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Theorem dchrisum0 26868

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 26842 and dchrvmasumif 26851. 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 4040	. . . 4 ⊢ 𝑊 ⊆ (𝐷 ∖ { 1 })
10		difss 4091	. . . 4 ⊢ (𝐷 ∖ { 1 }) ⊆ 𝐷
11	9, 10	sstri 3953	. . 3 ⊢ 𝑊 ⊆ 𝐷
12		dchrisum0.b	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑊)
13	11, 12	sselid 3942	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
14	1, 2, 3, 4, 5, 6, 8, 12	dchrisum0re 26861	. 2 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℝ)
15		fveq2 6842	. . . . . . . 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 12923	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
18	17	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
19	13	ad3antrrr 728	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘}) → 𝑋 ∈ 𝐷)
20		elrabi 3639	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘} → 𝑚 ∈ ℕ)
21	20	nnzd 12526	. . . . . . . . . . 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 26593	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘}) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
24		elfznn 13470	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1...(⌊‘𝑥)) → 𝑘 ∈ ℕ)
25	24	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ∈ ℕ)
26	25	nnrpd 12955	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ∈ ℝ⁺)
27	26	rpsqrtcld 15296	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (√‘𝑘) ∈ ℝ⁺)
28	27	rpcnd 12959	. . . . . . . . . 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 12962	. . . . . . . . . 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 11931	. . . . . . . 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 26537	. . . . . 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 26853	. . . . . . . . . 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 13878	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (1...𝑘) ∈ Fin)
39		dvdsssfz1 16200	. . . . . . . . . . 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 9211	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → {𝑖 ∈ ℕ ∣ 𝑖 ∥ 𝑘} ∈ Fin)
42	41, 28, 23, 30	fsumdivc 15671	. . . . . . . 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 2776	. . . . . . 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 15588	. . . . . 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 12930	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
46	45	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
47		resqrtth 15140	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → ((√‘𝑥)↑2) = 𝑥)
48	46, 47	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((√‘𝑥)↑2) = 𝑥)
49	48	fveq2d 6846	. . . . . . . 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 15586	. . . . . . . 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 15591	. . . . . 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 2786	. . . . 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 5205	. . . 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 15149	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (√‘𝑥) ∈ ℝ⁺)
59	58	adantl 482	. . . . 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 7364	. . . . . . . 8 ⊢ (𝑧 = (√‘𝑥) → (𝑧↑2) = ((√‘𝑥)↑2))
63	62	fveq2d 6846	. . . . . . 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 15586	. . . . . . 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 15591	. . . . 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 7075	. . . 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 2779	. . 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 26862	. . . . . . 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 769	. . . . . . . . . 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 779	. . . . . . . . . 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 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))))) → ∀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 26867	. . . . . . . . 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 3153	. . . . . . . 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 1936	. . . . . . 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 15411	. . . . . 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 15572	. . . . . . 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 6645	. . . . . 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 217	. . . 4 ⊢ (𝜑 → (𝑧 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘(𝑧↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑧↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑)))):ℝ⁺⟶ℂ)
90		rpssre 12922	. . . . 5 ⊢ ℝ⁺ ⊆ ℝ
91	90	a1i 11	. . . 4 ⊢ (𝜑 → ℝ⁺ ⊆ ℝ)
92		resqcl 14029	. . . . 5 ⊢ (𝑡 ∈ ℝ → (𝑡↑2) ∈ ℝ)
93		0red 11158	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) → 0 ∈ ℝ)
94		simplr 767	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) → 𝑡 ∈ ℝ)
95		simplrr 776	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 0 ≤ 𝑡) → (𝑡↑2) ≤ 𝑥)
96	45	ad2antrl 726	. . . . . . . . . . . 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 5133	. . . . . . . . 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 12957	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℝ)
102	101	ad2ant2r 745	. . . . . . . . . . 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 15142	. . . . . . . . . . . 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 14160	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 0 ≤ 𝑡) → (𝑡 ≤ (√‘𝑥) ↔ (𝑡↑2) ≤ ((√‘𝑥)↑2)))
109	99, 108	mpbird 256	. . . . . . . 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 11158	. . . . . . . . 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 11312	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑡 ∈ ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ (𝑡↑2) ≤ 𝑥)) ∧ 𝑡 ≤ 0) → 𝑡 ≤ (√‘𝑥))
116	93, 94, 109, 115	lecasei 11261	. . . . . . 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 3143	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → ∀𝑥 ∈ ℝ⁺ ((𝑡↑2) ≤ 𝑥 → 𝑡 ≤ (√‘𝑥)))
119		breq1 5108	. . . . . 6 ⊢ (𝑐 = (𝑡↑2) → (𝑐 ≤ 𝑥 ↔ (𝑡↑2) ≤ 𝑥))
120	119	rspceaimv 3585	. . . . 5 ⊢ (((𝑡↑2) ∈ ℝ ∧ ∀𝑥 ∈ ℝ⁺ ((𝑡↑2) ≤ 𝑥 → 𝑡 ≤ (√‘𝑥))) → ∃𝑐 ∈ ℝ ∀𝑥 ∈ ℝ⁺ (𝑐 ≤ 𝑥 → 𝑡 ≤ (√‘𝑥)))
121	92, 118, 120	syl2an2 684	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ ℝ) → ∃𝑐 ∈ ℝ ∀𝑥 ∈ ℝ⁺ (𝑐 ≤ 𝑥 → 𝑡 ≤ (√‘𝑥)))
122	89, 82, 59, 91, 121	o1compt 15469	. . 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 2838	. 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 26859	1 ⊢ ¬ 𝜑

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3073 {crab 3407 ∖ cdif 3907 ⊆ wss 3910 {csn 4586 class class class wbr 5105 ↦ cmpt 5188 dom cdm 5633 ∘ ccom 5637 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 · cmul 11056 +∞cpnf 11186 ≤ cle 11190 − cmin 11385 / cdiv 11812 ℕcn 12153 2c2 12208 ℤcz 12499 ℝ⁺crp 12915 [,)cico 13266 ...cfz 13424 ⌊cfl 13695 seqcseq 13906 ↑cexp 13967 √csqrt 15118 abscabs 15119 ⇝ cli 15366 𝑂(1)co1 15368 Σcsu 15570 ∥ cdvds 16136 Basecbs 17083 0_gc0g 17321 ℤRHomczrh 20900 ℤ/nℤczn 20903 DChrcdchr 26580

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-disj 5071 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-rpss 7660 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-tpos 8157 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-oadd 8416 df-omul 8417 df-er 8648 df-ec 8650 df-qs 8654 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-dju 9837 df-card 9875 df-acn 9878 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-xnn0 12486 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ioc 13269 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-fac 14174 df-bc 14203 df-hash 14231 df-word 14403 df-concat 14459 df-s1 14484 df-shft 14952 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-limsup 15353 df-clim 15370 df-rlim 15371 df-o1 15372 df-lo1 15373 df-sum 15571 df-ef 15950 df-e 15951 df-sin 15952 df-cos 15953 df-tan 15954 df-pi 15955 df-dvds 16137 df-gcd 16375 df-prm 16548 df-numer 16610 df-denom 16611 df-phi 16638 df-pc 16709 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-qus 17391 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-mhm 18601 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-mulg 18873 df-subg 18925 df-nsg 18926 df-eqg 18927 df-ghm 19006 df-gim 19049 df-ga 19070 df-cntz 19097 df-oppg 19124 df-od 19310 df-gex 19311 df-pgp 19312 df-lsm 19418 df-pj1 19419 df-cmn 19564 df-abl 19565 df-cyg 19655 df-dprd 19774 df-dpj 19775 df-mgp 19897 df-ur 19914 df-ring 19966 df-cring 19967 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-invr 20101 df-dvr 20112 df-rnghom 20146 df-drng 20187 df-subrg 20220 df-lmod 20324 df-lss 20393 df-lsp 20433 df-sra 20633 df-rgmod 20634 df-lidl 20635 df-rsp 20636 df-2idl 20702 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-zring 20870 df-zrh 20904 df-zn 20907 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-cmp 22738 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-0p 25034 df-limc 25230 df-dv 25231 df-ply 25549 df-idp 25550 df-coe 25551 df-dgr 25552 df-quot 25651 df-ulm 25736 df-log 25912 df-cxp 25913 df-atan 26217 df-em 26342 df-cht 26446 df-vma 26447 df-chp 26448 df-ppi 26449 df-mu 26450 df-dchr 26581

This theorem is referenced by: dchrisumn0 26869 rpvmasum 26874

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