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Theorem dchrisum0 27481

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 27455 and dchrvmasumif 27464. 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 2735	. 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 4057	. . . 4 ⊢ 𝑊 ⊆ (𝐷 ∖ { 1 })
10		difss 4111	. . . 4 ⊢ (𝐷 ∖ { 1 }) ⊆ 𝐷
11	9, 10	sstri 3968	. . 3 ⊢ 𝑊 ⊆ 𝐷
12		dchrisum0.b	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑊)
13	11, 12	sselid 3956	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
14	1, 2, 3, 4, 5, 6, 8, 12	dchrisum0re 27474	. 2 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℝ)
15		fveq2 6875	. . . . . . . 8 ⊢ (𝑘 = (𝑚 · 𝑑) → (√‘𝑘) = (√‘(𝑚 · 𝑑)))
16	15	oveq2d 7419	. . . . . . 7 ⊢ (𝑘 = (𝑚 · 𝑑) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑘)) = ((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))))
17		rpre 13015	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
18	17	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
19	13	ad3antrrr 730	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘}) → 𝑋 ∈ 𝐷)
20		elrabi 3666	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘} → 𝑚 ∈ ℕ)
21	20	nnzd 12613	. . . . . . . . . . 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 27206	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘}) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
24		elfznn 13568	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1...(⌊‘𝑥)) → 𝑘 ∈ ℕ)
25	24	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ∈ ℕ)
26	25	nnrpd 13047	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ∈ ℝ⁺)
27	26	rpsqrtcld 15428	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (√‘𝑘) ∈ ℝ⁺)
28	27	rpcnd 13051	. . . . . . . . . 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 13054	. . . . . . . . . 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 12015	. . . . . . . 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 27148	. . . . . 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 27466	. . . . . . . . . 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 7418	. . . . . . . 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 13989	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (1...𝑘) ∈ Fin)
39		dvdsssfz1 16335	. . . . . . . . . . 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 9271	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘} ∈ Fin)
42	41, 28, 23, 30	fsumdivc 15800	. . . . . . . 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 2770	. . . . . . 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 15716	. . . . . 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 13022	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
46	45	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
47		resqrtth 15272	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → ((√‘𝑥)↑2) = 𝑥)
48	46, 47	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((√‘𝑥)↑2) = 𝑥)
49	48	fveq2d 6879	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (⌊‘((√‘𝑥)↑2)) = (⌊‘𝑥))
50	49	oveq2d 7419	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘((√‘𝑥)↑2))) = (1...(⌊‘𝑥)))
51	48	fvoveq1d 7425	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (⌊‘(((√‘𝑥)↑2) / 𝑚)) = (⌊‘(𝑥 / 𝑚)))
52	51	oveq2d 7419	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘(((√‘𝑥)↑2) / 𝑚))) = (1...(⌊‘(𝑥 / 𝑚))))
53	52	sumeq1d 15714	. . . . . . . 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 15720	. . . . . 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 2780	. . . . 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 5214	. . . 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 15281	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (√‘𝑥) ∈ ℝ⁺)
59	58	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℝ⁺)
60		eqidd 2736	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (√‘𝑥)) = (𝑥 ∈ ℝ⁺ ↦ (√‘𝑥)))
61		eqidd 2736	. . . . 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 7410	. . . . . . . 8 ⊢ (𝑧 = (√‘𝑥) → (𝑧↑2) = ((√‘𝑥)↑2))
63	62	fveq2d 6879	. . . . . . 7 ⊢ (𝑧 = (√‘𝑥) → (⌊‘(𝑧↑2)) = (⌊‘((√‘𝑥)↑2)))
64	63	oveq2d 7419	. . . . . 6 ⊢ (𝑧 = (√‘𝑥) → (1...(⌊‘(𝑧↑2))) = (1...(⌊‘((√‘𝑥)↑2))))
65	62	fvoveq1d 7425	. . . . . . . . 9 ⊢ (𝑧 = (√‘𝑥) → (⌊‘((𝑧↑2) / 𝑚)) = (⌊‘(((√‘𝑥)↑2) / 𝑚)))
66	65	oveq2d 7419	. . . . . . . 8 ⊢ (𝑧 = (√‘𝑥) → (1...(⌊‘((𝑧↑2) / 𝑚))) = (1...(⌊‘(((√‘𝑥)↑2) / 𝑚))))
67	66	sumeq1d 15714	. . . . . . 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 15720	. . . . 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 7118	. . . 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 2773	. . 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 2735	. . . . . . . 8 ⊢ (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎))) = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)))
73	1, 2, 3, 4, 5, 6, 8, 12, 72	dchrisum0lema 27475	. . . . . . 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 770	. . . . . . . . . 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 780	. . . . . . . . . 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 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))))) → ∀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 27480	. . . . . . . . 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 3142	. . . . . . . 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 1933	. . . . . . 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 15543	. . . . . 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 15702	. . . . . . 7 ⊢ Σ𝑚 ∈ (1...(⌊‘(𝑧↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑧↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))) ∈ V
86		eqid 2735	. . . . . . 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 6681	. . . . . 6 ⊢ dom (𝑧 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘(𝑧↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑧↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑)))) = ℝ⁺
88	87	feq2i 6697	. . . . 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 13014	. . . . 5 ⊢ ℝ⁺ ⊆ ℝ
91	90	a1i 11	. . . 4 ⊢ (𝜑 → ℝ⁺ ⊆ ℝ)
92		resqcl 14140	. . . . 5 ⊢ (𝑡 ∈ ℝ → (𝑡↑2) ∈ ℝ)
93		0red 11236	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) → 0 ∈ ℝ)
94		simplr 768	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) → 𝑡 ∈ ℝ)
95		simplrr 777	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 0 ≤ 𝑡) → (𝑡↑2) ≤ 𝑥)
96	45	ad2antrl 728	. . . . . . . . . . . 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 5147	. . . . . . . . 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 13049	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℝ)
102	101	ad2ant2r 747	. . . . . . . . . . 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 15274	. . . . . . . . . . . 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 14273	. . . . . . . . 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 11236	. . . . . . . . 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 11390	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 𝑡 ≤ 0) → 𝑡 ≤ (√‘𝑥))
116	93, 94, 109, 115	lecasei 11339	. . . . . . 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 3132	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → ∀𝑥 ∈ ℝ⁺ ((𝑡↑2) ≤ 𝑥 → 𝑡 ≤ (√‘𝑥)))
119		breq1 5122	. . . . . 6 ⊢ (𝑐 = (𝑡↑2) → (𝑐 ≤ 𝑥 ↔ (𝑡↑2) ≤ 𝑥))
120	119	rspceaimv 3607	. . . . 5 ⊢ (((𝑡↑2) ∈ ℝ ∧ ∀𝑥 ∈ ℝ⁺ ((𝑡↑2) ≤ 𝑥 → 𝑡 ≤ (√‘𝑥))) → ∃𝑐 ∈ ℝ ∀𝑥 ∈ ℝ⁺ (𝑐 ≤ 𝑥 → 𝑡 ≤ (√‘𝑥)))
121	92, 118, 120	syl2an2 686	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → ∃𝑐 ∈ ℝ ∀𝑥 ∈ ℝ⁺ (𝑐 ≤ 𝑥 → 𝑡 ≤ (√‘𝑥)))
122	89, 82, 59, 91, 121	o1compt 15601	. . 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 2834	. 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 27472	1 ⊢ ¬ 𝜑

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2932 ∀wral 3051 ∃wrex 3060 {crab 3415 ∖ cdif 3923 ⊆ wss 3926 {csn 4601 class class class wbr 5119 ↦ cmpt 5201 dom cdm 5654 ∘ ccom 5658 ⟶wf 6526 ‘cfv 6530 (class class class)co 7403 ℂcc 11125 ℝcr 11126 0cc0 11127 1c1 11128 + caddc 11130 · cmul 11132 +∞cpnf 11264 ≤ cle 11268 − cmin 11464 / cdiv 11892 ℕcn 12238 2c2 12293 ℤcz 12586 ℝ⁺crp 13006 [,)cico 13362 ...cfz 13522 ⌊cfl 13805 seqcseq 14017 ↑cexp 14077 √csqrt 15250 abscabs 15251 ⇝ cli 15498 𝑂(1)co1 15500 Σcsu 15700 ∥ cdvds 16270 Basecbs 17226 0_gc0g 17451 ℤRHomczrh 21458 ℤ/nℤczn 21461 DChrcdchr 27193

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-inf2 9653 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206 ax-mulf 11207

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-disj 5087 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-isom 6539 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-of 7669 df-rpss 7715 df-om 7860 df-1st 7986 df-2nd 7987 df-supp 8158 df-tpos 8223 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-oadd 8482 df-omul 8483 df-er 8717 df-ec 8719 df-qs 8723 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9372 df-fi 9421 df-sup 9452 df-inf 9453 df-oi 9522 df-dju 9913 df-card 9951 df-acn 9954 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-div 11893 df-nn 12239 df-2 12301 df-3 12302 df-4 12303 df-5 12304 df-6 12305 df-7 12306 df-8 12307 df-9 12308 df-n0 12500 df-xnn0 12573 df-z 12587 df-dec 12707 df-uz 12851 df-q 12963 df-rp 13007 df-xneg 13126 df-xadd 13127 df-xmul 13128 df-ioo 13364 df-ioc 13365 df-ico 13366 df-icc 13367 df-fz 13523 df-fzo 13670 df-fl 13807 df-mod 13885 df-seq 14018 df-exp 14078 df-fac 14290 df-bc 14319 df-hash 14347 df-word 14530 df-concat 14587 df-s1 14612 df-shft 15084 df-cj 15116 df-re 15117 df-im 15118 df-sqrt 15252 df-abs 15253 df-limsup 15485 df-clim 15502 df-rlim 15503 df-o1 15504 df-lo1 15505 df-sum 15701 df-ef 16081 df-e 16082 df-sin 16083 df-cos 16084 df-tan 16085 df-pi 16086 df-dvds 16271 df-gcd 16512 df-prm 16689 df-numer 16752 df-denom 16753 df-phi 16783 df-pc 16855 df-struct 17164 df-sets 17181 df-slot 17199 df-ndx 17211 df-base 17227 df-ress 17250 df-plusg 17282 df-mulr 17283 df-starv 17284 df-sca 17285 df-vsca 17286 df-ip 17287 df-tset 17288 df-ple 17289 df-ds 17291 df-unif 17292 df-hom 17293 df-cco 17294 df-rest 17434 df-topn 17435 df-0g 17453 df-gsum 17454 df-topgen 17455 df-pt 17456 df-prds 17459 df-xrs 17514 df-qtop 17519 df-imas 17520 df-qus 17521 df-xps 17522 df-mre 17596 df-mrc 17597 df-acs 17599 df-mgm 18616 df-sgrp 18695 df-mnd 18711 df-mhm 18759 df-submnd 18760 df-grp 18917 df-minusg 18918 df-sbg 18919 df-mulg 19049 df-subg 19104 df-nsg 19105 df-eqg 19106 df-ghm 19194 df-gim 19240 df-ga 19271 df-cntz 19298 df-oppg 19327 df-od 19507 df-gex 19508 df-pgp 19509 df-lsm 19615 df-pj1 19616 df-cmn 19761 df-abl 19762 df-cyg 19857 df-dprd 19976 df-dpj 19977 df-mgp 20099 df-rng 20111 df-ur 20140 df-ring 20193 df-cring 20194 df-oppr 20295 df-dvdsr 20315 df-unit 20316 df-invr 20346 df-dvr 20359 df-rhm 20430 df-subrng 20504 df-subrg 20528 df-drng 20689 df-lmod 20817 df-lss 20887 df-lsp 20927 df-sra 21129 df-rgmod 21130 df-lidl 21167 df-rsp 21168 df-2idl 21209 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-fbas 21310 df-fg 21311 df-cnfld 21314 df-zring 21406 df-zrh 21462 df-zn 21465 df-top 22830 df-topon 22847 df-topsp 22869 df-bases 22882 df-cld 22955 df-ntr 22956 df-cls 22957 df-nei 23034 df-lp 23072 df-perf 23073 df-cn 23163 df-cnp 23164 df-haus 23251 df-cmp 23323 df-tx 23498 df-hmeo 23691 df-fil 23782 df-fm 23874 df-flim 23875 df-flf 23876 df-xms 24257 df-ms 24258 df-tms 24259 df-cncf 24820 df-0p 25621 df-limc 25817 df-dv 25818 df-ply 26143 df-idp 26144 df-coe 26145 df-dgr 26146 df-quot 26249 df-ulm 26336 df-log 26515 df-cxp 26516 df-atan 26827 df-em 26953 df-cht 27057 df-vma 27058 df-chp 27059 df-ppi 27060 df-mu 27061 df-dchr 27194

This theorem is referenced by: dchrisumn0 27482 rpvmasum 27487

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