Theorem mertenslem2 11679

Description: Lemma for mertensabs 11680. (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 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnuz 9628	. . 3 ⊢ ℕ = (ℤ≥‘1)
2		1zzd 9344	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
3		mertens.9	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ ℝ+)
4	3	rphalfcld 9775	. . . 4 ⊢ (𝜑 → (𝐸 / 2) ∈ ℝ+)
5		nn0uz 9627	. . . . . 6 ⊢ ℕ0 = (ℤ≥‘0)
6		0zd 9329	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℤ)
7		eqidd 2194	. . . . . 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 11325	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (abs‘𝐴) ∈ ℝ)
11	8, 10	eqeltrd 2270	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐾‘𝑗) ∈ ℝ)
12		mertens.7	. . . . . 6 ⊢ (𝜑 → seq0( + , 𝐾) ∈ dom ⇝ )
13	5, 6, 7, 11, 12	isumrecl 11572	. . . . 5 ⊢ (𝜑 → Σ𝑗 ∈ ℕ0 (𝐾‘𝑗) ∈ ℝ)
14	9	absge0d 11328	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → 0 ≤ (abs‘𝐴))
15	14, 8	breqtrrd 4057	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → 0 ≤ (𝐾‘𝑗))
16	5, 6, 7, 11, 12, 15	isumge0 11573	. . . . 5 ⊢ (𝜑 → 0 ≤ Σ𝑗 ∈ ℕ0 (𝐾‘𝑗))
17	13, 16	ge0p1rpd 9793	. . . 4 ⊢ (𝜑 → (Σ𝑗 ∈ ℕ0 (𝐾‘𝑗) + 1) ∈ ℝ+)
18	4, 17	rpdivcld 9780	. . 3 ⊢ (𝜑 → ((𝐸 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾‘𝑗) + 1)) ∈ ℝ+)
19		eqidd 2194	. . 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 11565	. . 3 ⊢ (𝜑 → seq0( + , 𝐺) ⇝ Σ𝑘 ∈ ℕ0 𝐵)
24	1, 2, 18, 19, 23	climi2 11431	. 2 ⊢ (𝜑 → ∃𝑠 ∈ ℕ ∀𝑚 ∈ (ℤ≥‘𝑠)(abs‘((seq0( + , 𝐺)‘𝑚) − Σ𝑘 ∈ ℕ0 𝐵)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾‘𝑗) + 1)))
25		eluznn 9665	. . . . . . . 8 ⊢ ((𝑠 ∈ ℕ ∧ 𝑚 ∈ (ℤ≥‘𝑠)) → 𝑚 ∈ ℕ)
26	20, 21	eqeltrd 2270	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) ∈ ℂ)
27	5, 6, 26	serf 10554	. . . . . . . . . . . 12 ⊢ (𝜑 → seq0( + , 𝐺):ℕ0⟶ℂ)
28		nnnn0 9247	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℕ0)
29		ffvelcdm 5691	. . . . . . . . . . . 12 ⊢ ((seq0( + , 𝐺):ℕ0⟶ℂ ∧ 𝑚 ∈ ℕ0) → (seq0( + , 𝐺)‘𝑚) ∈ ℂ)
30	27, 28, 29	syl2an 289	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (seq0( + , 𝐺)‘𝑚) ∈ ℂ)
31	5, 6, 20, 21, 22	isumcl 11568	. . . . . . . . . . . 12 ⊢ (𝜑 → Σ𝑘 ∈ ℕ0 𝐵 ∈ ℂ)
32	31	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ ℕ0 𝐵 ∈ ℂ)
33	30, 32	abssubd 11337	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (abs‘((seq0( + , 𝐺)‘𝑚) − Σ𝑘 ∈ ℕ0 𝐵)) = (abs‘(Σ𝑘 ∈ ℕ0 𝐵 − (seq0( + , 𝐺)‘𝑚))))
34		eqid 2193	. . . . . . . . . . . . . 14 ⊢ (ℤ≥‘(𝑚 + 1)) = (ℤ≥‘(𝑚 + 1))
35	28	adantl 277	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ0)
36		peano2nn0 9280	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℕ0)
37	35, 36	syl 14	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 + 1) ∈ ℕ0)
38	37	nn0zd 9437	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 + 1) ∈ ℤ)
39		simpll 527	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘(𝑚 + 1))) → 𝜑)
40		eluznn0 9664	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 + 1) ∈ ℕ0 ∧ 𝑘 ∈ (ℤ≥‘(𝑚 + 1))) → 𝑘 ∈ ℕ0)
41	37, 40	sylan 283	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘(𝑚 + 1))) → 𝑘 ∈ ℕ0)
42	39, 41, 20	syl2anc 411	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘(𝑚 + 1))) → (𝐺‘𝑘) = 𝐵)
43	39, 41, 21	syl2anc 411	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘(𝑚 + 1))) → 𝐵 ∈ ℂ)
44	22	adantr 276	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → seq0( + , 𝐺) ∈ dom ⇝ )
45	26	adantlr 477	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) ∈ ℂ)
46	5, 37, 45	iserex 11482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (seq0( + , 𝐺) ∈ dom ⇝ ↔ seq(𝑚 + 1)( + , 𝐺) ∈ dom ⇝ ))
47	44, 46	mpbid 147	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → seq(𝑚 + 1)( + , 𝐺) ∈ dom ⇝ )
48	34, 38, 42, 43, 47	isumcl 11568	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ (ℤ≥‘(𝑚 + 1))𝐵 ∈ ℂ)
49	30, 48	pncan2d 8332	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((seq0( + , 𝐺)‘𝑚) + Σ𝑘 ∈ (ℤ≥‘(𝑚 + 1))𝐵) − (seq0( + , 𝐺)‘𝑚)) = Σ𝑘 ∈ (ℤ≥‘(𝑚 + 1))𝐵)
50	20	adantlr 477	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) = 𝐵)
51	21	adantlr 477	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)
52	5, 34, 37, 50, 51, 44	isumsplit 11634	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ ℕ0 𝐵 = (Σ𝑘 ∈ (0...((𝑚 + 1) − 1))𝐵 + Σ𝑘 ∈ (ℤ≥‘(𝑚 + 1))𝐵))
53		nncn 8990	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
54	53	adantl 277	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
55		ax-1cn 7965	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℂ
56		pncan 8225	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑚 + 1) − 1) = 𝑚)
57	54, 55, 56	sylancl 413	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑚 + 1) − 1) = 𝑚)
58	57	oveq2d 5934	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (0...((𝑚 + 1) − 1)) = (0...𝑚))
59	58	sumeq1d 11509	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ (0...((𝑚 + 1) − 1))𝐵 = Σ𝑘 ∈ (0...𝑚)𝐵)
60		elnn0uz 9630	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℕ0 ↔ 𝑘 ∈ (ℤ≥‘0))
61	60, 50	sylan2br 288	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘0)) → (𝐺‘𝑘) = 𝐵)
62	35, 5	eleqtrdi 2286	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ (ℤ≥‘0))
63	60, 51	sylan2br 288	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘0)) → 𝐵 ∈ ℂ)
64	61, 62, 63	fsum3ser 11540	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ (0...𝑚)𝐵 = (seq0( + , 𝐺)‘𝑚))
65	59, 64	eqtrd 2226	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ (0...((𝑚 + 1) − 1))𝐵 = (seq0( + , 𝐺)‘𝑚))
66	65	oveq1d 5933	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (Σ𝑘 ∈ (0...((𝑚 + 1) − 1))𝐵 + Σ𝑘 ∈ (ℤ≥‘(𝑚 + 1))𝐵) = ((seq0( + , 𝐺)‘𝑚) + Σ𝑘 ∈ (ℤ≥‘(𝑚 + 1))𝐵))
67	52, 66	eqtrd 2226	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ ℕ0 𝐵 = ((seq0( + , 𝐺)‘𝑚) + Σ𝑘 ∈ (ℤ≥‘(𝑚 + 1))𝐵))
68	67	oveq1d 5933	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (Σ𝑘 ∈ ℕ0 𝐵 − (seq0( + , 𝐺)‘𝑚)) = (((seq0( + , 𝐺)‘𝑚) + Σ𝑘 ∈ (ℤ≥‘(𝑚 + 1))𝐵) − (seq0( + , 𝐺)‘𝑚)))
69	42	sumeq2dv 11511	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ (ℤ≥‘(𝑚 + 1))(𝐺‘𝑘) = Σ𝑘 ∈ (ℤ≥‘(𝑚 + 1))𝐵)
70	49, 68, 69	3eqtr4d 2236	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (Σ𝑘 ∈ ℕ0 𝐵 − (seq0( + , 𝐺)‘𝑚)) = Σ𝑘 ∈ (ℤ≥‘(𝑚 + 1))(𝐺‘𝑘))
71	70	fveq2d 5558	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (abs‘(Σ𝑘 ∈ ℕ0 𝐵 − (seq0( + , 𝐺)‘𝑚))) = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑚 + 1))(𝐺‘𝑘)))
72	33, 71	eqtrd 2226	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (abs‘((seq0( + , 𝐺)‘𝑚) − Σ𝑘 ∈ ℕ0 𝐵)) = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑚 + 1))(𝐺‘𝑘)))
73	72	breq1d 4039	. . . . . . . 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))))
74	25, 73	sylan2 286	. . . . . . 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))))
75	74	anassrs 400	. . . . . 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))))
76	75	ralbidva 2490	. . . . 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))))
77		fvoveq1 5941	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (ℤ≥‘(𝑚 + 1)) = (ℤ≥‘(𝑛 + 1)))
78	77	sumeq1d 11509	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → Σ𝑘 ∈ (ℤ≥‘(𝑚 + 1))(𝐺‘𝑘) = Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))
79	78	fveq2d 5558	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑚 + 1))(𝐺‘𝑘)) = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)))
80	79	breq1d 4039	. . . . . 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))))
81	80	cbvralv 2726	. . . . 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)))
82	76, 81	bitrdi 196	. . . 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))))
83		mertens.11	. . . . . 6 ⊢ (𝜓 ↔ (𝑠 ∈ ℕ ∧ ∀𝑛 ∈ (ℤ≥‘𝑠)(abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾‘𝑗) + 1))))
84		0zd 9329	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → 0 ∈ ℤ)
85	4	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → (𝐸 / 2) ∈ ℝ+)
86	83	simplbi 274	. . . . . . . . . . . . . 14 ⊢ (𝜓 → 𝑠 ∈ ℕ)
87	86	adantl 277	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → 𝑠 ∈ ℕ)
88	87	nnrpd 9760	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → 𝑠 ∈ ℝ+)
89	85, 88	rpdivcld 9780	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → ((𝐸 / 2) / 𝑠) ∈ ℝ+)
90	87	nnzd 9438	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → 𝑠 ∈ ℤ)
91		1zzd 9344	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → 1 ∈ ℤ)
92	90, 91	zsubcld 9444	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → (𝑠 − 1) ∈ ℤ)
93	84, 92	fzfigd 10502	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → (0...(𝑠 − 1)) ∈ Fin)
94		eqid 2193	. . . . . . . . . . . . . . 15 ⊢ (ℤ≥‘(𝑛 + 1)) = (ℤ≥‘(𝑛 + 1))
95		elfznn0 10180	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (0...(𝑠 − 1)) → 𝑛 ∈ ℕ0)
96	95	adantl 277	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (0...(𝑠 − 1))) → 𝑛 ∈ ℕ0)
97		peano2nn0 9280	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ0)
98	96, 97	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (0...(𝑠 − 1))) → (𝑛 + 1) ∈ ℕ0)
99	98	nn0zd 9437	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (0...(𝑠 − 1))) → (𝑛 + 1) ∈ ℤ)
100		eqidd 2194	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (0...(𝑠 − 1))) ∧ 𝑘 ∈ (ℤ≥‘(𝑛 + 1))) → (𝐺‘𝑘) = (𝐺‘𝑘))
101		simplll 533	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (0...(𝑠 − 1))) ∧ 𝑘 ∈ (ℤ≥‘(𝑛 + 1))) → 𝜑)
102		eluznn0 9664	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 + 1) ∈ ℕ0 ∧ 𝑘 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑘 ∈ ℕ0)
103	98, 102	sylan 283	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (0...(𝑠 − 1))) ∧ 𝑘 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑘 ∈ ℕ0)
104	101, 103, 26	syl2anc 411	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (0...(𝑠 − 1))) ∧ 𝑘 ∈ (ℤ≥‘(𝑛 + 1))) → (𝐺‘𝑘) ∈ ℂ)
105	22	ad2antrr 488	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (0...(𝑠 − 1))) → seq0( + , 𝐺) ∈ dom ⇝ )
106		simpll 527	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (0...(𝑠 − 1))) → 𝜑)
107	106, 26	sylan 283	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (0...(𝑠 − 1))) ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) ∈ ℂ)
108	5, 98, 107	iserex 11482	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (0...(𝑠 − 1))) → (seq0( + , 𝐺) ∈ dom ⇝ ↔ seq(𝑛 + 1)( + , 𝐺) ∈ dom ⇝ ))
109	105, 108	mpbid 147	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (0...(𝑠 − 1))) → seq(𝑛 + 1)( + , 𝐺) ∈ dom ⇝ )
110	94, 99, 100, 104, 109	isumcl 11568	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (0...(𝑠 − 1))) → Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘) ∈ ℂ)
111	110	abscld 11325	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑛 ∈ (0...(𝑠 − 1))) → (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) ∈ ℝ)
112	93, 111	fsumrecl 11544	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) ∈ ℝ)
113		0red 8020	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → 0 ∈ ℝ)
114		nnnn0 9247	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
115	114, 20	sylan2 286	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) = 𝐵)
116	114, 21	sylan2 286	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐵 ∈ ℂ)
117		1nn0 9256	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℕ0
118	117	a1i 9	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 1 ∈ ℕ0)
119	5, 118, 26	iserex 11482	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (seq0( + , 𝐺) ∈ dom ⇝ ↔ seq1( + , 𝐺) ∈ dom ⇝ ))
120	22, 119	mpbid 147	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → seq1( + , 𝐺) ∈ dom ⇝ )
121	1, 2, 115, 116, 120	isumcl 11568	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → Σ𝑘 ∈ ℕ 𝐵 ∈ ℂ)
122	121	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → Σ𝑘 ∈ ℕ 𝐵 ∈ ℂ)
123	122	abscld 11325	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → (abs‘Σ𝑘 ∈ ℕ 𝐵) ∈ ℝ)
124	122	absge0d 11328	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → 0 ≤ (abs‘Σ𝑘 ∈ ℕ 𝐵))
125	20	adantlr 477	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) = 𝐵)
126	21	adantlr 477	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)
127	22	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → seq0( + , 𝐺) ∈ dom ⇝ )
128		mertens.10	. . . . . . . . . . . . . 14 ⊢ 𝑇 = {𝑧 ∣ ∃𝑛 ∈ (0...(𝑠 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))}
129		nnm1nn0 9281	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ ℕ → (𝑠 − 1) ∈ ℕ0)
130	87, 129	syl 14	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜓) → (𝑠 − 1) ∈ ℕ0)
131	130, 5	eleqtrdi 2286	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜓) → (𝑠 − 1) ∈ (ℤ≥‘0))
132		eluzfz1 10097	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑠 − 1) ∈ (ℤ≥‘0) → 0 ∈ (0...(𝑠 − 1)))
133	131, 132	syl 14	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜓) → 0 ∈ (0...(𝑠 − 1)))
134	115	sumeq2dv 11511	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → Σ𝑘 ∈ ℕ (𝐺‘𝑘) = Σ𝑘 ∈ ℕ 𝐵)
135	134	adantr 276	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜓) → Σ𝑘 ∈ ℕ (𝐺‘𝑘) = Σ𝑘 ∈ ℕ 𝐵)
136	135	fveq2d 5558	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜓) → (abs‘Σ𝑘 ∈ ℕ (𝐺‘𝑘)) = (abs‘Σ𝑘 ∈ ℕ 𝐵))
137	136	eqcomd 2199	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜓) → (abs‘Σ𝑘 ∈ ℕ 𝐵) = (abs‘Σ𝑘 ∈ ℕ (𝐺‘𝑘)))
138		fv0p1e1 9097	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 0 → (ℤ≥‘(𝑛 + 1)) = (ℤ≥‘1))
139	138, 1	eqtr4di 2244	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 0 → (ℤ≥‘(𝑛 + 1)) = ℕ)
140	139	sumeq1d 11509	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 0 → Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘) = Σ𝑘 ∈ ℕ (𝐺‘𝑘))
141	140	fveq2d 5558	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 0 → (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) = (abs‘Σ𝑘 ∈ ℕ (𝐺‘𝑘)))
142	141	rspceeqv 2882	. . . . . . . . . . . . . . . 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)))
143	133, 137, 142	syl2anc 411	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → ∃𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑘 ∈ ℕ 𝐵) = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)))
144		eqeq1 2200	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (abs‘Σ𝑘 ∈ ℕ 𝐵) → (𝑧 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) ↔ (abs‘Σ𝑘 ∈ ℕ 𝐵) = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))))
145	144	rexbidv 2495	. . . . . . . . . . . . . . . . 17 ⊢ (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))))
146	145, 128	elab2g 2907	. . . . . . . . . . . . . . . 16 ⊢ ((abs‘Σ𝑘 ∈ ℕ 𝐵) ∈ ℝ → ((abs‘Σ𝑘 ∈ ℕ 𝐵) ∈ 𝑇 ↔ ∃𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑘 ∈ ℕ 𝐵) = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))))
147	123, 146	syl 14	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → ((abs‘Σ𝑘 ∈ ℕ 𝐵) ∈ 𝑇 ↔ ∃𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑘 ∈ ℕ 𝐵) = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))))
148	143, 147	mpbird 167	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → (abs‘Σ𝑘 ∈ ℕ 𝐵) ∈ 𝑇)
149	125, 126, 127, 128, 148, 87	mertenslemub 11677	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → (abs‘Σ𝑘 ∈ ℕ 𝐵) ≤ Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)))
150	113, 123, 112, 124, 149	letrd 8143	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → 0 ≤ Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)))
151	112, 150	ge0p1rpd 9793	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) + 1) ∈ ℝ+)
152	89, 151	rpdivcld 9780	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) + 1)) ∈ ℝ+)
153		simpr 110	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℕ0)
154		fveq2 5554	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑚 → (𝐾‘𝑗) = (𝐾‘𝑚))
155	154	eleq1d 2262	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑚 → ((𝐾‘𝑗) ∈ ℝ ↔ (𝐾‘𝑚) ∈ ℝ))
156	11	ralrimiva 2567	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑗 ∈ ℕ0 (𝐾‘𝑗) ∈ ℝ)
157	156	ad2antrr 488	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑚 ∈ ℕ0) → ∀𝑗 ∈ ℕ0 (𝐾‘𝑗) ∈ ℝ)
158	155, 157, 153	rspcdva 2869	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑚 ∈ ℕ0) → (𝐾‘𝑚) ∈ ℝ)
159		fveq2 5554	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → (𝐾‘𝑛) = (𝐾‘𝑚))
160		eqid 2193	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ0 ↦ (𝐾‘𝑛)) = (𝑛 ∈ ℕ0 ↦ (𝐾‘𝑛))
161	159, 160	fvmptg 5633	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ0 ∧ (𝐾‘𝑚) ∈ ℝ) → ((𝑛 ∈ ℕ0 ↦ (𝐾‘𝑛))‘𝑚) = (𝐾‘𝑚))
162	153, 158, 161	syl2anc 411	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑚 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (𝐾‘𝑛))‘𝑚) = (𝐾‘𝑚))
163		nn0ex 9246	. . . . . . . . . . . . . 14 ⊢ ℕ0 ∈ V
164	163	mptex 5784	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ0 ↦ (𝐾‘𝑛)) ∈ V
165	164	a1i 9	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑛 ∈ ℕ0 ↦ (𝐾‘𝑛)) ∈ V)
166	60	biimpri 133	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (ℤ≥‘0) → 𝑘 ∈ ℕ0)
167		fveq2 5554	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑘 → (𝐾‘𝑗) = (𝐾‘𝑘))
168	167	eleq1d 2262	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑘 → ((𝐾‘𝑗) ∈ ℝ ↔ (𝐾‘𝑘) ∈ ℝ))
169	156	adantr 276	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ∀𝑗 ∈ ℕ0 (𝐾‘𝑗) ∈ ℝ)
170		simpr 110	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
171	168, 169, 170	rspcdva 2869	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐾‘𝑘) ∈ ℝ)
172	60, 171	sylan2br 288	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘0)) → (𝐾‘𝑘) ∈ ℝ)
173		fveq2 5554	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → (𝐾‘𝑛) = (𝐾‘𝑘))
174	173, 160	fvmptg 5633	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝐾‘𝑘) ∈ ℝ) → ((𝑛 ∈ ℕ0 ↦ (𝐾‘𝑛))‘𝑘) = (𝐾‘𝑘))
175	166, 172, 174	syl2an2 594	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘0)) → ((𝑛 ∈ ℕ0 ↦ (𝐾‘𝑛))‘𝑘) = (𝐾‘𝑘))
176	175, 172	eqeltrd 2270	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘0)) → ((𝑛 ∈ ℕ0 ↦ (𝐾‘𝑛))‘𝑘) ∈ ℝ)
177		elnn0uz 9630	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℕ0 ↔ 𝑗 ∈ (ℤ≥‘0))
178		simpr 110	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → 𝑗 ∈ ℕ0)
179		fveq2 5554	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑗 → (𝐾‘𝑛) = (𝐾‘𝑗))
180	179, 160	fvmptg 5633	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ ℕ0 ∧ (𝐾‘𝑗) ∈ ℝ) → ((𝑛 ∈ ℕ0 ↦ (𝐾‘𝑛))‘𝑗) = (𝐾‘𝑗))
181	178, 11, 180	syl2anc 411	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (𝐾‘𝑛))‘𝑗) = (𝐾‘𝑗))
182	177, 181	sylan2br 288	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘0)) → ((𝑛 ∈ ℕ0 ↦ (𝐾‘𝑛))‘𝑗) = (𝐾‘𝑗))
183		readdcl 7998	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑘 + 𝑦) ∈ ℝ)
184	183	adantl 277	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑘 + 𝑦) ∈ ℝ)
185	6, 176, 182, 184	seq3feq 10551	. . . . . . . . . . . . 13 ⊢ (𝜑 → seq0( + , (𝑛 ∈ ℕ0 ↦ (𝐾‘𝑛))) = seq0( + , 𝐾))
186	185, 12	eqeltrd 2270	. . . . . . . . . . . 12 ⊢ (𝜑 → seq0( + , (𝑛 ∈ ℕ0 ↦ (𝐾‘𝑛))) ∈ dom ⇝ )
187	181, 11	eqeltrd 2270	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (𝐾‘𝑛))‘𝑗) ∈ ℝ)
188	187	recnd 8048	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (𝐾‘𝑛))‘𝑗) ∈ ℂ)
189	5, 6, 165, 186, 188	serf0 11495	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑛 ∈ ℕ0 ↦ (𝐾‘𝑛)) ⇝ 0)
190	189	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → (𝑛 ∈ ℕ0 ↦ (𝐾‘𝑛)) ⇝ 0)
191	5, 84, 152, 162, 190	climi0 11432	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → ∃𝑡 ∈ ℕ0 ∀𝑚 ∈ (ℤ≥‘𝑡)(abs‘(𝐾‘𝑚)) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) + 1)))
192		fveq2 5554	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑎 → (𝐺‘𝑘) = (𝐺‘𝑎))
193	192	cbvsumv 11504	. . . . . . . . . . . . . . . . 17 ⊢ Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘) = Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)
194	193	fveq2i 5557	. . . . . . . . . . . . . . . 16 ⊢ (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) = (abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎))
195	194	a1i 9	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (0...(𝑠 − 1)) → (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) = (abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)))
196	195	sumeq2i 11507	. . . . . . . . . . . . . 14 ⊢ Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) = Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎))
197	196	oveq1i 5928	. . . . . . . . . . . . 13 ⊢ (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) + 1) = (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)
198	197	oveq2i 5929	. . . . . . . . . . . 12 ⊢ (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) + 1)) = (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1))
199	198	breq2i 4037	. . . . . . . . . . 11 ⊢ ((abs‘(𝐾‘𝑚)) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) + 1)) ↔ (abs‘(𝐾‘𝑚)) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)))
200	199	ralbii 2500	. . . . . . . . . 10 ⊢ (∀𝑚 ∈ (ℤ≥‘𝑡)(abs‘(𝐾‘𝑚)) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) + 1)) ↔ ∀𝑚 ∈ (ℤ≥‘𝑡)(abs‘(𝐾‘𝑚)) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)))
201	200	rexbii 2501	. . . . . . . . 9 ⊢ (∃𝑡 ∈ ℕ0 ∀𝑚 ∈ (ℤ≥‘𝑡)(abs‘(𝐾‘𝑚)) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) + 1)) ↔ ∃𝑡 ∈ ℕ0 ∀𝑚 ∈ (ℤ≥‘𝑡)(abs‘(𝐾‘𝑚)) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)))
202	191, 201	sylib 122	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → ∃𝑡 ∈ ℕ0 ∀𝑚 ∈ (ℤ≥‘𝑡)(abs‘(𝐾‘𝑚)) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)))
203		simplll 533	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑡 ∈ ℕ0) ∧ 𝑚 ∈ (ℤ≥‘𝑡)) → 𝜑)
204		eluznn0 9664	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ ℕ0 ∧ 𝑚 ∈ (ℤ≥‘𝑡)) → 𝑚 ∈ ℕ0)
205	204	adantll 476	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑡 ∈ ℕ0) ∧ 𝑚 ∈ (ℤ≥‘𝑡)) → 𝑚 ∈ ℕ0)
206	11, 15	absidd 11311	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (abs‘(𝐾‘𝑗)) = (𝐾‘𝑗))
207	206	ralrimiva 2567	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑗 ∈ ℕ0 (abs‘(𝐾‘𝑗)) = (𝐾‘𝑗))
208	154	fveq2d 5558	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑚 → (abs‘(𝐾‘𝑗)) = (abs‘(𝐾‘𝑚)))
209	208, 154	eqeq12d 2208	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑚 → ((abs‘(𝐾‘𝑗)) = (𝐾‘𝑗) ↔ (abs‘(𝐾‘𝑚)) = (𝐾‘𝑚)))
210	209	rspccva 2863	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑗 ∈ ℕ0 (abs‘(𝐾‘𝑗)) = (𝐾‘𝑗) ∧ 𝑚 ∈ ℕ0) → (abs‘(𝐾‘𝑚)) = (𝐾‘𝑚))
211	207, 210	sylan 283	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (abs‘(𝐾‘𝑚)) = (𝐾‘𝑚))
212	203, 205, 211	syl2anc 411	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑡 ∈ ℕ0) ∧ 𝑚 ∈ (ℤ≥‘𝑡)) → (abs‘(𝐾‘𝑚)) = (𝐾‘𝑚))
213	212	breq1d 4039	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑡 ∈ ℕ0) ∧ 𝑚 ∈ (ℤ≥‘𝑡)) → ((abs‘(𝐾‘𝑚)) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)) ↔ (𝐾‘𝑚) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1))))
214	213	ralbidva 2490	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑡 ∈ ℕ0) → (∀𝑚 ∈ (ℤ≥‘𝑡)(abs‘(𝐾‘𝑚)) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)) ↔ ∀𝑚 ∈ (ℤ≥‘𝑡)(𝐾‘𝑚) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1))))
215		nfv 1539	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1))
216		nfcv 2336	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛(𝐾‘𝑚)
217		nfcv 2336	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛 <
218		nfcv 2336	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛((𝐸 / 2) / 𝑠)
219		nfcv 2336	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛 /
220		nfcv 2336	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑛(0...(𝑠 − 1))
221	220	nfsum1 11499	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎))
222		nfcv 2336	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛 +
223		nfcv 2336	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛1
224	221, 222, 223	nfov 5948	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛(Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)
225	218, 219, 224	nfov 5948	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛(((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1))
226	216, 217, 225	nfbr 4075	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(𝐾‘𝑚) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1))
227	159	breq1d 4039	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)) ↔ (𝐾‘𝑚) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1))))
228	215, 226, 227	cbvral 2722	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ (ℤ≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)) ↔ ∀𝑚 ∈ (ℤ≥‘𝑡)(𝐾‘𝑚) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)))
229	214, 228	bitr4di 198	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑡 ∈ ℕ0) → (∀𝑚 ∈ (ℤ≥‘𝑡)(abs‘(𝐾‘𝑚)) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)) ↔ ∀𝑛 ∈ (ℤ≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1))))
230		simpll 527	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑡 ∈ ℕ0 ∧ ∀𝑛 ∈ (ℤ≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)))) → 𝜑)
231		mertens.1	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐹‘𝑗) = 𝐴)
232	230, 231	sylan 283	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝜓) ∧ (𝑡 ∈ ℕ0 ∧ ∀𝑛 ∈ (ℤ≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)))) ∧ 𝑗 ∈ ℕ0) → (𝐹‘𝑗) = 𝐴)
233	230, 8	sylan 283	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝜓) ∧ (𝑡 ∈ ℕ0 ∧ ∀𝑛 ∈ (ℤ≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)))) ∧ 𝑗 ∈ ℕ0) → (𝐾‘𝑗) = (abs‘𝐴))
234	230, 9	sylan 283	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝜓) ∧ (𝑡 ∈ ℕ0 ∧ ∀𝑛 ∈ (ℤ≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)))) ∧ 𝑗 ∈ ℕ0) → 𝐴 ∈ ℂ)
235	230, 20	sylan 283	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝜓) ∧ (𝑡 ∈ ℕ0 ∧ ∀𝑛 ∈ (ℤ≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)))) ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) = 𝐵)
236	230, 21	sylan 283	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝜓) ∧ (𝑡 ∈ ℕ0 ∧ ∀𝑛 ∈ (ℤ≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)))) ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)
237		mertens.6	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐻‘𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗))))
238	230, 237	sylan 283	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝜓) ∧ (𝑡 ∈ ℕ0 ∧ ∀𝑛 ∈ (ℤ≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)))) ∧ 𝑘 ∈ ℕ0) → (𝐻‘𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗))))
239	12	ad2antrr 488	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑡 ∈ ℕ0 ∧ ∀𝑛 ∈ (ℤ≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)))) → seq0( + , 𝐾) ∈ dom ⇝ )
240	22	ad2antrr 488	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑡 ∈ ℕ0 ∧ ∀𝑛 ∈ (ℤ≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)))) → seq0( + , 𝐺) ∈ dom ⇝ )
241	3	ad2antrr 488	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑡 ∈ ℕ0 ∧ ∀𝑛 ∈ (ℤ≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)))) → 𝐸 ∈ ℝ+)
242	196, 112	eqeltrrid 2281	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) ∈ ℝ)
243	242	adantr 276	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑡 ∈ ℕ0 ∧ ∀𝑛 ∈ (ℤ≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)))) → Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) ∈ ℝ)
244	228	anbi2i 457	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ ℕ0 ∧ ∀𝑛 ∈ (ℤ≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1))) ↔ (𝑡 ∈ ℕ0 ∧ ∀𝑚 ∈ (ℤ≥‘𝑡)(𝐾‘𝑚) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1))))
245	244	anbi2i 457	. . . . . . . . . . . . . 14 ⊢ ((𝜓 ∧ (𝑡 ∈ ℕ0 ∧ ∀𝑛 ∈ (ℤ≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)))) ↔ (𝜓 ∧ (𝑡 ∈ ℕ0 ∧ ∀𝑚 ∈ (ℤ≥‘𝑡)(𝐾‘𝑚) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)))))
246	245	biimpi 120	. . . . . . . . . . . . 13 ⊢ ((𝜓 ∧ (𝑡 ∈ ℕ0 ∧ ∀𝑛 ∈ (ℤ≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)))) → (𝜓 ∧ (𝑡 ∈ ℕ0 ∧ ∀𝑚 ∈ (ℤ≥‘𝑡)(𝐾‘𝑚) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)))))
247	246	adantll 476	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑡 ∈ ℕ0 ∧ ∀𝑛 ∈ (ℤ≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)))) → (𝜓 ∧ (𝑡 ∈ ℕ0 ∧ ∀𝑚 ∈ (ℤ≥‘𝑡)(𝐾‘𝑚) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)))))
248	150, 196	breqtrdi 4070	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → 0 ≤ Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)))
249	248	adantr 276	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑡 ∈ ℕ0 ∧ ∀𝑛 ∈ (ℤ≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)))) → 0 ≤ Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)))
250		simpr 110	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑤 ∈ 𝑇) ∧ 𝑎 ∈ ℕ0) → 𝑎 ∈ ℕ0)
251	20	ralrimiva 2567	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 (𝐺‘𝑘) = 𝐵)
252	251	ad3antrrr 492	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑤 ∈ 𝑇) ∧ 𝑎 ∈ ℕ0) → ∀𝑘 ∈ ℕ0 (𝐺‘𝑘) = 𝐵)
253		nfcsb1v 3113	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘⦋𝑎 / 𝑘⦌𝐵
254	253	nfeq2 2348	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(𝐺‘𝑎) = ⦋𝑎 / 𝑘⦌𝐵
255		csbeq1a 3089	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑎 → 𝐵 = ⦋𝑎 / 𝑘⦌𝐵)
256	192, 255	eqeq12d 2208	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑎 → ((𝐺‘𝑘) = 𝐵 ↔ (𝐺‘𝑎) = ⦋𝑎 / 𝑘⦌𝐵))
257	254, 256	rspc 2858	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ ℕ0 → (∀𝑘 ∈ ℕ0 (𝐺‘𝑘) = 𝐵 → (𝐺‘𝑎) = ⦋𝑎 / 𝑘⦌𝐵))
258	250, 252, 257	sylc 62	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑤 ∈ 𝑇) ∧ 𝑎 ∈ ℕ0) → (𝐺‘𝑎) = ⦋𝑎 / 𝑘⦌𝐵)
259	21	ralrimiva 2567	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 𝐵 ∈ ℂ)
260	259	ad3antrrr 492	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑤 ∈ 𝑇) ∧ 𝑎 ∈ ℕ0) → ∀𝑘 ∈ ℕ0 𝐵 ∈ ℂ)
261	253	nfel1 2347	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘⦋𝑎 / 𝑘⦌𝐵 ∈ ℂ
262	255	eleq1d 2262	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑎 → (𝐵 ∈ ℂ ↔ ⦋𝑎 / 𝑘⦌𝐵 ∈ ℂ))
263	261, 262	rspc 2858	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ ℕ0 → (∀𝑘 ∈ ℕ0 𝐵 ∈ ℂ → ⦋𝑎 / 𝑘⦌𝐵 ∈ ℂ))
264	250, 260, 263	sylc 62	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝑤 ∈ 𝑇) ∧ 𝑎 ∈ ℕ0) → ⦋𝑎 / 𝑘⦌𝐵 ∈ ℂ)
265	22	ad2antrr 488	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑤 ∈ 𝑇) → seq0( + , 𝐺) ∈ dom ⇝ )
266	194	eqeq2i 2204	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) ↔ 𝑧 = (abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)))
267	266	rexbii 2501	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑛 ∈ (0...(𝑠 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) ↔ ∃𝑛 ∈ (0...(𝑠 − 1))𝑧 = (abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)))
268	267	abbii 2309	. . . . . . . . . . . . . . . 16 ⊢ {𝑧 ∣ ∃𝑛 ∈ (0...(𝑠 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))} = {𝑧 ∣ ∃𝑛 ∈ (0...(𝑠 − 1))𝑧 = (abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎))}
269	128, 268	eqtri 2214	. . . . . . . . . . . . . . 15 ⊢ 𝑇 = {𝑧 ∣ ∃𝑛 ∈ (0...(𝑠 − 1))𝑧 = (abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎))}
270		simpr 110	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑤 ∈ 𝑇) → 𝑤 ∈ 𝑇)
271	87	adantr 276	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑤 ∈ 𝑇) → 𝑠 ∈ ℕ)
272	258, 264, 265, 269, 270, 271	mertenslemub 11677	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑤 ∈ 𝑇) → 𝑤 ≤ Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)))
273	272	ralrimiva 2567	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑤 ∈ 𝑇 𝑤 ≤ Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)))
274	273	adantr 276	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑡 ∈ ℕ0 ∧ ∀𝑛 ∈ (ℤ≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)))) → ∀𝑤 ∈ 𝑇 𝑤 ≤ Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)))
275	232, 233, 234, 235, 236, 238, 239, 240, 241, 128, 83, 243, 247, 249, 274	mertenslemi1 11678	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑡 ∈ ℕ0 ∧ ∀𝑛 ∈ (ℤ≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)))) → ∃𝑦 ∈ ℕ0 ∀𝑚 ∈ (ℤ≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝐸)
276	275	expr 375	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑡 ∈ ℕ0) → (∀𝑛 ∈ (ℤ≥‘𝑡)(𝐾‘𝑛) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)) → ∃𝑦 ∈ ℕ0 ∀𝑚 ∈ (ℤ≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝐸))
277	229, 276	sylbid 150	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝑡 ∈ ℕ0) → (∀𝑚 ∈ (ℤ≥‘𝑡)(abs‘(𝐾‘𝑚)) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)) → ∃𝑦 ∈ ℕ0 ∀𝑚 ∈ (ℤ≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝐸))
278	277	rexlimdva 2611	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (∃𝑡 ∈ ℕ0 ∀𝑚 ∈ (ℤ≥‘𝑡)(abs‘(𝐾‘𝑚)) < (((𝐸 / 2) / 𝑠) / (Σ𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑎 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑎)) + 1)) → ∃𝑦 ∈ ℕ0 ∀𝑚 ∈ (ℤ≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝐸))
279	202, 278	mpd 13	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → ∃𝑦 ∈ ℕ0 ∀𝑚 ∈ (ℤ≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝐸)
280	279	ex 115	. . . . . 6 ⊢ (𝜑 → (𝜓 → ∃𝑦 ∈ ℕ0 ∀𝑚 ∈ (ℤ≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝐸))
281	83, 280	biimtrrid 153	. . . . 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))
282	281	expdimp 259	. . . 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))
283	82, 282	sylbid 150	. . 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))
284	283	rexlimdva 2611	. 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))
285	24, 284	mpd 13	1 ⊢ (𝜑 → ∃𝑦 ∈ ℕ0 ∀𝑚 ∈ (ℤ≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝐸)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2164  {cab 2179  ∀wral 2472  ∃wrex 2473  Vcvv 2760  ⦋csb 3080   class class class wbr 4029   ↦ cmpt 4090  dom cdm 4659  ⟶wf 5250  ‘cfv 5254  (class class class)co 5918  ℂcc 7870  ℝcr 7871  0cc0 7872  1c1 7873   + caddc 7875   · cmul 7877   < clt 8054   ≤ cle 8055   − cmin 8190   / cdiv 8691  ℕcn 8982  2c2 9033  ℕ₀cn0 9240  ℤ_≥cuz 9592  ℝ⁺crp 9719  ...cfz 10074  seqcseq 10518  abscabs 11141   ⇝ cli 11421  Σcsu 11496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-frec 6444  df-1o 6469  df-oadd 6473  df-er 6587  df-en 6795  df-dom 6796  df-fin 6797  df-sup 7043  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-ico 9960  df-fz 10075  df-fzo 10209  df-seqfrec 10519  df-exp 10610  df-ihash 10847  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-clim 11422  df-sumdc 11497

This theorem is referenced by:  mertensabs  11680