1 = (0_g‘𝐺)
dchrisum.b	⊢ (𝜑 → 𝑋 ∈ 𝐷)
dchrisum.n1	⊢ (𝜑 → 𝑋 ≠ 1 )
dchrvmasumif.f	⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))
dchrvmasumif.c	⊢ (𝜑 → 𝐶 ∈ (0[,)+∞))
dchrvmasumif.s	⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)
dchrvmasumif.1	⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / 𝑦))
dchrvmaeq0.w	⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}

**Assertion**
Ref	Expression
dchrvmaeq0	⊢ (𝜑 → (𝑋 ∈ 𝑊 ↔ 𝑆 = 0))

Distinct variable groups: 𝑦,𝑚, 1 𝐶,𝑚,𝑦 𝑦,𝐹 𝑚,𝑎,𝑦 𝑚,𝑁,𝑦 𝜑,𝑚 𝑆,𝑚,𝑦 𝑚,𝑍,𝑦 𝐷,𝑚,𝑦 𝐿,𝑎,𝑚,𝑦 𝑋,𝑎,𝑚,𝑦 𝑚,𝐹 Allowed substitution hints: 𝜑(𝑦, 𝑎) 𝐶(𝑎) 𝐷(𝑎) 𝑆(𝑎) 1 (𝑎) 𝐹(𝑎) 𝐺(𝑦, 𝑚, 𝑎) 𝑁(𝑎) 𝑊(𝑦, 𝑚, 𝑎) 𝑍(𝑎)

**Proof of Theorem dchrvmaeq0**
Step	Hyp	Ref	Expression
1		dchrisum.b	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
2		dchrisum.n1	. . . 4 ⊢ (𝜑 → 𝑋 ≠ 1 )
3		eldifsn 4748	. . . 4 ⊢ (𝑋 ∈ (𝐷 ∖ { 1 }) ↔ (𝑋 ∈ 𝐷 ∧ 𝑋 ≠ 1 ))
4	1, 2, 3	sylanbrc 595	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐷 ∖ { 1 }))
5		fveq1 6884	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → (𝑦‘(𝐿‘𝑚)) = (𝑋‘(𝐿‘𝑚)))
6	5	oveq1d 7435	. . . . . . 7 ⊢ (𝑦 = 𝑋 → ((𝑦‘(𝐿‘𝑚)) / 𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
7	6	sumeq2sdv 15870	. . . . . 6 ⊢ (𝑦 = 𝑋 → Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚))
8	7	eqeq1d 2763	. . . . 5 ⊢ (𝑦 = 𝑋 → (Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0 ↔ Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚) = 0))
9		dchrvmaeq0.w	. . . . 5 ⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}
10	8, 9	elrab2 3649	. . . 4 ⊢ (𝑋 ∈ 𝑊 ↔ (𝑋 ∈ (𝐷 ∖ { 1 }) ∧ Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚) = 0))
11	10	baib 545	. . 3 ⊢ (𝑋 ∈ (𝐷 ∖ { 1 }) → (𝑋 ∈ 𝑊 ↔ Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚) = 0))
12	4, 11	syl 18	. 2 ⊢ (𝜑 → (𝑋 ∈ 𝑊 ↔ Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚) = 0))
13		nnuz 13004	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
14		1zzd 12727	. . . 4 ⊢ (𝜑 → 1 ∈ ℤ)
15		2fveq3 6890	. . . . . . 7 ⊢ (𝑎 = 𝑚 → (𝑋‘(𝐿‘𝑎)) = (𝑋‘(𝐿‘𝑚)))
16		id 23	. . . . . . 7 ⊢ (𝑎 = 𝑚 → 𝑎 = 𝑚)
17	15, 16	oveq12d 7438	. . . . . 6 ⊢ (𝑎 = 𝑚 → ((𝑋‘(𝐿‘𝑎)) / 𝑎) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
18		dchrvmasumif.f	. . . . . 6 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))
19		ovex 7453	. . . . . 6 ⊢ ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ V
20	17, 18, 19	fvmpt 6993	. . . . 5 ⊢ (𝑚 ∈ ℕ → (𝐹‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
21	20	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
22		rpvmasum.g	. . . . . 6 ⊢ 𝐺 = (DChr‘𝑁)
23		rpvmasum.z	. . . . . 6 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
24		rpvmasum.d	. . . . . 6 ⊢ 𝐷 = (Base‘𝐺)
25		rpvmasum.l	. . . . . 6 ⊢ 𝐿 = (ℤRHom‘𝑍)
26	1	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑋 ∈ 𝐷)
27		nnz 12714	. . . . . . 7 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℤ)
28	27	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℤ)
29	22, 23, 24, 25, 26, 28	dchrzrhcl 27572	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
30		nncn 12343	. . . . . 6 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
31	30	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
32		nnne0 12372	. . . . . 6 ⊢ (𝑚 ∈ ℕ → 𝑚 ≠ 0)
33	32	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ≠ 0)
34	29, 31, 33	divcld 12093	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
35		dchrvmasumif.s	. . . 4 ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)
36	13, 14, 21, 34, 35	isumclim 15923	. . 3 ⊢ (𝜑 → Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚) = 𝑆)
37	36	eqeq1d 2763	. 2 ⊢ (𝜑 → (Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚) = 0 ↔ 𝑆 = 0))
38	12, 37	bitrd 282	1 ⊢ (𝜑 → (𝑋 ∈ 𝑊 ↔ 𝑆 = 0))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ∀wral 3077 {crab 3413 ∖ cdif 3896 {csn 4584 class class class wbr 5103 ↦ cmpt 5186 ‘cfv 6538 (class class class)co 7420 ℂcc 11198 0cc0 11200 1c1 11201 + caddc 11203 +∞cpnf 11340 ≤ cle 11344 − cmin 11541 / cdiv 11973 ℕcn 12335 ℤcz 12693 [,)cico 13478 ⌊cfl 13930 seqcseq 14144 abscabs 15401 ⇝ cli 15651 Σcsu 15853 Basecbs 17387 0_gc0g 17610 ℤRHomczrh 21805 ℤ/nℤczn 21808 DChrcdchr 27559

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-inf2 9642 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278 ax-addf 11279 ax-mulf 11280

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-tpos 8243 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-er 8717 df-ec 8719 df-qs 8723 df-map 8849 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-sup 9434 df-inf 9435 df-oi 9504 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-z 12694 df-dec 12815 df-uz 12966 df-rp 13121 df-fz 13640 df-fzo 13789 df-seq 14145 df-exp 14205 df-hash 14475 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-clim 15655 df-sum 15854 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-starv 17443 df-sca 17444 df-vsca 17445 df-ip 17446 df-tset 17447 df-ple 17448 df-ds 17450 df-unif 17451 df-0g 17612 df-imas 17680 df-qus 17681 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-mhm 18978 df-grp 19147 df-minusg 19148 df-sbg 19149 df-mulg 19278 df-subg 19333 df-nsg 19334 df-eqg 19335 df-ghm 19428 df-cmn 19996 df-abl 19997 df-mgp 20361 df-rng 20375 df-ur 20408 df-ring 20461 df-cring 20462 df-oppr 20567 df-dvdsr 20587 df-unit 20588 df-rhm 20702 df-subrng 20798 df-subrg 20822 df-lmod 21137 df-lss 21207 df-lsp 21247 df-sra 21448 df-rgmod 21449 df-lidl 21486 df-rsp 21487 df-2idl 21543 df-cnfld 21679 df-zring 21753 df-zrh 21809 df-zn 21812 df-dchr 27560

This theorem is used by: rpvmasum2 27839 dchrisum0re 27840 dchrisum0lem2 27845 dchrisumn0 27848

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