dchrisum0 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dchrisum0		Structured version Visualization version GIF version

Theorem dchrisum0 27483

Description: The sum Σ𝑛 ∈ ℕ, 𝑋(𝑛) / 𝑛 is nonzero for all non-principal Dirichlet characters (i.e. the assumption 𝑋 ∈ 𝑊 is contradictory). This is the key result that allows to eliminate the conditionals from dchrmusum2 27457 and dchrvmasumif 27466. Lemma 9.4.4 of [Shapiro], p. 382. (Contributed by Mario Carneiro, 12-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)
rpvmasum2.w	⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}
dchrisum0.b	⊢ (𝜑 → 𝑋 ∈ 𝑊)

**Assertion**
Ref	Expression
dchrisum0	⊢ ¬ 𝜑

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

Proof of Theorem dchrisum0 Dummy variables 𝑘 𝑥 𝑧 𝑐 𝑖 𝑡 𝑑 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rpvmasum.z	. 2 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
2		rpvmasum.l	. 2 ⊢ 𝐿 = (ℤRHom‘𝑍)
3		rpvmasum.a	. 2 ⊢ (𝜑 → 𝑁 ∈ ℕ)
4		rpvmasum2.g	. 2 ⊢ 𝐺 = (DChr‘𝑁)
5		rpvmasum2.d	. 2 ⊢ 𝐷 = (Base‘𝐺)
6		rpvmasum2.1	. 2 ⊢ 1 = (0_g‘𝐺)
7		eqid 2736	. 2 ⊢ (𝑏 ∈ ℕ ↦ Σ𝑦 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑏} (𝑋‘(𝐿‘𝑦))) = (𝑏 ∈ ℕ ↦ Σ𝑦 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑏} (𝑋‘(𝐿‘𝑦)))
8		rpvmasum2.w	. . . . 5 ⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}
9	8	ssrab3 4022	. . . 4 ⊢ 𝑊 ⊆ (𝐷 ∖ { 1 })
10		difss 4076	. . . 4 ⊢ (𝐷 ∖ { 1 }) ⊆ 𝐷
11	9, 10	sstri 3931	. . 3 ⊢ 𝑊 ⊆ 𝐷
12		dchrisum0.b	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑊)
13	11, 12	sselid 3919	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
14	1, 2, 3, 4, 5, 6, 8, 12	dchrisum0re 27476	. 2 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℝ)
15		fveq2 6840	. . . . . . . 8 ⊢ (𝑘 = (𝑚 · 𝑑) → (√‘𝑘) = (√‘(𝑚 · 𝑑)))
16	15	oveq2d 7383	. . . . . . 7 ⊢ (𝑘 = (𝑚 · 𝑑) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑘)) = ((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))))
17		rpre 12951	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
18	17	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
19	13	ad3antrrr 731	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘}) → 𝑋 ∈ 𝐷)
20		elrabi 3630	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘} → 𝑚 ∈ ℕ)
21	20	nnzd 12550	. . . . . . . . . . 11 ⊢ (𝑚 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘} → 𝑚 ∈ ℤ)
22	21	adantl 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘}) → 𝑚 ∈ ℤ)
23	4, 1, 5, 2, 19, 22	dchrzrhcl 27208	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘}) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
24		elfznn 13507	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1...(⌊‘𝑥)) → 𝑘 ∈ ℕ)
25	24	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ∈ ℕ)
26	25	nnrpd 12984	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ∈ ℝ⁺)
27	26	rpsqrtcld 15374	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (√‘𝑘) ∈ ℝ⁺)
28	27	rpcnd 12988	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (√‘𝑘) ∈ ℂ)
29	28	adantr 480	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘}) → (√‘𝑘) ∈ ℂ)
30	27	rpne0d 12991	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (√‘𝑘) ≠ 0)
31	30	adantr 480	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘}) → (√‘𝑘) ≠ 0)
32	23, 29, 31	divcld 11931	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘}) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑘)) ∈ ℂ)
33	32	anasss 466	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘})) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑘)) ∈ ℂ)
34	16, 18, 33	dvdsflsumcom 27151	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑘 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘} ((𝑋‘(𝐿‘𝑚)) / (√‘𝑘)) = Σ𝑚 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘(𝑥 / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))))
35	1, 2, 3, 4, 5, 6, 7	dchrisum0fval 27468	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → ((𝑏 ∈ ℕ ↦ Σ𝑦 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑏} (𝑋‘(𝐿‘𝑦)))‘𝑘) = Σ𝑚 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘} (𝑋‘(𝐿‘𝑚)))
36	25, 35	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → ((𝑏 ∈ ℕ ↦ Σ𝑦 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑏} (𝑋‘(𝐿‘𝑦)))‘𝑘) = Σ𝑚 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘} (𝑋‘(𝐿‘𝑚)))
37	36	oveq1d 7382	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (((𝑏 ∈ ℕ ↦ Σ𝑦 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑏} (𝑋‘(𝐿‘𝑦)))‘𝑘) / (√‘𝑘)) = (Σ𝑚 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘} (𝑋‘(𝐿‘𝑚)) / (√‘𝑘)))
38		fzfid 13935	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (1...𝑘) ∈ Fin)
39		dvdsssfz1 16287	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘} ⊆ (1...𝑘))
40	25, 39	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘} ⊆ (1...𝑘))
41	38, 40	ssfid 9179	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘} ∈ Fin)
42	41, 28, 23, 30	fsumdivc 15748	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘} (𝑋‘(𝐿‘𝑚)) / (√‘𝑘)) = Σ𝑚 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘} ((𝑋‘(𝐿‘𝑚)) / (√‘𝑘)))
43	37, 42	eqtrd 2771	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (((𝑏 ∈ ℕ ↦ Σ𝑦 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑏} (𝑋‘(𝐿‘𝑦)))‘𝑘) / (√‘𝑘)) = Σ𝑚 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘} ((𝑋‘(𝐿‘𝑚)) / (√‘𝑘)))
44	43	sumeq2dv 15664	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑘 ∈ (1...(⌊‘𝑥))(((𝑏 ∈ ℕ ↦ Σ𝑦 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑏} (𝑋‘(𝐿‘𝑦)))‘𝑘) / (√‘𝑘)) = Σ𝑘 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘} ((𝑋‘(𝐿‘𝑚)) / (√‘𝑘)))
45		rprege0 12958	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
46	45	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
47		resqrtth 15217	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → ((√‘𝑥)↑2) = 𝑥)
48	46, 47	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((√‘𝑥)↑2) = 𝑥)
49	48	fveq2d 6844	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (⌊‘((√‘𝑥)↑2)) = (⌊‘𝑥))
50	49	oveq2d 7383	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘((√‘𝑥)↑2))) = (1...(⌊‘𝑥)))
51	48	fvoveq1d 7389	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (⌊‘(((√‘𝑥)↑2) / 𝑚)) = (⌊‘(𝑥 / 𝑚)))
52	51	oveq2d 7383	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘(((√‘𝑥)↑2) / 𝑚))) = (1...(⌊‘(𝑥 / 𝑚))))
53	52	sumeq1d 15662	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘(((√‘𝑥)↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))) = Σ𝑑 ∈ (1...(⌊‘(𝑥 / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))))
54	53	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘((√‘𝑥)↑2)))) → Σ𝑑 ∈ (1...(⌊‘(((√‘𝑥)↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))) = Σ𝑑 ∈ (1...(⌊‘(𝑥 / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))))
55	50, 54	sumeq12dv 15668	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘((√‘𝑥)↑2)))Σ𝑑 ∈ (1...(⌊‘(((√‘𝑥)↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))) = Σ𝑚 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘(𝑥 / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))))
56	34, 44, 55	3eqtr4d 2781	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑘 ∈ (1...(⌊‘𝑥))(((𝑏 ∈ ℕ ↦ Σ𝑦 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑏} (𝑋‘(𝐿‘𝑦)))‘𝑘) / (√‘𝑘)) = Σ𝑚 ∈ (1...(⌊‘((√‘𝑥)↑2)))Σ𝑑 ∈ (1...(⌊‘(((√‘𝑥)↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))))
57	56	mpteq2dva 5178	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑘 ∈ (1...(⌊‘𝑥))(((𝑏 ∈ ℕ ↦ Σ𝑦 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑏} (𝑋‘(𝐿‘𝑦)))‘𝑘) / (√‘𝑘))) = (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘((√‘𝑥)↑2)))Σ𝑑 ∈ (1...(⌊‘(((√‘𝑥)↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑)))))
58		rpsqrtcl 15226	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (√‘𝑥) ∈ ℝ⁺)
59	58	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℝ⁺)
60		eqidd 2737	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (√‘𝑥)) = (𝑥 ∈ ℝ⁺ ↦ (√‘𝑥)))
61		eqidd 2737	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘(𝑧↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑧↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑)))) = (𝑧 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘(𝑧↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑧↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑)))))
62		oveq1 7374	. . . . . . . 8 ⊢ (𝑧 = (√‘𝑥) → (𝑧↑2) = ((√‘𝑥)↑2))
63	62	fveq2d 6844	. . . . . . 7 ⊢ (𝑧 = (√‘𝑥) → (⌊‘(𝑧↑2)) = (⌊‘((√‘𝑥)↑2)))
64	63	oveq2d 7383	. . . . . 6 ⊢ (𝑧 = (√‘𝑥) → (1...(⌊‘(𝑧↑2))) = (1...(⌊‘((√‘𝑥)↑2))))
65	62	fvoveq1d 7389	. . . . . . . . 9 ⊢ (𝑧 = (√‘𝑥) → (⌊‘((𝑧↑2) / 𝑚)) = (⌊‘(((√‘𝑥)↑2) / 𝑚)))
66	65	oveq2d 7383	. . . . . . . 8 ⊢ (𝑧 = (√‘𝑥) → (1...(⌊‘((𝑧↑2) / 𝑚))) = (1...(⌊‘(((√‘𝑥)↑2) / 𝑚))))
67	66	sumeq1d 15662	. . . . . . 7 ⊢ (𝑧 = (√‘𝑥) → Σ𝑑 ∈ (1...(⌊‘((𝑧↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))) = Σ𝑑 ∈ (1...(⌊‘(((√‘𝑥)↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))))
68	67	adantr 480	. . . . . 6 ⊢ ((𝑧 = (√‘𝑥) ∧ 𝑚 ∈ (1...(⌊‘(𝑧↑2)))) → Σ𝑑 ∈ (1...(⌊‘((𝑧↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))) = Σ𝑑 ∈ (1...(⌊‘(((√‘𝑥)↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))))
69	64, 68	sumeq12dv 15668	. . . . 5 ⊢ (𝑧 = (√‘𝑥) → Σ𝑚 ∈ (1...(⌊‘(𝑧↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑧↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))) = Σ𝑚 ∈ (1...(⌊‘((√‘𝑥)↑2)))Σ𝑑 ∈ (1...(⌊‘(((√‘𝑥)↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))))
70	59, 60, 61, 69	fmptco 7082	. . . 4 ⊢ (𝜑 → ((𝑧 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘(𝑧↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑧↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑)))) ∘ (𝑥 ∈ ℝ⁺ ↦ (√‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘((√‘𝑥)↑2)))Σ𝑑 ∈ (1...(⌊‘(((√‘𝑥)↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑)))))
71	57, 70	eqtr4d 2774	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑘 ∈ (1...(⌊‘𝑥))(((𝑏 ∈ ℕ ↦ Σ𝑦 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑏} (𝑋‘(𝐿‘𝑦)))‘𝑘) / (√‘𝑘))) = ((𝑧 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘(𝑧↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑧↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑)))) ∘ (𝑥 ∈ ℝ⁺ ↦ (√‘𝑥))))
72		eqid 2736	. . . . . . . 8 ⊢ (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎))) = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)))
73	1, 2, 3, 4, 5, 6, 8, 12, 72	dchrisum0lema 27477	. . . . . . 7 ⊢ (𝜑 → ∃𝑡∃𝑐 ∈ (0[,)+∞)(seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎))))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / (√‘𝑦))))
74	3	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎))))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / (√‘𝑦))))) → 𝑁 ∈ ℕ)
75	12	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎))))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / (√‘𝑦))))) → 𝑋 ∈ 𝑊)
76		simprl 771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎))))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / (√‘𝑦))))) → 𝑐 ∈ (0[,)+∞))
77		simprrl 781	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎))))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / (√‘𝑦))))) → seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)))) ⇝ 𝑡)
78		simprrr 782	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎))))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / (√‘𝑦))))) → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎))))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / (√‘𝑦)))
79	1, 2, 74, 4, 5, 6, 8, 75, 72, 76, 77, 78	dchrisum0lem3 27482	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎))))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / (√‘𝑦))))) → (𝑧 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘(𝑧↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑧↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑)))) ∈ 𝑂(1))
80	79	rexlimdvaa 3139	. . . . . . . 8 ⊢ (𝜑 → (∃𝑐 ∈ (0[,)+∞)(seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎))))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / (√‘𝑦))) → (𝑧 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘(𝑧↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑧↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑)))) ∈ 𝑂(1)))
81	80	exlimdv 1935	. . . . . . 7 ⊢ (𝜑 → (∃𝑡∃𝑐 ∈ (0[,)+∞)(seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎))))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / (√‘𝑦))) → (𝑧 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘(𝑧↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑧↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑)))) ∈ 𝑂(1)))
82	73, 81	mpd 15	. . . . . 6 ⊢ (𝜑 → (𝑧 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘(𝑧↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑧↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑)))) ∈ 𝑂(1))
83		o1f 15491	. . . . . 6 ⊢ ((𝑧 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘(𝑧↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑧↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑)))) ∈ 𝑂(1) → (𝑧 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘(𝑧↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑧↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑)))):dom (𝑧 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘(𝑧↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑧↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))))⟶ℂ)
84	82, 83	syl 17	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘(𝑧↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑧↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑)))):dom (𝑧 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘(𝑧↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑧↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))))⟶ℂ)
85		sumex 15650	. . . . . . 7 ⊢ Σ𝑚 ∈ (1...(⌊‘(𝑧↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑧↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))) ∈ V
86		eqid 2736	. . . . . . 7 ⊢ (𝑧 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘(𝑧↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑧↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑)))) = (𝑧 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘(𝑧↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑧↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))))
87	85, 86	dmmpti 6642	. . . . . 6 ⊢ dom (𝑧 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘(𝑧↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑧↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑)))) = ℝ⁺
88	87	feq2i 6660	. . . . 5 ⊢ ((𝑧 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘(𝑧↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑧↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑)))):dom (𝑧 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘(𝑧↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑧↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))))⟶ℂ ↔ (𝑧 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘(𝑧↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑧↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑)))):ℝ⁺⟶ℂ)
89	84, 88	sylib 218	. . . 4 ⊢ (𝜑 → (𝑧 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘(𝑧↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑧↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑)))):ℝ⁺⟶ℂ)
90		rpssre 12950	. . . . 5 ⊢ ℝ⁺ ⊆ ℝ
91	90	a1i 11	. . . 4 ⊢ (𝜑 → ℝ⁺ ⊆ ℝ)
92		resqcl 14086	. . . . 5 ⊢ (𝑡 ∈ ℝ → (𝑡↑2) ∈ ℝ)
93		0red 11147	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) → 0 ∈ ℝ)
94		simplr 769	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) → 𝑡 ∈ ℝ)
95		simplrr 778	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 0 ≤ 𝑡) → (𝑡↑2) ≤ 𝑥)
96	45	ad2antrl 729	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
97	96	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 0 ≤ 𝑡) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
98	97, 47	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 0 ≤ 𝑡) → ((√‘𝑥)↑2) = 𝑥)
99	95, 98	breqtrrd 5113	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 0 ≤ 𝑡) → (𝑡↑2) ≤ ((√‘𝑥)↑2))
100	94	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 0 ≤ 𝑡) → 𝑡 ∈ ℝ)
101	59	rpred 12986	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℝ)
102	101	ad2ant2r 748	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) → (√‘𝑥) ∈ ℝ)
103	102	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 0 ≤ 𝑡) → (√‘𝑥) ∈ ℝ)
104		simpr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 0 ≤ 𝑡) → 0 ≤ 𝑡)
105		sqrtge0 15219	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → 0 ≤ (√‘𝑥))
106	96, 105	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) → 0 ≤ (√‘𝑥))
107	106	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 0 ≤ 𝑡) → 0 ≤ (√‘𝑥))
108	100, 103, 104, 107	le2sqd 14219	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 0 ≤ 𝑡) → (𝑡 ≤ (√‘𝑥) ↔ (𝑡↑2) ≤ ((√‘𝑥)↑2)))
109	99, 108	mpbird 257	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 0 ≤ 𝑡) → 𝑡 ≤ (√‘𝑥))
110	94	adantr 480	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 𝑡 ≤ 0) → 𝑡 ∈ ℝ)
111		0red 11147	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 𝑡 ≤ 0) → 0 ∈ ℝ)
112	102	adantr 480	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 𝑡 ≤ 0) → (√‘𝑥) ∈ ℝ)
113		simpr 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 𝑡 ≤ 0) → 𝑡 ≤ 0)
114	106	adantr 480	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 𝑡 ≤ 0) → 0 ≤ (√‘𝑥))
115	110, 111, 112, 113, 114	letrd 11303	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 𝑡 ≤ 0) → 𝑡 ≤ (√‘𝑥))
116	93, 94, 109, 115	lecasei 11252	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) → 𝑡 ≤ (√‘𝑥))
117	116	expr 456	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → ((𝑡↑2) ≤ 𝑥 → 𝑡 ≤ (√‘𝑥)))
118	117	ralrimiva 3129	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → ∀𝑥 ∈ ℝ⁺ ((𝑡↑2) ≤ 𝑥 → 𝑡 ≤ (√‘𝑥)))
119		breq1 5088	. . . . . 6 ⊢ (𝑐 = (𝑡↑2) → (𝑐 ≤ 𝑥 ↔ (𝑡↑2) ≤ 𝑥))
120	119	rspceaimv 3570	. . . . 5 ⊢ (((𝑡↑2) ∈ ℝ ∧ ∀𝑥 ∈ ℝ⁺ ((𝑡↑2) ≤ 𝑥 → 𝑡 ≤ (√‘𝑥))) → ∃𝑐 ∈ ℝ ∀𝑥 ∈ ℝ⁺ (𝑐 ≤ 𝑥 → 𝑡 ≤ (√‘𝑥)))
121	92, 118, 120	syl2an2 687	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → ∃𝑐 ∈ ℝ ∀𝑥 ∈ ℝ⁺ (𝑐 ≤ 𝑥 → 𝑡 ≤ (√‘𝑥)))
122	89, 82, 59, 91, 121	o1compt 15549	. . 3 ⊢ (𝜑 → ((𝑧 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘(𝑧↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑧↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑)))) ∘ (𝑥 ∈ ℝ⁺ ↦ (√‘𝑥))) ∈ 𝑂(1))
123	71, 122	eqeltrd 2836	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑘 ∈ (1...(⌊‘𝑥))(((𝑏 ∈ ℕ ↦ Σ𝑦 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑏} (𝑋‘(𝐿‘𝑦)))‘𝑘) / (√‘𝑘))) ∈ 𝑂(1))
124	1, 2, 3, 4, 5, 6, 7, 13, 14, 123	dchrisum0fno1 27474	1 ⊢ ¬ 𝜑

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1542 ∃wex 1781 ∈ wcel 2114 ≠ wne 2932 ∀wral 3051 ∃wrex 3061 {crab 3389 ∖ cdif 3886 ⊆ wss 3889 {csn 4567 class class class wbr 5085 ↦ cmpt 5166 dom cdm 5631 ∘ ccom 5635 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 +∞cpnf 11176 ≤ cle 11180 − cmin 11377 / cdiv 11807 ℕcn 12174 2c2 12236 ℤcz 12524 ℝ⁺crp 12942 [,)cico 13300 ...cfz 13461 ⌊cfl 13749 seqcseq 13963 ↑cexp 14023 √csqrt 15195 abscabs 15196 ⇝ cli 15446 𝑂(1)co1 15448 Σcsu 15648 ∥ cdvds 16221 Basecbs 17179 0_gc0g 17402 ℤRHomczrh 21479 ℤ/nℤczn 21482 DChrcdchr 27195

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-disj 5053 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-rpss 7677 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-tpos 8176 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-omul 8410 df-er 8643 df-ec 8645 df-qs 8649 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-dju 9825 df-card 9863 df-acn 9866 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-xnn0 12511 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-fac 14236 df-bc 14265 df-hash 14293 df-word 14476 df-concat 14533 df-s1 14559 df-shft 15029 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433 df-clim 15450 df-rlim 15451 df-o1 15452 df-lo1 15453 df-sum 15649 df-ef 16032 df-e 16033 df-sin 16034 df-cos 16035 df-tan 16036 df-pi 16037 df-dvds 16222 df-gcd 16464 df-prm 16641 df-numer 16705 df-denom 16706 df-phi 16736 df-pc 16808 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-qus 17473 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-mhm 18751 df-submnd 18752 df-grp 18912 df-minusg 18913 df-sbg 18914 df-mulg 19044 df-subg 19099 df-nsg 19100 df-eqg 19101 df-ghm 19188 df-gim 19234 df-ga 19265 df-cntz 19292 df-oppg 19321 df-od 19503 df-gex 19504 df-pgp 19505 df-lsm 19611 df-pj1 19612 df-cmn 19757 df-abl 19758 df-cyg 19853 df-dprd 19972 df-dpj 19973 df-mgp 20122 df-rng 20134 df-ur 20163 df-ring 20216 df-cring 20217 df-oppr 20317 df-dvdsr 20337 df-unit 20338 df-invr 20368 df-dvr 20381 df-rhm 20452 df-subrng 20523 df-subrg 20547 df-drng 20708 df-lmod 20857 df-lss 20927 df-lsp 20967 df-sra 21168 df-rgmod 21169 df-lidl 21206 df-rsp 21207 df-2idl 21248 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-zring 21427 df-zrh 21483 df-zn 21486 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-cmp 23352 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-0p 25637 df-limc 25833 df-dv 25834 df-ply 26153 df-idp 26154 df-coe 26155 df-dgr 26156 df-quot 26257 df-ulm 26342 df-log 26520 df-cxp 26521 df-atan 26831 df-em 26956 df-cht 27060 df-vma 27061 df-chp 27062 df-ppi 27063 df-mu 27064 df-dchr 27196

This theorem is referenced by: dchrisumn0 27484 rpvmasum 27489

W3C validator