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Theorem dchrisum0 27502

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 27476 and dchrvmasumif 27485. 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 2739	. 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 4014	. . . 4 ⊢ 𝑊 ⊆ (𝐷 ∖ { 1 })
10		difss 4067	. . . 4 ⊢ (𝐷 ∖ { 1 }) ⊆ 𝐷
11	9, 10	sstri 3924	. . 3 ⊢ 𝑊 ⊆ 𝐷
12		dchrisum0.b	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑊)
13	11, 12	sselid 3913	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
14	1, 2, 3, 4, 5, 6, 8, 12	dchrisum0re 27495	. 2 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℝ)
15		fveq2 6828	. . . . . . . 8 ⊢ (𝑘 = (𝑚 · 𝑑) → (√‘𝑘) = (√‘(𝑚 · 𝑑)))
16	15	oveq2d 7373	. . . . . . 7 ⊢ (𝑘 = (𝑚 · 𝑑) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑘)) = ((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))))
17		rpre 12943	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
18	17	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
19	13	ad3antrrr 736	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘}) → 𝑋 ∈ 𝐷)
20		elrabi 3625	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘} → 𝑚 ∈ ℕ)
21	20	nnzd 12542	. . . . . . . . . . 11 ⊢ (𝑚 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘} → 𝑚 ∈ ℤ)
22	21	adantl 482	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘}) → 𝑚 ∈ ℤ)
23	4, 1, 5, 2, 19, 22	dchrzrhcl 27227	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘}) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
24		elfznn 13499	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1...(⌊‘𝑥)) → 𝑘 ∈ ℕ)
25	24	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ∈ ℕ)
26	25	nnrpd 12976	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ∈ ℝ⁺)
27	26	rpsqrtcld 15366	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (√‘𝑘) ∈ ℝ⁺)
28	27	rpcnd 12980	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (√‘𝑘) ∈ ℂ)
29	28	adantr 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘}) → (√‘𝑘) ∈ ℂ)
30	27	rpne0d 12983	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (√‘𝑘) ≠ 0)
31	30	adantr 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘}) → (√‘𝑘) ≠ 0)
32	23, 29, 31	divcld 11923	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘}) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑘)) ∈ ℂ)
33	32	anasss 467	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘})) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑘)) ∈ ℂ)
34	16, 18, 33	dvdsflsumcom 27170	. . . . . 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 27487	. . . . . . . . . 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 7372	. . . . . . . 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 13927	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (1...𝑘) ∈ Fin)
39		dvdsssfz1 16279	. . . . . . . . . . 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 9170	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘} ∈ Fin)
42	41, 28, 23, 30	fsumdivc 15740	. . . . . . . 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 2774	. . . . . . 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 15656	. . . . . 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 12950	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
46	45	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
47		resqrtth 15209	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → ((√‘𝑥)↑2) = 𝑥)
48	46, 47	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((√‘𝑥)↑2) = 𝑥)
49	48	fveq2d 6832	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (⌊‘((√‘𝑥)↑2)) = (⌊‘𝑥))
50	49	oveq2d 7373	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘((√‘𝑥)↑2))) = (1...(⌊‘𝑥)))
51	48	fvoveq1d 7379	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (⌊‘(((√‘𝑥)↑2) / 𝑚)) = (⌊‘(𝑥 / 𝑚)))
52	51	oveq2d 7373	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘(((√‘𝑥)↑2) / 𝑚))) = (1...(⌊‘(𝑥 / 𝑚))))
53	52	sumeq1d 15654	. . . . . . . 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 481	. . . . . . 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 15660	. . . . . 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 2784	. . . . 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 5166	. . . 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 15218	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (√‘𝑥) ∈ ℝ⁺)
59	58	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℝ⁺)
60		eqidd 2740	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (√‘𝑥)) = (𝑥 ∈ ℝ⁺ ↦ (√‘𝑥)))
61		eqidd 2740	. . . . 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 7364	. . . . . . . 8 ⊢ (𝑧 = (√‘𝑥) → (𝑧↑2) = ((√‘𝑥)↑2))
63	62	fveq2d 6832	. . . . . . 7 ⊢ (𝑧 = (√‘𝑥) → (⌊‘(𝑧↑2)) = (⌊‘((√‘𝑥)↑2)))
64	63	oveq2d 7373	. . . . . 6 ⊢ (𝑧 = (√‘𝑥) → (1...(⌊‘(𝑧↑2))) = (1...(⌊‘((√‘𝑥)↑2))))
65	62	fvoveq1d 7379	. . . . . . . . 9 ⊢ (𝑧 = (√‘𝑥) → (⌊‘((𝑧↑2) / 𝑚)) = (⌊‘(((√‘𝑥)↑2) / 𝑚)))
66	65	oveq2d 7373	. . . . . . . 8 ⊢ (𝑧 = (√‘𝑥) → (1...(⌊‘((𝑧↑2) / 𝑚))) = (1...(⌊‘(((√‘𝑥)↑2) / 𝑚))))
67	66	sumeq1d 15654	. . . . . . 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 481	. . . . . 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 15660	. . . . 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 7072	. . . 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 2777	. . 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 2739	. . . . . . . 8 ⊢ (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎))) = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)))
73	1, 2, 3, 4, 5, 6, 8, 12, 72	dchrisum0lema 27496	. . . . . . 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 481	. . . . . . . . . 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 481	. . . . . . . . . 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 776	. . . . . . . . . 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 786	. . . . . . . . . 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 787	. . . . . . . . . 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 27501	. . . . . . . . 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 3141	. . . . . . . 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 1940	. . . . . . 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 15483	. . . . . 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 15642	. . . . . . 7 ⊢ Σ𝑚 ∈ (1...(⌊‘(𝑧↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑧↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))) ∈ V
86		eqid 2739	. . . . . . 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 6630	. . . . . 6 ⊢ dom (𝑧 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘(𝑧↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑧↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑)))) = ℝ⁺
88	87	feq2i 6648	. . . . 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 219	. . . 4 ⊢ (𝜑 → (𝑧 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘(𝑧↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑧↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑)))):ℝ⁺⟶ℂ)
90		rpssre 12942	. . . . 5 ⊢ ℝ⁺ ⊆ ℝ
91	90	a1i 11	. . . 4 ⊢ (𝜑 → ℝ⁺ ⊆ ℝ)
92		resqcl 14078	. . . . 5 ⊢ (𝑡 ∈ ℝ → (𝑡↑2) ∈ ℝ)
93		0red 11139	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) → 0 ∈ ℝ)
94		simplr 774	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) → 𝑡 ∈ ℝ)
95		simplrr 783	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 0 ≤ 𝑡) → (𝑡↑2) ≤ 𝑥)
96	45	ad2antrl 734	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
97	96	adantr 481	. . . . . . . . . . 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 5101	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 0 ≤ 𝑡) → (𝑡↑2) ≤ ((√‘𝑥)↑2))
100	94	adantr 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 0 ≤ 𝑡) → 𝑡 ∈ ℝ)
101	59	rpred 12978	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℝ)
102	101	ad2ant2r 753	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) → (√‘𝑥) ∈ ℝ)
103	102	adantr 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 0 ≤ 𝑡) → (√‘𝑥) ∈ ℝ)
104		simpr 485	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 0 ≤ 𝑡) → 0 ≤ 𝑡)
105		sqrtge0 15211	. . . . . . . . . . . 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 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 0 ≤ 𝑡) → 0 ≤ (√‘𝑥))
108	100, 103, 104, 107	le2sqd 14211	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 0 ≤ 𝑡) → (𝑡 ≤ (√‘𝑥) ↔ (𝑡↑2) ≤ ((√‘𝑥)↑2)))
109	99, 108	mpbird 258	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 0 ≤ 𝑡) → 𝑡 ≤ (√‘𝑥))
110	94	adantr 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 𝑡 ≤ 0) → 𝑡 ∈ ℝ)
111		0red 11139	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 𝑡 ≤ 0) → 0 ∈ ℝ)
112	102	adantr 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 𝑡 ≤ 0) → (√‘𝑥) ∈ ℝ)
113		simpr 485	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 𝑡 ≤ 0) → 𝑡 ≤ 0)
114	106	adantr 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 𝑡 ≤ 0) → 0 ≤ (√‘𝑥))
115	110, 111, 112, 113, 114	letrd 11295	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 𝑡 ≤ 0) → 𝑡 ≤ (√‘𝑥))
116	93, 94, 109, 115	lecasei 11244	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) → 𝑡 ≤ (√‘𝑥))
117	116	expr 457	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → ((𝑡↑2) ≤ 𝑥 → 𝑡 ≤ (√‘𝑥)))
118	117	ralrimiva 3131	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → ∀𝑥 ∈ ℝ⁺ ((𝑡↑2) ≤ 𝑥 → 𝑡 ≤ (√‘𝑥)))
119		breq1 5076	. . . . . 6 ⊢ (𝑐 = (𝑡↑2) → (𝑐 ≤ 𝑥 ↔ (𝑡↑2) ≤ 𝑥))
120	119	rspceaimv 3566	. . . . 5 ⊢ (((𝑡↑2) ∈ ℝ ∧ ∀𝑥 ∈ ℝ⁺ ((𝑡↑2) ≤ 𝑥 → 𝑡 ≤ (√‘𝑥))) → ∃𝑐 ∈ ℝ ∀𝑥 ∈ ℝ⁺ (𝑐 ≤ 𝑥 → 𝑡 ≤ (√‘𝑥)))
121	92, 118, 120	syl2an2 692	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → ∃𝑐 ∈ ℝ ∀𝑥 ∈ ℝ⁺ (𝑐 ≤ 𝑥 → 𝑡 ≤ (√‘𝑥)))
122	89, 82, 59, 91, 121	o1compt 15541	. . 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 2839	. 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 27493	1 ⊢ ¬ 𝜑

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1547 ∃wex 1786 ∈ wcel 2119 ≠ wne 2934 ∀wral 3053 ∃wrex 3063 {crab 3391 ∖ cdif 3880 ⊆ wss 3883 {csn 4556 class class class wbr 5073 ↦ cmpt 5154 dom cdm 5619 ∘ ccom 5623 ⟶wf 6482 ‘cfv 6486 (class class class)co 7357 ℂcc 11028 ℝcr 11029 0cc0 11030 1c1 11031 + caddc 11033 · cmul 11035 +∞cpnf 11168 ≤ cle 11172 − cmin 11369 / cdiv 11799 ℕcn 12166 2c2 12228 ℤcz 12516 ℝ⁺crp 12934 [,)cico 13292 ...cfz 13453 ⌊cfl 13741 seqcseq 13955 ↑cexp 14015 √csqrt 15187 abscabs 15188 ⇝ cli 15438 𝑂(1)co1 15440 Σcsu 15640 ∥ cdvds 16213 Basecbs 17171 0_gc0g 17394 ℤRHomczrh 21475 ℤ/nℤczn 21478 DChrcdchr 27214

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5200 ax-sep 5219 ax-nul 5229 ax-pow 5295 ax-pr 5363 ax-un 7679 ax-inf2 9554 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108 ax-addf 11109 ax-mulf 11110

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4263 df-if 4456 df-pw 4532 df-sn 4557 df-pr 4559 df-tp 4561 df-op 4563 df-uni 4840 df-int 4879 df-iun 4924 df-iin 4925 df-disj 5041 df-br 5074 df-opab 5136 df-mpt 5155 df-tr 5181 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7314 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7621 df-rpss 7667 df-om 7808 df-1st 7932 df-2nd 7933 df-supp 8102 df-tpos 8167 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-oadd 8400 df-omul 8401 df-er 8634 df-ec 8636 df-qs 8640 df-map 8766 df-pm 8767 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-fi 9315 df-sup 9346 df-inf 9347 df-oi 9416 df-dju 9817 df-card 9855 df-acn 9858 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-div 11800 df-nn 12167 df-2 12236 df-3 12237 df-4 12238 df-5 12239 df-6 12240 df-7 12241 df-8 12242 df-9 12243 df-n0 12430 df-xnn0 12503 df-z 12517 df-dec 12637 df-uz 12781 df-q 12891 df-rp 12935 df-xneg 13055 df-xadd 13056 df-xmul 13057 df-ioo 13294 df-ioc 13295 df-ico 13296 df-icc 13297 df-fz 13454 df-fzo 13601 df-fl 13743 df-mod 13821 df-seq 13956 df-exp 14016 df-fac 14228 df-bc 14257 df-hash 14285 df-word 14468 df-concat 14525 df-s1 14551 df-shft 15021 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-limsup 15425 df-clim 15442 df-rlim 15443 df-o1 15444 df-lo1 15445 df-sum 15641 df-ef 16024 df-e 16025 df-sin 16026 df-cos 16027 df-tan 16028 df-pi 16029 df-dvds 16214 df-gcd 16456 df-prm 16633 df-numer 16697 df-denom 16698 df-phi 16728 df-pc 16800 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17172 df-ress 17193 df-plusg 17225 df-mulr 17226 df-starv 17227 df-sca 17228 df-vsca 17229 df-ip 17230 df-tset 17231 df-ple 17232 df-ds 17234 df-unif 17235 df-hom 17236 df-cco 17237 df-rest 17377 df-topn 17378 df-0g 17396 df-gsum 17397 df-topgen 17398 df-pt 17399 df-prds 17402 df-xrs 17458 df-qtop 17463 df-imas 17464 df-qus 17465 df-xps 17466 df-mre 17540 df-mrc 17541 df-acs 17543 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-mhm 18743 df-submnd 18744 df-grp 18904 df-minusg 18905 df-sbg 18906 df-mulg 19036 df-subg 19091 df-nsg 19092 df-eqg 19093 df-ghm 19180 df-gim 19226 df-ga 19257 df-cntz 19284 df-oppg 19313 df-od 19495 df-gex 19496 df-pgp 19497 df-lsm 19603 df-pj1 19604 df-cmn 19749 df-abl 19750 df-cyg 19845 df-dprd 19964 df-dpj 19965 df-mgp 20114 df-rng 20126 df-ur 20155 df-ring 20208 df-cring 20209 df-oppr 20309 df-dvdsr 20329 df-unit 20330 df-invr 20360 df-dvr 20373 df-rhm 20444 df-subrng 20519 df-subrg 20543 df-drng 20704 df-lmod 20853 df-lss 20923 df-lsp 20963 df-sra 21164 df-rgmod 21165 df-lidl 21202 df-rsp 21203 df-2idl 21244 df-psmet 21340 df-xmet 21341 df-met 21342 df-bl 21343 df-mopn 21344 df-fbas 21345 df-fg 21346 df-cnfld 21349 df-zring 21423 df-zrh 21479 df-zn 21482 df-top 22878 df-topon 22895 df-topsp 22917 df-bases 22930 df-cld 23003 df-ntr 23004 df-cls 23005 df-nei 23082 df-lp 23120 df-perf 23121 df-cn 23211 df-cnp 23212 df-haus 23299 df-cmp 23371 df-tx 23546 df-hmeo 23739 df-fil 23830 df-fm 23922 df-flim 23923 df-flf 23924 df-xms 24304 df-ms 24305 df-tms 24306 df-cncf 24864 df-0p 25656 df-limc 25852 df-dv 25853 df-ply 26172 df-idp 26173 df-coe 26174 df-dgr 26175 df-quot 26276 df-ulm 26361 df-log 26539 df-cxp 26540 df-atan 26850 df-em 26975 df-cht 27079 df-vma 27080 df-chp 27081 df-ppi 27082 df-mu 27083 df-dchr 27215

This theorem is referenced by: dchrisumn0 27503 rpvmasum 27508

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