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Theorem mertenslem2 15837

Description: Lemma for mertens 15838. (Contributed by Mario Carneiro, 28-Apr-2014.)

**Hypotheses**
Ref	Expression
mertens.1	⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝐹‘𝑗) = 𝐴)
mertens.2	⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝐾‘𝑗) = (abs‘𝐴))
mertens.3	⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → 𝐴 ∈ ℂ)
mertens.4	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝐺‘𝑘) = 𝐵)
mertens.5	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → 𝐵 ∈ ℂ)
mertens.6	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝐻‘𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗))))
mertens.7	⊢ (𝜑 → seq0( + , 𝐾) ∈ dom ⇝ )
mertens.8	⊢ (𝜑 → seq0( + , 𝐺) ∈ dom ⇝ )
mertens.9	⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
mertens.10	⊢ 𝑇 = {𝑧 ∣ ∃𝑛 ∈ (0...(𝑠 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘))}
mertens.11	⊢ (𝜓 ↔ (𝑠 ∈ ℕ ∧ ∀𝑛 ∈ (ℤ_≥‘𝑠)(abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1))))

**Assertion**
Ref	Expression
mertenslem2	⊢ (𝜑 → ∃𝑦 ∈ ℕ₀ ∀𝑚 ∈ (ℤ_≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝐸)

Distinct variable groups: 𝑗,𝑚,𝑛,𝑠,𝑦,𝑧,𝐵 𝑗,𝑘,𝐺,𝑚,𝑛,𝑠,𝑦,𝑧 𝜑,𝑗,𝑘,𝑚,𝑦,𝑧 𝐴,𝑘,𝑚,𝑛,𝑠,𝑦 𝑗,𝐸,𝑘,𝑚,𝑛,𝑠,𝑦,𝑧 𝑗,𝐾,𝑘,𝑚,𝑛,𝑠,𝑦,𝑧 𝑗,𝐹,𝑚,𝑛,𝑦 𝜓,𝑗,𝑘,𝑚,𝑛,𝑦,𝑧 𝑇,𝑗,𝑘,𝑚,𝑛,𝑦,𝑧 𝑘,𝐻,𝑚,𝑦 𝜑,𝑛,𝑠 Allowed substitution hints: 𝜓(𝑠) 𝐴(𝑧,𝑗) 𝐵(𝑘) 𝑇(𝑠) 𝐹(𝑧,𝑘,𝑠) 𝐻(𝑧,𝑗,𝑛,𝑠)

Proof of Theorem mertenslem2 Dummy variables 𝑡 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnuz 12869	. . 3 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 12597	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
3		mertens.9	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
4	3	rphalfcld 13034	. . . 4 ⊢ (𝜑 → (𝐸 / 2) ∈ ℝ⁺)
5		nn0uz 12868	. . . . . 6 ⊢ ℕ₀ = (ℤ_≥‘0)
6		0zd 12574	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℤ)
7		eqidd 2727	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝐾‘𝑗) = (𝐾‘𝑗))
8		mertens.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝐾‘𝑗) = (abs‘𝐴))
9		mertens.3	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → 𝐴 ∈ ℂ)
10	9	abscld 15389	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (abs‘𝐴) ∈ ℝ)
11	8, 10	eqeltrd 2827	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝐾‘𝑗) ∈ ℝ)
12		mertens.7	. . . . . 6 ⊢ (𝜑 → seq0( + , 𝐾) ∈ dom ⇝ )
13	5, 6, 7, 11, 12	isumrecl 15717	. . . . 5 ⊢ (𝜑 → Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) ∈ ℝ)
14	9	absge0d 15397	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → 0 ≤ (abs‘𝐴))
15	14, 8	breqtrrd 5169	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → 0 ≤ (𝐾‘𝑗))
16	5, 6, 7, 11, 12, 15	isumge0 15718	. . . . 5 ⊢ (𝜑 → 0 ≤ Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗))
17	13, 16	ge0p1rpd 13052	. . . 4 ⊢ (𝜑 → (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1) ∈ ℝ⁺)
18	4, 17	rpdivcld 13039	. . 3 ⊢ (𝜑 → ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)) ∈ ℝ⁺)
19		eqidd 2727	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (seq0( + , 𝐺)‘𝑚) = (seq0( + , 𝐺)‘𝑚))
20		mertens.4	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝐺‘𝑘) = 𝐵)
21		mertens.5	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → 𝐵 ∈ ℂ)
22		mertens.8	. . . 4 ⊢ (𝜑 → seq0( + , 𝐺) ∈ dom ⇝ )
23	5, 6, 20, 21, 22	isumclim2 15710	. . 3 ⊢ (𝜑 → seq0( + , 𝐺) ⇝ Σ𝑘 ∈ ℕ₀ 𝐵)
24	1, 2, 18, 19, 23	climi2 15461	. 2 ⊢ (𝜑 → ∃𝑠 ∈ ℕ ∀𝑚 ∈ (ℤ_≥‘𝑠)(abs‘((seq0( + , 𝐺)‘𝑚) − Σ𝑘 ∈ ℕ₀ 𝐵)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)))
25		eluznn 12906	. . . . . . . 8 ⊢ ((𝑠 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘𝑠)) → 𝑚 ∈ ℕ)
26	20, 21	eqeltrd 2827	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝐺‘𝑘) ∈ ℂ)
27	5, 6, 26	serf 14001	. . . . . . . . . . . 12 ⊢ (𝜑 → seq0( + , 𝐺):ℕ₀⟶ℂ)
28		nnnn0 12483	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℕ₀)
29		ffvelcdm 7077	. . . . . . . . . . . 12 ⊢ ((seq0( + , 𝐺):ℕ₀⟶ℂ ∧ 𝑚 ∈ ℕ₀) → (seq0( + , 𝐺)‘𝑚) ∈ ℂ)
30	27, 28, 29	syl2an 595	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (seq0( + , 𝐺)‘𝑚) ∈ ℂ)
31	5, 6, 20, 21, 22	isumcl 15713	. . . . . . . . . . . 12 ⊢ (𝜑 → Σ𝑘 ∈ ℕ₀ 𝐵 ∈ ℂ)
32	31	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ ℕ₀ 𝐵 ∈ ℂ)
33	30, 32	abssubd 15406	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (abs‘((seq0( + , 𝐺)‘𝑚) − Σ𝑘 ∈ ℕ₀ 𝐵)) = (abs‘(Σ𝑘 ∈ ℕ₀ 𝐵 − (seq0( + , 𝐺)‘𝑚))))
34		eqid 2726	. . . . . . . . . . . . . 14 ⊢ (ℤ_≥‘(𝑚 + 1)) = (ℤ_≥‘(𝑚 + 1))
35	28	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ₀)
36		peano2nn0 12516	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ₀ → (𝑚 + 1) ∈ ℕ₀)
37	35, 36	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 + 1) ∈ ℕ₀)
38	37	nn0zd 12588	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 + 1) ∈ ℤ)
39		simpll 764	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘(𝑚 + 1))) → 𝜑)
40		eluznn0 12905	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 + 1) ∈ ℕ₀ ∧ 𝑘 ∈ (ℤ_≥‘(𝑚 + 1))) → 𝑘 ∈ ℕ₀)
41	37, 40	sylan 579	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘(𝑚 + 1))) → 𝑘 ∈ ℕ₀)
42	39, 41, 20	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘(𝑚 + 1))) → (𝐺‘𝑘) = 𝐵)
43	39, 41, 21	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘(𝑚 + 1))) → 𝐵 ∈ ℂ)
44	22	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → seq0( + , 𝐺) ∈ dom ⇝ )
45	26	adantlr 712	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → (𝐺‘𝑘) ∈ ℂ)
46	5, 37, 45	iserex 15609	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (seq0( + , 𝐺) ∈ dom ⇝ ↔ seq(𝑚 + 1)( + , 𝐺) ∈ dom ⇝ ))
47	44, 46	mpbid 231	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → seq(𝑚 + 1)( + , 𝐺) ∈ dom ⇝ )
48	34, 38, 42, 43, 47	isumcl 15713	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ (ℤ_≥‘(𝑚 + 1))𝐵 ∈ ℂ)
49	30, 48	pncan2d 11577	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((seq0( + , 𝐺)‘𝑚) + Σ𝑘 ∈ (ℤ_≥‘(𝑚 + 1))𝐵) − (seq0( + , 𝐺)‘𝑚)) = Σ𝑘 ∈ (ℤ_≥‘(𝑚 + 1))𝐵)
50	20	adantlr 712	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → (𝐺‘𝑘) = 𝐵)
51	21	adantlr 712	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ ℕ₀) → 𝐵 ∈ ℂ)
52	5, 34, 37, 50, 51, 44	isumsplit 15792	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ ℕ₀ 𝐵 = (Σ𝑘 ∈ (0...((𝑚 + 1) − 1))𝐵 + Σ𝑘 ∈ (ℤ_≥‘(𝑚 + 1))𝐵))
53		nncn 12224	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
54	53	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
55		ax-1cn 11170	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℂ
56		pncan 11470	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑚 + 1) − 1) = 𝑚)
57	54, 55, 56	sylancl 585	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑚 + 1) − 1) = 𝑚)
58	57	oveq2d 7421	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (0...((𝑚 + 1) − 1)) = (0...𝑚))
59	58	sumeq1d 15653	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ (0...((𝑚 + 1) − 1))𝐵 = Σ𝑘 ∈ (0...𝑚)𝐵)
60		simpl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝜑)
61		elfznn0 13600	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (0...𝑚) → 𝑘 ∈ ℕ₀)
62	60, 61, 20	syl2an 595	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑚)) → (𝐺‘𝑘) = 𝐵)
63	35, 5	eleqtrdi 2837	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ (ℤ_≥‘0))
64	60, 61, 21	syl2an 595	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (0...𝑚)) → 𝐵 ∈ ℂ)
65	62, 63, 64	fsumser 15682	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ (0...𝑚)𝐵 = (seq0( + , 𝐺)‘𝑚))
66	59, 65	eqtrd 2766	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ (0...((𝑚 + 1) − 1))𝐵 = (seq0( + , 𝐺)‘𝑚))
67	66	oveq1d 7420	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (Σ𝑘 ∈ (0...((𝑚 + 1) − 1))𝐵 + Σ𝑘 ∈ (ℤ_≥‘(𝑚 + 1))𝐵) = ((seq0( + , 𝐺)‘𝑚) + Σ𝑘 ∈ (ℤ_≥‘(𝑚 + 1))𝐵))
68	52, 67	eqtrd 2766	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ ℕ₀ 𝐵 = ((seq0( + , 𝐺)‘𝑚) + Σ𝑘 ∈ (ℤ_≥‘(𝑚 + 1))𝐵))
69	68	oveq1d 7420	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (Σ𝑘 ∈ ℕ₀ 𝐵 − (seq0( + , 𝐺)‘𝑚)) = (((seq0( + , 𝐺)‘𝑚) + Σ𝑘 ∈ (ℤ_≥‘(𝑚 + 1))𝐵) − (seq0( + , 𝐺)‘𝑚)))
70	42	sumeq2dv 15655	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ (ℤ_≥‘(𝑚 + 1))(𝐺‘𝑘) = Σ𝑘 ∈ (ℤ_≥‘(𝑚 + 1))𝐵)
71	49, 69, 70	3eqtr4d 2776	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (Σ𝑘 ∈ ℕ₀ 𝐵 − (seq0( + , 𝐺)‘𝑚)) = Σ𝑘 ∈ (ℤ_≥‘(𝑚 + 1))(𝐺‘𝑘))
72	71	fveq2d 6889	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (abs‘(Σ𝑘 ∈ ℕ₀ 𝐵 − (seq0( + , 𝐺)‘𝑚))) = (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑚 + 1))(𝐺‘𝑘)))
73	33, 72	eqtrd 2766	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (abs‘((seq0( + , 𝐺)‘𝑚) − Σ𝑘 ∈ ℕ₀ 𝐵)) = (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑚 + 1))(𝐺‘𝑘)))
74	73	breq1d 5151	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((abs‘((seq0( + , 𝐺)‘𝑚) − Σ𝑘 ∈ ℕ₀ 𝐵)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)) ↔ (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑚 + 1))(𝐺‘𝑘)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1))))
75	25, 74	sylan2 592	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑠 ∈ ℕ ∧ 𝑚 ∈ (ℤ_≥‘𝑠))) → ((abs‘((seq0( + , 𝐺)‘𝑚) − Σ𝑘 ∈ ℕ₀ 𝐵)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)) ↔ (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑚 + 1))(𝐺‘𝑘)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1))))
76	75	anassrs 467	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ) ∧ 𝑚 ∈ (ℤ_≥‘𝑠)) → ((abs‘((seq0( + , 𝐺)‘𝑚) − Σ𝑘 ∈ ℕ₀ 𝐵)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)) ↔ (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑚 + 1))(𝐺‘𝑘)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1))))
77	76	ralbidva 3169	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ) → (∀𝑚 ∈ (ℤ_≥‘𝑠)(abs‘((seq0( + , 𝐺)‘𝑚) − Σ𝑘 ∈ ℕ₀ 𝐵)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)) ↔ ∀𝑚 ∈ (ℤ_≥‘𝑠)(abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑚 + 1))(𝐺‘𝑘)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1))))
78		fvoveq1 7428	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (ℤ_≥‘(𝑚 + 1)) = (ℤ_≥‘(𝑛 + 1)))
79	78	sumeq1d 15653	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → Σ𝑘 ∈ (ℤ_≥‘(𝑚 + 1))(𝐺‘𝑘) = Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘))
80	79	fveq2d 6889	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑚 + 1))(𝐺‘𝑘)) = (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘)))
81	80	breq1d 5151	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑚 + 1))(𝐺‘𝑘)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)) ↔ (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1))))
82	81	cbvralvw 3228	. . . . 5 ⊢ (∀𝑚 ∈ (ℤ_≥‘𝑠)(abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑚 + 1))(𝐺‘𝑘)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)) ↔ ∀𝑛 ∈ (ℤ_≥‘𝑠)(abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)))
83	77, 82	bitrdi 287	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ) → (∀𝑚 ∈ (ℤ_≥‘𝑠)(abs‘((seq0( + , 𝐺)‘𝑚) − Σ𝑘 ∈ ℕ₀ 𝐵)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)) ↔ ∀𝑛 ∈ (ℤ_≥‘𝑠)(abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1))))
84		mertens.11	. . . . . 6 ⊢ (𝜓 ↔ (𝑠 ∈ ℕ ∧ ∀𝑛 ∈ (ℤ_≥‘𝑠)(abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1))))
85		0zd 12574	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → 0 ∈ ℤ)
86	4	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (𝐸 / 2) ∈ ℝ⁺)
87	84	simplbi 497	. . . . . . . . . . . . 13 ⊢ (𝜓 → 𝑠 ∈ ℕ)
88	87	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → 𝑠 ∈ ℕ)
89	88	nnrpd 13020	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → 𝑠 ∈ ℝ⁺)
90	86, 89	rpdivcld 13039	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → ((𝐸 / 2) / 𝑠) ∈ ℝ⁺)
91		mertens.10	. . . . . . . . . . . . 13 ⊢ 𝑇 = {𝑧 ∣ ∃𝑛 ∈ (0...(𝑠 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘))}
92		eqid 2726	. . . . . . . . . . . . . . . . . 18 ⊢ (ℤ_≥‘(𝑛 + 1)) = (ℤ_≥‘(𝑛 + 1))
93		elfznn0 13600	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ (0...(𝑠 − 1)) → 𝑛 ∈ ℕ₀)
94	93	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (0...(𝑠 − 1))) → 𝑛 ∈ ℕ₀)
95		peano2nn0 12516	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ₀)
96	94, 95	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (0...(𝑠 − 1))) → (𝑛 + 1) ∈ ℕ₀)
97	96	nn0zd 12588	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (0...(𝑠 − 1))) → (𝑛 + 1) ∈ ℤ)
98		eqidd 2727	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (0...(𝑠 − 1))) ∧ 𝑘 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝐺‘𝑘) = (𝐺‘𝑘))
99		simplll 772	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (0...(𝑠 − 1))) ∧ 𝑘 ∈ (ℤ_≥‘(𝑛 + 1))) → 𝜑)
100		eluznn0 12905	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑛 + 1) ∈ ℕ₀ ∧ 𝑘 ∈ (ℤ_≥‘(𝑛 + 1))) → 𝑘 ∈ ℕ₀)
101	96, 100	sylan 579	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (0...(𝑠 − 1))) ∧ 𝑘 ∈ (ℤ_≥‘(𝑛 + 1))) → 𝑘 ∈ ℕ₀)
102	99, 101, 26	syl2anc 583	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (0...(𝑠 − 1))) ∧ 𝑘 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝐺‘𝑘) ∈ ℂ)
103	22	ad2antrr 723	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (0...(𝑠 − 1))) → seq0( + , 𝐺) ∈ dom ⇝ )
104	26	ad4ant14 749	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (0...(𝑠 − 1))) ∧ 𝑘 ∈ ℕ₀) → (𝐺‘𝑘) ∈ ℂ)
105	5, 96, 104	iserex 15609	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (0...(𝑠 − 1))) → (seq0( + , 𝐺) ∈ dom ⇝ ↔ seq(𝑛 + 1)( + , 𝐺) ∈ dom ⇝ ))
106	103, 105	mpbid 231	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (0...(𝑠 − 1))) → seq(𝑛 + 1)( + , 𝐺) ∈ dom ⇝ )
107	92, 97, 98, 102, 106	isumcl 15713	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (0...(𝑠 − 1))) → Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘) ∈ ℂ)
108	107	abscld 15389	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (0...(𝑠 − 1))) → (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘)) ∈ ℝ)
109		eleq1a 2822	. . . . . . . . . . . . . . . 16 ⊢ ((abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘)) ∈ ℝ → (𝑧 = (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘)) → 𝑧 ∈ ℝ))
110	108, 109	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (0...(𝑠 − 1))) → (𝑧 = (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘)) → 𝑧 ∈ ℝ))
111	110	rexlimdva 3149	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → (∃𝑛 ∈ (0...(𝑠 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘)) → 𝑧 ∈ ℝ))
112	111	abssdv 4060	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → {𝑧 ∣ ∃𝑛 ∈ (0...(𝑠 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘))} ⊆ ℝ)
113	91, 112	eqsstrid 4025	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → 𝑇 ⊆ ℝ)
114		fzfid 13944	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → (0...(𝑠 − 1)) ∈ Fin)
115		abrexfi 9354	. . . . . . . . . . . . . . 15 ⊢ ((0...(𝑠 − 1)) ∈ Fin → {𝑧 ∣ ∃𝑛 ∈ (0...(𝑠 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘))} ∈ Fin)
116	114, 115	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → {𝑧 ∣ ∃𝑛 ∈ (0...(𝑠 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘))} ∈ Fin)
117	91, 116	eqeltrid 2831	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → 𝑇 ∈ Fin)
118		nnm1nn0 12517	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ ℕ → (𝑠 − 1) ∈ ℕ₀)
119	88, 118	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜓) → (𝑠 − 1) ∈ ℕ₀)
120	119, 5	eleqtrdi 2837	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜓) → (𝑠 − 1) ∈ (ℤ_≥‘0))
121		eluzfz1 13514	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑠 − 1) ∈ (ℤ_≥‘0) → 0 ∈ (0...(𝑠 − 1)))
122	120, 121	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜓) → 0 ∈ (0...(𝑠 − 1)))
123		nnnn0 12483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
124	123, 20	sylan2 592	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) = 𝐵)
125	124	sumeq2dv 15655	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → Σ𝑘 ∈ ℕ (𝐺‘𝑘) = Σ𝑘 ∈ ℕ 𝐵)
126	125	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜓) → Σ𝑘 ∈ ℕ (𝐺‘𝑘) = Σ𝑘 ∈ ℕ 𝐵)
127	126	fveq2d 6889	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜓) → (abs‘Σ𝑘 ∈ ℕ (𝐺‘𝑘)) = (abs‘Σ𝑘 ∈ ℕ 𝐵))
128	127	eqcomd 2732	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜓) → (abs‘Σ𝑘 ∈ ℕ 𝐵) = (abs‘Σ𝑘 ∈ ℕ (𝐺‘𝑘)))
129		fv0p1e1 12339	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 0 → (ℤ_≥‘(𝑛 + 1)) = (ℤ_≥‘1))
130	129, 1	eqtr4di 2784	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 0 → (ℤ_≥‘(𝑛 + 1)) = ℕ)
131	130	sumeq1d 15653	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 0 → Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘) = Σ𝑘 ∈ ℕ (𝐺‘𝑘))
132	131	fveq2d 6889	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 0 → (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘)) = (abs‘Σ𝑘 ∈ ℕ (𝐺‘𝑘)))
133	132	rspceeqv 3628	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ (0...(𝑠 − 1)) ∧ (abs‘Σ𝑘 ∈ ℕ 𝐵) = (abs‘Σ𝑘 ∈ ℕ (𝐺‘𝑘))) → ∃𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑘 ∈ ℕ 𝐵) = (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘)))
134	122, 128, 133	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → ∃𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑘 ∈ ℕ 𝐵) = (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘)))
135		fvex 6898	. . . . . . . . . . . . . . . 16 ⊢ (abs‘Σ𝑘 ∈ ℕ 𝐵) ∈ V
136		eqeq1 2730	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (abs‘Σ𝑘 ∈ ℕ 𝐵) → (𝑧 = (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘)) ↔ (abs‘Σ𝑘 ∈ ℕ 𝐵) = (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘))))
137	136	rexbidv 3172	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (abs‘Σ𝑘 ∈ ℕ 𝐵) → (∃𝑛 ∈ (0...(𝑠 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘)) ↔ ∃𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑘 ∈ ℕ 𝐵) = (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘))))
138	135, 137, 91	elab2 3667	. . . . . . . . . . . . . . 15 ⊢ ((abs‘Σ𝑘 ∈ ℕ 𝐵) ∈ 𝑇 ↔ ∃𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑘 ∈ ℕ 𝐵) = (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘)))
139	134, 138	sylibr 233	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → (abs‘Σ𝑘 ∈ ℕ 𝐵) ∈ 𝑇)
140	139	ne0d 4330	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → 𝑇 ≠ ∅)
141		ltso 11298	. . . . . . . . . . . . . 14 ⊢ < Or ℝ
142		fisupcl 9466	. . . . . . . . . . . . . 14 ⊢ (( < Or ℝ ∧ (𝑇 ∈ Fin ∧ 𝑇 ≠ ∅ ∧ 𝑇 ⊆ ℝ)) → sup(𝑇, ℝ, < ) ∈ 𝑇)
143	141, 142	mpan 687	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ Fin ∧ 𝑇 ≠ ∅ ∧ 𝑇 ⊆ ℝ) → sup(𝑇, ℝ, < ) ∈ 𝑇)
144	117, 140, 113, 143	syl3anc 1368	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → sup(𝑇, ℝ, < ) ∈ 𝑇)
145	113, 144	sseldd 3978	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → sup(𝑇, ℝ, < ) ∈ ℝ)
146		0red 11221	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → 0 ∈ ℝ)
147	123, 21	sylan2 592	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐵 ∈ ℂ)
148		1nn0 12492	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℕ₀
149	148	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ∈ ℕ₀)
150	5, 149, 26	iserex 15609	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (seq0( + , 𝐺) ∈ dom ⇝ ↔ seq1( + , 𝐺) ∈ dom ⇝ ))
151	22, 150	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → seq1( + , 𝐺) ∈ dom ⇝ )
152	1, 2, 124, 147, 151	isumcl 15713	. . . . . . . . . . . . . 14 ⊢ (𝜑 → Σ𝑘 ∈ ℕ 𝐵 ∈ ℂ)
153	152	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → Σ𝑘 ∈ ℕ 𝐵 ∈ ℂ)
154	153	abscld 15389	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → (abs‘Σ𝑘 ∈ ℕ 𝐵) ∈ ℝ)
155	153	absge0d 15397	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → 0 ≤ (abs‘Σ𝑘 ∈ ℕ 𝐵))
156		fimaxre2 12163	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ⊆ ℝ ∧ 𝑇 ∈ Fin) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ 𝑇 𝑤 ≤ 𝑧)
157	113, 117, 156	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ 𝑇 𝑤 ≤ 𝑧)
158	113, 140, 157, 139	suprubd 12180	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → (abs‘Σ𝑘 ∈ ℕ 𝐵) ≤ sup(𝑇, ℝ, < ))
159	146, 154, 145, 155, 158	letrd 11375	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → 0 ≤ sup(𝑇, ℝ, < ))
160	145, 159	ge0p1rpd 13052	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → (sup(𝑇, ℝ, < ) + 1) ∈ ℝ⁺)
161	90, 160	rpdivcld 13039	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) ∈ ℝ⁺)
162		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → (𝐾‘𝑛) = (𝐾‘𝑚))
163		eqid 2726	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ ↦ (𝐾‘𝑛)) = (𝑛 ∈ ℕ₀ ↦ (𝐾‘𝑛))
164		fvex 6898	. . . . . . . . . . 11 ⊢ (𝐾‘𝑚) ∈ V
165	162, 163, 164	fvmpt 6992	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ₀ → ((𝑛 ∈ ℕ₀ ↦ (𝐾‘𝑛))‘𝑚) = (𝐾‘𝑚))
166	165	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑚 ∈ ℕ₀) → ((𝑛 ∈ ℕ₀ ↦ (𝐾‘𝑛))‘𝑚) = (𝐾‘𝑚))
167		nn0ex 12482	. . . . . . . . . . . . 13 ⊢ ℕ₀ ∈ V
168	167	mptex 7220	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ₀ ↦ (𝐾‘𝑛)) ∈ V
169	168	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑛 ∈ ℕ₀ ↦ (𝐾‘𝑛)) ∈ V)
170		elnn0uz 12871	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ℕ₀ ↔ 𝑗 ∈ (ℤ_≥‘0))
171		fveq2 6885	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑗 → (𝐾‘𝑛) = (𝐾‘𝑗))
172		fvex 6898	. . . . . . . . . . . . . . . 16 ⊢ (𝐾‘𝑗) ∈ V
173	171, 163, 172	fvmpt 6992	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℕ₀ → ((𝑛 ∈ ℕ₀ ↦ (𝐾‘𝑛))‘𝑗) = (𝐾‘𝑗))
174	173	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((𝑛 ∈ ℕ₀ ↦ (𝐾‘𝑛))‘𝑗) = (𝐾‘𝑗))
175	170, 174	sylan2br 594	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘0)) → ((𝑛 ∈ ℕ₀ ↦ (𝐾‘𝑛))‘𝑗) = (𝐾‘𝑗))
176	6, 175	seqfeq 13998	. . . . . . . . . . . 12 ⊢ (𝜑 → seq0( + , (𝑛 ∈ ℕ₀ ↦ (𝐾‘𝑛))) = seq0( + , 𝐾))
177	176, 12	eqeltrd 2827	. . . . . . . . . . 11 ⊢ (𝜑 → seq0( + , (𝑛 ∈ ℕ₀ ↦ (𝐾‘𝑛))) ∈ dom ⇝ )
178	174, 8	eqtrd 2766	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((𝑛 ∈ ℕ₀ ↦ (𝐾‘𝑛))‘𝑗) = (abs‘𝐴))
179	178, 10	eqeltrd 2827	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((𝑛 ∈ ℕ₀ ↦ (𝐾‘𝑛))‘𝑗) ∈ ℝ)
180	179	recnd 11246	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((𝑛 ∈ ℕ₀ ↦ (𝐾‘𝑛))‘𝑗) ∈ ℂ)
181	5, 6, 169, 177, 180	serf0 15633	. . . . . . . . . 10 ⊢ (𝜑 → (𝑛 ∈ ℕ₀ ↦ (𝐾‘𝑛)) ⇝ 0)
182	181	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → (𝑛 ∈ ℕ₀ ↦ (𝐾‘𝑛)) ⇝ 0)
183	5, 85, 161, 166, 182	climi0 15462	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → ∃𝑡 ∈ ℕ₀ ∀𝑚 ∈ (ℤ_≥‘𝑡)(abs‘(𝐾‘𝑚)) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)))
184		simplll 772	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑡 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑡)) → 𝜑)
185		eluznn0 12905	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ ℕ₀ ∧ 𝑚 ∈ (ℤ_≥‘𝑡)) → 𝑚 ∈ ℕ₀)
186	185	adantll 711	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑡 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑡)) → 𝑚 ∈ ℕ₀)
187	11, 15	absidd 15375	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (abs‘(𝐾‘𝑗)) = (𝐾‘𝑗))
188	187	ralrimiva 3140	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑗 ∈ ℕ₀ (abs‘(𝐾‘𝑗)) = (𝐾‘𝑗))
189		fveq2 6885	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑚 → (𝐾‘𝑗) = (𝐾‘𝑚))
190	189	fveq2d 6889	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑚 → (abs‘(𝐾‘𝑗)) = (abs‘(𝐾‘𝑚)))
191	190, 189	eqeq12d 2742	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑚 → ((abs‘(𝐾‘𝑗)) = (𝐾‘𝑗) ↔ (abs‘(𝐾‘𝑚)) = (𝐾‘𝑚)))
192	191	rspccva 3605	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑗 ∈ ℕ₀ (abs‘(𝐾‘𝑗)) = (𝐾‘𝑗) ∧ 𝑚 ∈ ℕ₀) → (abs‘(𝐾‘𝑚)) = (𝐾‘𝑚))
193	188, 192	sylan 579	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (abs‘(𝐾‘𝑚)) = (𝐾‘𝑚))
194	184, 186, 193	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑡 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑡)) → (abs‘(𝐾‘𝑚)) = (𝐾‘𝑚))
195	194	breq1d 5151	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑡 ∈ ℕ₀) ∧ 𝑚 ∈ (ℤ_≥‘𝑡)) → ((abs‘(𝐾‘𝑚)) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) ↔ (𝐾‘𝑚) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1))))
196	195	ralbidva 3169	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑡 ∈ ℕ₀) → (∀𝑚 ∈ (ℤ_≥‘𝑡)(abs‘(𝐾‘𝑚)) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) ↔ ∀𝑚 ∈ (ℤ_≥‘𝑡)(𝐾‘𝑚) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1))))
197	162	breq1d 5151	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) ↔ (𝐾‘𝑚) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1))))
198	197	cbvralvw 3228	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ (ℤ_≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) ↔ ∀𝑚 ∈ (ℤ_≥‘𝑡)(𝐾‘𝑚) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)))
199	196, 198	bitr4di 289	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑡 ∈ ℕ₀) → (∀𝑚 ∈ (ℤ_≥‘𝑡)(abs‘(𝐾‘𝑚)) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) ↔ ∀𝑛 ∈ (ℤ_≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1))))
200		mertens.1	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝐹‘𝑗) = 𝐴)
201	200	ad4ant14 749	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝜓) ∧ (𝑡 ∈ ℕ₀ ∧ ∀𝑛 ∈ (ℤ_≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)))) ∧ 𝑗 ∈ ℕ₀) → (𝐹‘𝑗) = 𝐴)
202	8	ad4ant14 749	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝜓) ∧ (𝑡 ∈ ℕ₀ ∧ ∀𝑛 ∈ (ℤ_≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)))) ∧ 𝑗 ∈ ℕ₀) → (𝐾‘𝑗) = (abs‘𝐴))
203	9	ad4ant14 749	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝜓) ∧ (𝑡 ∈ ℕ₀ ∧ ∀𝑛 ∈ (ℤ_≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)))) ∧ 𝑗 ∈ ℕ₀) → 𝐴 ∈ ℂ)
204	20	ad4ant14 749	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝜓) ∧ (𝑡 ∈ ℕ₀ ∧ ∀𝑛 ∈ (ℤ_≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)))) ∧ 𝑘 ∈ ℕ₀) → (𝐺‘𝑘) = 𝐵)
205	21	ad4ant14 749	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝜓) ∧ (𝑡 ∈ ℕ₀ ∧ ∀𝑛 ∈ (ℤ_≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)))) ∧ 𝑘 ∈ ℕ₀) → 𝐵 ∈ ℂ)
206		mertens.6	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝐻‘𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗))))
207	206	ad4ant14 749	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝜓) ∧ (𝑡 ∈ ℕ₀ ∧ ∀𝑛 ∈ (ℤ_≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)))) ∧ 𝑘 ∈ ℕ₀) → (𝐻‘𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗))))
208	12	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑡 ∈ ℕ₀ ∧ ∀𝑛 ∈ (ℤ_≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)))) → seq0( + , 𝐾) ∈ dom ⇝ )
209	22	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑡 ∈ ℕ₀ ∧ ∀𝑛 ∈ (ℤ_≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)))) → seq0( + , 𝐺) ∈ dom ⇝ )
210	3	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑡 ∈ ℕ₀ ∧ ∀𝑛 ∈ (ℤ_≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)))) → 𝐸 ∈ ℝ⁺)
211	198	anbi2i 622	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ ℕ₀ ∧ ∀𝑛 ∈ (ℤ_≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1))) ↔ (𝑡 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑡)(𝐾‘𝑚) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1))))
212	211	anbi2i 622	. . . . . . . . . . . . . 14 ⊢ ((𝜓 ∧ (𝑡 ∈ ℕ₀ ∧ ∀𝑛 ∈ (ℤ_≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)))) ↔ (𝜓 ∧ (𝑡 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑡)(𝐾‘𝑚) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)))))
213	212	biimpi 215	. . . . . . . . . . . . 13 ⊢ ((𝜓 ∧ (𝑡 ∈ ℕ₀ ∧ ∀𝑛 ∈ (ℤ_≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)))) → (𝜓 ∧ (𝑡 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑡)(𝐾‘𝑚) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)))))
214	213	adantll 711	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑡 ∈ ℕ₀ ∧ ∀𝑛 ∈ (ℤ_≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)))) → (𝜓 ∧ (𝑡 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑡)(𝐾‘𝑚) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)))))
215	113, 140, 157	3jca 1125	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → (𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤 ∈ 𝑇 𝑤 ≤ 𝑧))
216	159, 215	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → (0 ≤ sup(𝑇, ℝ, < ) ∧ (𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤 ∈ 𝑇 𝑤 ≤ 𝑧)))
217	216	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑡 ∈ ℕ₀ ∧ ∀𝑛 ∈ (ℤ_≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)))) → (0 ≤ sup(𝑇, ℝ, < ) ∧ (𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤 ∈ 𝑇 𝑤 ≤ 𝑧)))
218	201, 202, 203, 204, 205, 207, 208, 209, 210, 91, 84, 214, 217	mertenslem1 15836	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑡 ∈ ℕ₀ ∧ ∀𝑛 ∈ (ℤ_≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)))) → ∃𝑦 ∈ ℕ₀ ∀𝑚 ∈ (ℤ_≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝐸)
219	218	expr 456	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑡 ∈ ℕ₀) → (∀𝑛 ∈ (ℤ_≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) → ∃𝑦 ∈ ℕ₀ ∀𝑚 ∈ (ℤ_≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝐸))
220	199, 219	sylbid 239	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑡 ∈ ℕ₀) → (∀𝑚 ∈ (ℤ_≥‘𝑡)(abs‘(𝐾‘𝑚)) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) → ∃𝑦 ∈ ℕ₀ ∀𝑚 ∈ (ℤ_≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝐸))
221	220	rexlimdva 3149	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (∃𝑡 ∈ ℕ₀ ∀𝑚 ∈ (ℤ_≥‘𝑡)(abs‘(𝐾‘𝑚)) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) → ∃𝑦 ∈ ℕ₀ ∀𝑚 ∈ (ℤ_≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝐸))
222	183, 221	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → ∃𝑦 ∈ ℕ₀ ∀𝑚 ∈ (ℤ_≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝐸)
223	222	ex 412	. . . . . 6 ⊢ (𝜑 → (𝜓 → ∃𝑦 ∈ ℕ₀ ∀𝑚 ∈ (ℤ_≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝐸))
224	84, 223	biimtrrid 242	. . . . 5 ⊢ (𝜑 → ((𝑠 ∈ ℕ ∧ ∀𝑛 ∈ (ℤ_≥‘𝑠)(abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1))) → ∃𝑦 ∈ ℕ₀ ∀𝑚 ∈ (ℤ_≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝐸))
225	224	expdimp 452	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ) → (∀𝑛 ∈ (ℤ_≥‘𝑠)(abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)) → ∃𝑦 ∈ ℕ₀ ∀𝑚 ∈ (ℤ_≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝐸))
226	83, 225	sylbid 239	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ) → (∀𝑚 ∈ (ℤ_≥‘𝑠)(abs‘((seq0( + , 𝐺)‘𝑚) − Σ𝑘 ∈ ℕ₀ 𝐵)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)) → ∃𝑦 ∈ ℕ₀ ∀𝑚 ∈ (ℤ_≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝐸))
227	226	rexlimdva 3149	. 2 ⊢ (𝜑 → (∃𝑠 ∈ ℕ ∀𝑚 ∈ (ℤ_≥‘𝑠)(abs‘((seq0( + , 𝐺)‘𝑚) − Σ𝑘 ∈ ℕ₀ 𝐵)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)) → ∃𝑦 ∈ ℕ₀ ∀𝑚 ∈ (ℤ_≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝐸))
228	24, 227	mpd 15	1 ⊢ (𝜑 → ∃𝑦 ∈ ℕ₀ ∀𝑚 ∈ (ℤ_≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 {cab 2703 ≠ wne 2934 ∀wral 3055 ∃wrex 3064 Vcvv 3468 ⊆ wss 3943 ∅c0 4317 class class class wbr 5141 ↦ cmpt 5224 Or wor 5580 dom cdm 5669 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 Fincfn 8941 supcsup 9437 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 + caddc 11115 · cmul 11117 < clt 11252 ≤ cle 11253 − cmin 11448 / cdiv 11875 ℕcn 12216 2c2 12271 ℕ₀cn0 12476 ℤ_≥cuz 12826 ℝ⁺crp 12980 ...cfz 13490 seqcseq 13972 abscabs 15187 ⇝ cli 15434 Σcsu 15638

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12981 df-ico 13336 df-fz 13491 df-fzo 13634 df-fl 13763 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15421 df-clim 15438 df-rlim 15439 df-sum 15639

This theorem is referenced by: mertens 15838

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