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Theorem mertenslem1 14455

Description: Lemma for mertens 14457. (Contributed by Mario Carneiro, 29-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))))
mertens.12	⊢ (𝜑 → (𝜓 ∧ (𝑡 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑡)(𝐾‘𝑚) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)))))
mertens.13	⊢ (𝜑 → (0 ≤ sup(𝑇, ℝ, < ) ∧ (𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤 ∈ 𝑇 𝑤 ≤ 𝑧)))

**Assertion**
Ref	Expression
mertenslem1	⊢ (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,𝑡,𝑦,𝑧,𝐵 𝑗,𝑘,𝐺,𝑚,𝑛,𝑠,𝑦,𝑧 𝜑,𝑗,𝑘,𝑚,𝑦,𝑧 𝑡,𝑘,𝐴,𝑚,𝑛,𝑠,𝑦 𝑗,𝐸,𝑘,𝑚,𝑛,𝑠,𝑡,𝑦,𝑧 𝑗,𝐾,𝑘,𝑚,𝑛,𝑠,𝑡,𝑦,𝑧 𝑗,𝐹,𝑚,𝑛,𝑦 𝜓,𝑗,𝑘,𝑚,𝑛,𝑡,𝑦,𝑧 𝑤,𝑗,𝑇,𝑘,𝑚,𝑛,𝑡,𝑦,𝑧 𝑘,𝐻,𝑚,𝑦 Allowed substitution hints: 𝜑(𝑤,𝑡,𝑛,𝑠) 𝜓(𝑤,𝑠) 𝐴(𝑧,𝑤,𝑗) 𝐵(𝑤,𝑘) 𝑇(𝑠) 𝐸(𝑤) 𝐹(𝑧,𝑤,𝑡,𝑘,𝑠) 𝐺(𝑤,𝑡) 𝐻(𝑧,𝑤,𝑡,𝑗,𝑛,𝑠) 𝐾(𝑤)

**Proof of Theorem mertenslem1**
Step	Hyp	Ref	Expression
1		mertens.12	. . . . . . 7 ⊢ (𝜑 → (𝜓 ∧ (𝑡 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑡)(𝐾‘𝑚) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)))))
2	1	simpld 474	. . . . . 6 ⊢ (𝜑 → 𝜓)
3		mertens.11	. . . . . 6 ⊢ (𝜓 ↔ (𝑠 ∈ ℕ ∧ ∀𝑛 ∈ (ℤ_≥‘𝑠)(abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1))))
4	2, 3	sylib 207	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ ℕ ∧ ∀𝑛 ∈ (ℤ_≥‘𝑠)(abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1))))
5	4	simpld 474	. . . 4 ⊢ (𝜑 → 𝑠 ∈ ℕ)
6	5	nnnn0d 11228	. . 3 ⊢ (𝜑 → 𝑠 ∈ ℕ₀)
7	1	simprd 478	. . . 4 ⊢ (𝜑 → (𝑡 ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘𝑡)(𝐾‘𝑚) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1))))
8	7	simpld 474	. . 3 ⊢ (𝜑 → 𝑡 ∈ ℕ₀)
9	6, 8	nn0addcld 11232	. 2 ⊢ (𝜑 → (𝑠 + 𝑡) ∈ ℕ₀)
10		fzfid 12634	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (0...𝑚) ∈ Fin)
11		simpl 472	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → 𝜑)
12		elfznn0 12302	. . . . . . . 8 ⊢ (𝑗 ∈ (0...𝑚) → 𝑗 ∈ ℕ₀)
13		mertens.3	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → 𝐴 ∈ ℂ)
14	11, 12, 13	syl2an 493	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...𝑚)) → 𝐴 ∈ ℂ)
15		eqid 2610	. . . . . . . 8 ⊢ (ℤ_≥‘((𝑚 − 𝑗) + 1)) = (ℤ_≥‘((𝑚 − 𝑗) + 1))
16		fznn0sub 12244	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0...𝑚) → (𝑚 − 𝑗) ∈ ℕ₀)
17	16	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...𝑚)) → (𝑚 − 𝑗) ∈ ℕ₀)
18		peano2nn0 11210	. . . . . . . . . 10 ⊢ ((𝑚 − 𝑗) ∈ ℕ₀ → ((𝑚 − 𝑗) + 1) ∈ ℕ₀)
19	17, 18	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...𝑚)) → ((𝑚 − 𝑗) + 1) ∈ ℕ₀)
20	19	nn0zd 11356	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...𝑚)) → ((𝑚 − 𝑗) + 1) ∈ ℤ)
21		simplll 794	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))) → 𝜑)
22		eluznn0 11633	. . . . . . . . . 10 ⊢ ((((𝑚 − 𝑗) + 1) ∈ ℕ₀ ∧ 𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))) → 𝑘 ∈ ℕ₀)
23	19, 22	sylan 487	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))) → 𝑘 ∈ ℕ₀)
24		mertens.4	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝐺‘𝑘) = 𝐵)
25	21, 23, 24	syl2anc 691	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))) → (𝐺‘𝑘) = 𝐵)
26		mertens.5	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → 𝐵 ∈ ℂ)
27	21, 23, 26	syl2anc 691	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))) → 𝐵 ∈ ℂ)
28		mertens.8	. . . . . . . . . 10 ⊢ (𝜑 → seq0( + , 𝐺) ∈ dom ⇝ )
29	28	ad2antrr 758	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...𝑚)) → seq0( + , 𝐺) ∈ dom ⇝ )
30		nn0uz 11598	. . . . . . . . . 10 ⊢ ℕ₀ = (ℤ_≥‘0)
31		simpll 786	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...𝑚)) → 𝜑)
32	24, 26	eqeltrd 2688	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ₀) → (𝐺‘𝑘) ∈ ℂ)
33	31, 32	sylan 487	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ ℕ₀) → (𝐺‘𝑘) ∈ ℂ)
34	30, 19, 33	iserex 14235	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...𝑚)) → (seq0( + , 𝐺) ∈ dom ⇝ ↔ seq((𝑚 − 𝑗) + 1)( + , 𝐺) ∈ dom ⇝ ))
35	29, 34	mpbid 221	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...𝑚)) → seq((𝑚 − 𝑗) + 1)( + , 𝐺) ∈ dom ⇝ )
36	15, 20, 25, 27, 35	isumcl 14334	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵 ∈ ℂ)
37	14, 36	mulcld 9939	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...𝑚)) → (𝐴 · Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵) ∈ ℂ)
38	10, 37	fsumcl 14311	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵) ∈ ℂ)
39	38	abscld 14023	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) ∈ ℝ)
40	37	abscld 14023	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...𝑚)) → (abs‘(𝐴 · Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) ∈ ℝ)
41	10, 40	fsumrecl 14312	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → Σ𝑗 ∈ (0...𝑚)(abs‘(𝐴 · Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) ∈ ℝ)
42		mertens.9	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
43	42	rpred 11748	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ ℝ)
44	43	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → 𝐸 ∈ ℝ)
45	10, 37	fsumabs 14374	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) ≤ Σ𝑗 ∈ (0...𝑚)(abs‘(𝐴 · Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)))
46		fzfid 12634	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (0...(𝑚 − 𝑠)) ∈ Fin)
47	6	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → 𝑠 ∈ ℕ₀)
48	47	nn0ge0d 11231	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → 0 ≤ 𝑠)
49		eluzelz 11573	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡)) → 𝑚 ∈ ℤ)
50	49	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → 𝑚 ∈ ℤ)
51	50	zred 11358	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → 𝑚 ∈ ℝ)
52	47	nn0red 11229	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → 𝑠 ∈ ℝ)
53	51, 52	subge02d 10498	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (0 ≤ 𝑠 ↔ (𝑚 − 𝑠) ≤ 𝑚))
54	48, 53	mpbid 221	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (𝑚 − 𝑠) ≤ 𝑚)
55	47, 30	syl6eleq 2698	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → 𝑠 ∈ (ℤ_≥‘0))
56	5	nnzd 11357	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑠 ∈ ℤ)
57		uzid 11578	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ ℤ → 𝑠 ∈ (ℤ_≥‘𝑠))
58	56, 57	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑠 ∈ (ℤ_≥‘𝑠))
59		uzaddcl 11620	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑠 ∈ (ℤ_≥‘𝑠) ∧ 𝑡 ∈ ℕ₀) → (𝑠 + 𝑡) ∈ (ℤ_≥‘𝑠))
60	58, 8, 59	syl2anc 691	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑠 + 𝑡) ∈ (ℤ_≥‘𝑠))
61		eqid 2610	. . . . . . . . . . . . . . . . 17 ⊢ (ℤ_≥‘𝑠) = (ℤ_≥‘𝑠)
62	61	uztrn2 11581	. . . . . . . . . . . . . . . 16 ⊢ (((𝑠 + 𝑡) ∈ (ℤ_≥‘𝑠) ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → 𝑚 ∈ (ℤ_≥‘𝑠))
63	60, 62	sylan 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → 𝑚 ∈ (ℤ_≥‘𝑠))
64		elfzuzb 12207	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ (0...𝑚) ↔ (𝑠 ∈ (ℤ_≥‘0) ∧ 𝑚 ∈ (ℤ_≥‘𝑠)))
65	55, 63, 64	sylanbrc 695	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → 𝑠 ∈ (0...𝑚))
66		fznn0sub2 12315	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ (0...𝑚) → (𝑚 − 𝑠) ∈ (0...𝑚))
67	65, 66	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (𝑚 − 𝑠) ∈ (0...𝑚))
68		elfzelz 12213	. . . . . . . . . . . . 13 ⊢ ((𝑚 − 𝑠) ∈ (0...𝑚) → (𝑚 − 𝑠) ∈ ℤ)
69	67, 68	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (𝑚 − 𝑠) ∈ ℤ)
70		eluz 11577	. . . . . . . . . . . 12 ⊢ (((𝑚 − 𝑠) ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑚 ∈ (ℤ_≥‘(𝑚 − 𝑠)) ↔ (𝑚 − 𝑠) ≤ 𝑚))
71	69, 50, 70	syl2anc 691	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (𝑚 ∈ (ℤ_≥‘(𝑚 − 𝑠)) ↔ (𝑚 − 𝑠) ≤ 𝑚))
72	54, 71	mpbird 246	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → 𝑚 ∈ (ℤ_≥‘(𝑚 − 𝑠)))
73		fzss2 12252	. . . . . . . . . 10 ⊢ (𝑚 ∈ (ℤ_≥‘(𝑚 − 𝑠)) → (0...(𝑚 − 𝑠)) ⊆ (0...𝑚))
74	72, 73	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (0...(𝑚 − 𝑠)) ⊆ (0...𝑚))
75	74	sselda 3568	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...(𝑚 − 𝑠))) → 𝑗 ∈ (0...𝑚))
76	13	abscld 14023	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (abs‘𝐴) ∈ ℝ)
77	11, 12, 76	syl2an 493	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...𝑚)) → (abs‘𝐴) ∈ ℝ)
78	36	abscld 14023	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...𝑚)) → (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵) ∈ ℝ)
79	77, 78	remulcld 9949	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...𝑚)) → ((abs‘𝐴) · (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) ∈ ℝ)
80	75, 79	syldan 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...(𝑚 − 𝑠))) → ((abs‘𝐴) · (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) ∈ ℝ)
81	46, 80	fsumrecl 14312	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → Σ𝑗 ∈ (0...(𝑚 − 𝑠))((abs‘𝐴) · (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) ∈ ℝ)
82		fzfid 12634	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (((𝑚 − 𝑠) + 1)...𝑚) ∈ Fin)
83		elfznn0 12302	. . . . . . . . . . . . 13 ⊢ ((𝑚 − 𝑠) ∈ (0...𝑚) → (𝑚 − 𝑠) ∈ ℕ₀)
84	67, 83	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (𝑚 − 𝑠) ∈ ℕ₀)
85		peano2nn0 11210	. . . . . . . . . . . 12 ⊢ ((𝑚 − 𝑠) ∈ ℕ₀ → ((𝑚 − 𝑠) + 1) ∈ ℕ₀)
86	84, 85	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → ((𝑚 − 𝑠) + 1) ∈ ℕ₀)
87	86, 30	syl6eleq 2698	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → ((𝑚 − 𝑠) + 1) ∈ (ℤ_≥‘0))
88		fzss1 12251	. . . . . . . . . 10 ⊢ (((𝑚 − 𝑠) + 1) ∈ (ℤ_≥‘0) → (((𝑚 − 𝑠) + 1)...𝑚) ⊆ (0...𝑚))
89	87, 88	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (((𝑚 − 𝑠) + 1)...𝑚) ⊆ (0...𝑚))
90	89	sselda 3568	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → 𝑗 ∈ (0...𝑚))
91	90, 79	syldan 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → ((abs‘𝐴) · (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) ∈ ℝ)
92	82, 91	fsumrecl 14312	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → Σ𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)((abs‘𝐴) · (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) ∈ ℝ)
93	42	rphalfcld 11760	. . . . . . . 8 ⊢ (𝜑 → (𝐸 / 2) ∈ ℝ⁺)
94	93	rpred 11748	. . . . . . 7 ⊢ (𝜑 → (𝐸 / 2) ∈ ℝ)
95	94	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (𝐸 / 2) ∈ ℝ)
96		elfznn0 12302	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0...(𝑚 − 𝑠)) → 𝑗 ∈ ℕ₀)
97	11, 96, 76	syl2an 493	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...(𝑚 − 𝑠))) → (abs‘𝐴) ∈ ℝ)
98	46, 97	fsumrecl 14312	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → Σ𝑗 ∈ (0...(𝑚 − 𝑠))(abs‘𝐴) ∈ ℝ)
99	98, 95	remulcld 9949	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (Σ𝑗 ∈ (0...(𝑚 − 𝑠))(abs‘𝐴) · (𝐸 / 2)) ∈ ℝ)
100		0zd 11266	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ∈ ℤ)
101		eqidd 2611	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝐾‘𝑗) = (𝐾‘𝑗))
102		mertens.2	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝐾‘𝑗) = (abs‘𝐴))
103	102, 76	eqeltrd 2688	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (𝐾‘𝑗) ∈ ℝ)
104		mertens.7	. . . . . . . . . . 11 ⊢ (𝜑 → seq0( + , 𝐾) ∈ dom ⇝ )
105	30, 100, 101, 103, 104	isumrecl 14338	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) ∈ ℝ)
106	13	absge0d 14031	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → 0 ≤ (abs‘𝐴))
107	106, 102	breqtrrd 4611	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → 0 ≤ (𝐾‘𝑗))
108	30, 100, 101, 103, 104, 107	isumge0 14339	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗))
109	105, 108	ge0p1rpd 11778	. . . . . . . . 9 ⊢ (𝜑 → (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1) ∈ ℝ⁺)
110	109	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1) ∈ ℝ⁺)
111	99, 110	rerpdivcld 11779	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → ((Σ𝑗 ∈ (0...(𝑚 − 𝑠))(abs‘𝐴) · (𝐸 / 2)) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)) ∈ ℝ)
112	93, 109	rpdivcld 11765	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)) ∈ ℝ⁺)
113	112	rpred 11748	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)) ∈ ℝ)
114	113	ad2antrr 758	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...(𝑚 − 𝑠))) → ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)) ∈ ℝ)
115	97, 114	remulcld 9949	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...(𝑚 − 𝑠))) → ((abs‘𝐴) · ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1))) ∈ ℝ)
116	75, 20	syldan 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...(𝑚 − 𝑠))) → ((𝑚 − 𝑗) + 1) ∈ ℤ)
117		simplll 794	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...(𝑚 − 𝑠))) ∧ 𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))) → 𝜑)
118	75, 19	syldan 486	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...(𝑚 − 𝑠))) → ((𝑚 − 𝑗) + 1) ∈ ℕ₀)
119	118, 22	sylan 487	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...(𝑚 − 𝑠))) ∧ 𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))) → 𝑘 ∈ ℕ₀)
120	117, 119, 24	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...(𝑚 − 𝑠))) ∧ 𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))) → (𝐺‘𝑘) = 𝐵)
121	117, 119, 26	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...(𝑚 − 𝑠))) ∧ 𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))) → 𝐵 ∈ ℂ)
122	75, 35	syldan 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...(𝑚 − 𝑠))) → seq((𝑚 − 𝑗) + 1)( + , 𝐺) ∈ dom ⇝ )
123	15, 116, 120, 121, 122	isumcl 14334	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...(𝑚 − 𝑠))) → Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵 ∈ ℂ)
124	123	abscld 14023	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...(𝑚 − 𝑠))) → (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵) ∈ ℝ)
125	76, 106	jca 553	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)))
126	11, 96, 125	syl2an 493	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...(𝑚 − 𝑠))) → ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)))
127	120	sumeq2dv 14281	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...(𝑚 − 𝑠))) → Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))(𝐺‘𝑘) = Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)
128	127	fveq2d 6107	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...(𝑚 − 𝑠))) → (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))(𝐺‘𝑘)) = (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵))
129		elfzelz 12213	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (0...(𝑚 − 𝑠)) → 𝑗 ∈ ℤ)
130	129	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...(𝑚 − 𝑠))) → 𝑗 ∈ ℤ)
131	130	zred 11358	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...(𝑚 − 𝑠))) → 𝑗 ∈ ℝ)
132	49	ad2antlr 759	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...(𝑚 − 𝑠))) → 𝑚 ∈ ℤ)
133	132	zred 11358	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...(𝑚 − 𝑠))) → 𝑚 ∈ ℝ)
134	56	ad2antrr 758	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...(𝑚 − 𝑠))) → 𝑠 ∈ ℤ)
135	134	zred 11358	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...(𝑚 − 𝑠))) → 𝑠 ∈ ℝ)
136		elfzle2 12216	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (0...(𝑚 − 𝑠)) → 𝑗 ≤ (𝑚 − 𝑠))
137	136	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...(𝑚 − 𝑠))) → 𝑗 ≤ (𝑚 − 𝑠))
138	131, 133, 135, 137	lesubd 10510	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...(𝑚 − 𝑠))) → 𝑠 ≤ (𝑚 − 𝑗))
139	132, 130	zsubcld 11363	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...(𝑚 − 𝑠))) → (𝑚 − 𝑗) ∈ ℤ)
140		eluz 11577	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ ℤ ∧ (𝑚 − 𝑗) ∈ ℤ) → ((𝑚 − 𝑗) ∈ (ℤ_≥‘𝑠) ↔ 𝑠 ≤ (𝑚 − 𝑗)))
141	134, 139, 140	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...(𝑚 − 𝑠))) → ((𝑚 − 𝑗) ∈ (ℤ_≥‘𝑠) ↔ 𝑠 ≤ (𝑚 − 𝑗)))
142	138, 141	mpbird 246	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...(𝑚 − 𝑠))) → (𝑚 − 𝑗) ∈ (ℤ_≥‘𝑠))
143	4	simprd 478	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑛 ∈ (ℤ_≥‘𝑠)(abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)))
144	143	ad2antrr 758	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...(𝑚 − 𝑠))) → ∀𝑛 ∈ (ℤ_≥‘𝑠)(abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)))
145		oveq1 6556	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = (𝑚 − 𝑗) → (𝑛 + 1) = ((𝑚 − 𝑗) + 1))
146	145	fveq2d 6107	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = (𝑚 − 𝑗) → (ℤ_≥‘(𝑛 + 1)) = (ℤ_≥‘((𝑚 − 𝑗) + 1)))
147	146	sumeq1d 14279	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝑚 − 𝑗) → Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘) = Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))(𝐺‘𝑘))
148	147	fveq2d 6107	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑚 − 𝑗) → (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘)) = (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))(𝐺‘𝑘)))
149	148	breq1d 4593	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑚 − 𝑗) → ((abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)) ↔ (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))(𝐺‘𝑘)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1))))
150	149	rspcv 3278	. . . . . . . . . . . . 13 ⊢ ((𝑚 − 𝑗) ∈ (ℤ_≥‘𝑠) → (∀𝑛 ∈ (ℤ_≥‘𝑠)(abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)) → (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))(𝐺‘𝑘)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1))))
151	142, 144, 150	sylc 63	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...(𝑚 − 𝑠))) → (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))(𝐺‘𝑘)) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)))
152	128, 151	eqbrtrrd 4607	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...(𝑚 − 𝑠))) → (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵) < ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)))
153	124, 114, 152	ltled 10064	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...(𝑚 − 𝑠))) → (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵) ≤ ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)))
154		lemul2a 10757	. . . . . . . . . 10 ⊢ ((((abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵) ∈ ℝ ∧ ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)) ∈ ℝ ∧ ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴))) ∧ (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵) ≤ ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1))) → ((abs‘𝐴) · (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) ≤ ((abs‘𝐴) · ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1))))
155	124, 114, 126, 153, 154	syl31anc 1321	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...(𝑚 − 𝑠))) → ((abs‘𝐴) · (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) ≤ ((abs‘𝐴) · ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1))))
156	46, 80, 115, 155	fsumle 14372	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → Σ𝑗 ∈ (0...(𝑚 − 𝑠))((abs‘𝐴) · (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) ≤ Σ𝑗 ∈ (0...(𝑚 − 𝑠))((abs‘𝐴) · ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1))))
157	98	recnd 9947	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → Σ𝑗 ∈ (0...(𝑚 − 𝑠))(abs‘𝐴) ∈ ℂ)
158	93	rpcnd 11750	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 / 2) ∈ ℂ)
159	158	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (𝐸 / 2) ∈ ℂ)
160		peano2re 10088	. . . . . . . . . . . . 13 ⊢ (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) ∈ ℝ → (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1) ∈ ℝ)
161	105, 160	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1) ∈ ℝ)
162	161	recnd 9947	. . . . . . . . . . 11 ⊢ (𝜑 → (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1) ∈ ℂ)
163	162	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1) ∈ ℂ)
164	109	rpne0d 11753	. . . . . . . . . . 11 ⊢ (𝜑 → (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1) ≠ 0)
165	164	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1) ≠ 0)
166	157, 159, 163, 165	divassd 10715	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → ((Σ𝑗 ∈ (0...(𝑚 − 𝑠))(abs‘𝐴) · (𝐸 / 2)) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)) = (Σ𝑗 ∈ (0...(𝑚 − 𝑠))(abs‘𝐴) · ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1))))
167		fveq2 6103	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑗 → (𝐾‘𝑛) = (𝐾‘𝑗))
168	167	cbvsumv 14274	. . . . . . . . . . . . . . . 16 ⊢ Σ𝑛 ∈ ℕ₀ (𝐾‘𝑛) = Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗)
169	168	oveq1i 6559	. . . . . . . . . . . . . . 15 ⊢ (Σ𝑛 ∈ ℕ₀ (𝐾‘𝑛) + 1) = (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)
170	169	oveq2i 6560	. . . . . . . . . . . . . 14 ⊢ ((𝐸 / 2) / (Σ𝑛 ∈ ℕ₀ (𝐾‘𝑛) + 1)) = ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1))
171	170, 112	syl5eqel 2692	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐸 / 2) / (Σ𝑛 ∈ ℕ₀ (𝐾‘𝑛) + 1)) ∈ ℝ⁺)
172	171	rpcnd 11750	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐸 / 2) / (Σ𝑛 ∈ ℕ₀ (𝐾‘𝑛) + 1)) ∈ ℂ)
173	172	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → ((𝐸 / 2) / (Σ𝑛 ∈ ℕ₀ (𝐾‘𝑛) + 1)) ∈ ℂ)
174	76	recnd 9947	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ₀) → (abs‘𝐴) ∈ ℂ)
175	11, 96, 174	syl2an 493	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...(𝑚 − 𝑠))) → (abs‘𝐴) ∈ ℂ)
176	46, 173, 175	fsummulc1 14359	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (Σ𝑗 ∈ (0...(𝑚 − 𝑠))(abs‘𝐴) · ((𝐸 / 2) / (Σ𝑛 ∈ ℕ₀ (𝐾‘𝑛) + 1))) = Σ𝑗 ∈ (0...(𝑚 − 𝑠))((abs‘𝐴) · ((𝐸 / 2) / (Σ𝑛 ∈ ℕ₀ (𝐾‘𝑛) + 1))))
177	170	oveq2i 6560	. . . . . . . . . 10 ⊢ (Σ𝑗 ∈ (0...(𝑚 − 𝑠))(abs‘𝐴) · ((𝐸 / 2) / (Σ𝑛 ∈ ℕ₀ (𝐾‘𝑛) + 1))) = (Σ𝑗 ∈ (0...(𝑚 − 𝑠))(abs‘𝐴) · ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)))
178	170	oveq2i 6560	. . . . . . . . . . . 12 ⊢ ((abs‘𝐴) · ((𝐸 / 2) / (Σ𝑛 ∈ ℕ₀ (𝐾‘𝑛) + 1))) = ((abs‘𝐴) · ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)))
179	178	a1i 11	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0...(𝑚 − 𝑠)) → ((abs‘𝐴) · ((𝐸 / 2) / (Σ𝑛 ∈ ℕ₀ (𝐾‘𝑛) + 1))) = ((abs‘𝐴) · ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1))))
180	179	sumeq2i 14277	. . . . . . . . . 10 ⊢ Σ𝑗 ∈ (0...(𝑚 − 𝑠))((abs‘𝐴) · ((𝐸 / 2) / (Σ𝑛 ∈ ℕ₀ (𝐾‘𝑛) + 1))) = Σ𝑗 ∈ (0...(𝑚 − 𝑠))((abs‘𝐴) · ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)))
181	176, 177, 180	3eqtr3g 2667	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (Σ𝑗 ∈ (0...(𝑚 − 𝑠))(abs‘𝐴) · ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1))) = Σ𝑗 ∈ (0...(𝑚 − 𝑠))((abs‘𝐴) · ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1))))
182	166, 181	eqtrd 2644	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → ((Σ𝑗 ∈ (0...(𝑚 − 𝑠))(abs‘𝐴) · (𝐸 / 2)) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)) = Σ𝑗 ∈ (0...(𝑚 − 𝑠))((abs‘𝐴) · ((𝐸 / 2) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1))))
183	156, 182	breqtrrd 4611	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → Σ𝑗 ∈ (0...(𝑚 − 𝑠))((abs‘𝐴) · (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) ≤ ((Σ𝑗 ∈ (0...(𝑚 − 𝑠))(abs‘𝐴) · (𝐸 / 2)) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)))
184	105	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) ∈ ℝ)
185	161	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1) ∈ ℝ)
186		0zd 11266	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → 0 ∈ ℤ)
187		fz0ssnn0 12304	. . . . . . . . . . . . 13 ⊢ (0...(𝑚 − 𝑠)) ⊆ ℕ₀
188	187	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (0...(𝑚 − 𝑠)) ⊆ ℕ₀)
189	102	adantlr 747	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ ℕ₀) → (𝐾‘𝑗) = (abs‘𝐴))
190	76	adantlr 747	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ ℕ₀) → (abs‘𝐴) ∈ ℝ)
191	106	adantlr 747	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ ℕ₀) → 0 ≤ (abs‘𝐴))
192	104	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → seq0( + , 𝐾) ∈ dom ⇝ )
193	30, 186, 46, 188, 189, 190, 191, 192	isumless 14416	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → Σ𝑗 ∈ (0...(𝑚 − 𝑠))(abs‘𝐴) ≤ Σ𝑗 ∈ ℕ₀ (abs‘𝐴))
194	102	sumeq2dv 14281	. . . . . . . . . . . 12 ⊢ (𝜑 → Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) = Σ𝑗 ∈ ℕ₀ (abs‘𝐴))
195	194	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) = Σ𝑗 ∈ ℕ₀ (abs‘𝐴))
196	193, 195	breqtrrd 4611	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → Σ𝑗 ∈ (0...(𝑚 − 𝑠))(abs‘𝐴) ≤ Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗))
197	105	ltp1d 10833	. . . . . . . . . . 11 ⊢ (𝜑 → Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) < (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1))
198	197	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) < (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1))
199	98, 184, 185, 196, 198	lelttrd 10074	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → Σ𝑗 ∈ (0...(𝑚 − 𝑠))(abs‘𝐴) < (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1))
200	93	rpregt0d 11754	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐸 / 2) ∈ ℝ ∧ 0 < (𝐸 / 2)))
201	200	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → ((𝐸 / 2) ∈ ℝ ∧ 0 < (𝐸 / 2)))
202		ltmul1 10752	. . . . . . . . . 10 ⊢ ((Σ𝑗 ∈ (0...(𝑚 − 𝑠))(abs‘𝐴) ∈ ℝ ∧ (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1) ∈ ℝ ∧ ((𝐸 / 2) ∈ ℝ ∧ 0 < (𝐸 / 2))) → (Σ𝑗 ∈ (0...(𝑚 − 𝑠))(abs‘𝐴) < (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1) ↔ (Σ𝑗 ∈ (0...(𝑚 − 𝑠))(abs‘𝐴) · (𝐸 / 2)) < ((Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1) · (𝐸 / 2))))
203	98, 185, 201, 202	syl3anc 1318	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (Σ𝑗 ∈ (0...(𝑚 − 𝑠))(abs‘𝐴) < (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1) ↔ (Σ𝑗 ∈ (0...(𝑚 − 𝑠))(abs‘𝐴) · (𝐸 / 2)) < ((Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1) · (𝐸 / 2))))
204	199, 203	mpbid 221	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (Σ𝑗 ∈ (0...(𝑚 − 𝑠))(abs‘𝐴) · (𝐸 / 2)) < ((Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1) · (𝐸 / 2)))
205	109	rpregt0d 11754	. . . . . . . . . 10 ⊢ (𝜑 → ((Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1) ∈ ℝ ∧ 0 < (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)))
206	205	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → ((Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1) ∈ ℝ ∧ 0 < (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)))
207		ltdivmul 10777	. . . . . . . . 9 ⊢ (((Σ𝑗 ∈ (0...(𝑚 − 𝑠))(abs‘𝐴) · (𝐸 / 2)) ∈ ℝ ∧ (𝐸 / 2) ∈ ℝ ∧ ((Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1) ∈ ℝ ∧ 0 < (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1))) → (((Σ𝑗 ∈ (0...(𝑚 − 𝑠))(abs‘𝐴) · (𝐸 / 2)) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)) < (𝐸 / 2) ↔ (Σ𝑗 ∈ (0...(𝑚 − 𝑠))(abs‘𝐴) · (𝐸 / 2)) < ((Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1) · (𝐸 / 2))))
208	99, 95, 206, 207	syl3anc 1318	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (((Σ𝑗 ∈ (0...(𝑚 − 𝑠))(abs‘𝐴) · (𝐸 / 2)) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)) < (𝐸 / 2) ↔ (Σ𝑗 ∈ (0...(𝑚 − 𝑠))(abs‘𝐴) · (𝐸 / 2)) < ((Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1) · (𝐸 / 2))))
209	204, 208	mpbird 246	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → ((Σ𝑗 ∈ (0...(𝑚 − 𝑠))(abs‘𝐴) · (𝐸 / 2)) / (Σ𝑗 ∈ ℕ₀ (𝐾‘𝑗) + 1)) < (𝐸 / 2))
210	81, 111, 95, 183, 209	lelttrd 10074	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → Σ𝑗 ∈ (0...(𝑚 − 𝑠))((abs‘𝐴) · (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) < (𝐸 / 2))
211		mertens.13	. . . . . . . . . . . 12 ⊢ (𝜑 → (0 ≤ sup(𝑇, ℝ, < ) ∧ (𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤 ∈ 𝑇 𝑤 ≤ 𝑧)))
212	211	simprd 478	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤 ∈ 𝑇 𝑤 ≤ 𝑧))
213		suprcl 10862	. . . . . . . . . . 11 ⊢ ((𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤 ∈ 𝑇 𝑤 ≤ 𝑧) → sup(𝑇, ℝ, < ) ∈ ℝ)
214	212, 213	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → sup(𝑇, ℝ, < ) ∈ ℝ)
215	94, 214	remulcld 9949	. . . . . . . . 9 ⊢ (𝜑 → ((𝐸 / 2) · sup(𝑇, ℝ, < )) ∈ ℝ)
216	211	simpld 474	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ sup(𝑇, ℝ, < ))
217	214, 216	ge0p1rpd 11778	. . . . . . . . 9 ⊢ (𝜑 → (sup(𝑇, ℝ, < ) + 1) ∈ ℝ⁺)
218	215, 217	rerpdivcld 11779	. . . . . . . 8 ⊢ (𝜑 → (((𝐸 / 2) · sup(𝑇, ℝ, < )) / (sup(𝑇, ℝ, < ) + 1)) ∈ ℝ)
219	218	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (((𝐸 / 2) · sup(𝑇, ℝ, < )) / (sup(𝑇, ℝ, < ) + 1)) ∈ ℝ)
220	5	nnrpd 11746	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑠 ∈ ℝ⁺)
221	93, 220	rpdivcld 11765	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐸 / 2) / 𝑠) ∈ ℝ⁺)
222	221, 217	rpdivcld 11765	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) ∈ ℝ⁺)
223	222	rpred 11748	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) ∈ ℝ)
224	223, 214	remulcld 9949	. . . . . . . . . 10 ⊢ (𝜑 → ((((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) · sup(𝑇, ℝ, < )) ∈ ℝ)
225	224	ad2antrr 758	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → ((((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) · sup(𝑇, ℝ, < )) ∈ ℝ)
226		simpll 786	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → 𝜑)
227	90, 12	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → 𝑗 ∈ ℕ₀)
228	226, 227, 76	syl2anc 691	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → (abs‘𝐴) ∈ ℝ)
229	223	ad2antrr 758	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) ∈ ℝ)
230	226, 227, 102	syl2anc 691	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → (𝐾‘𝑗) = (abs‘𝐴))
231		elfzuz 12209	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚) → 𝑗 ∈ (ℤ_≥‘((𝑚 − 𝑠) + 1)))
232		eluzle 11576	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡)) → (𝑠 + 𝑡) ≤ 𝑚)
233	232	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (𝑠 + 𝑡) ≤ 𝑚)
234	8	nn0zd 11356	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑡 ∈ ℤ)
235	234	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → 𝑡 ∈ ℤ)
236	235	zred 11358	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → 𝑡 ∈ ℝ)
237	52, 236, 51	leaddsub2d 10508	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → ((𝑠 + 𝑡) ≤ 𝑚 ↔ 𝑡 ≤ (𝑚 − 𝑠)))
238	233, 237	mpbid 221	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → 𝑡 ≤ (𝑚 − 𝑠))
239		eluz 11577	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑡 ∈ ℤ ∧ (𝑚 − 𝑠) ∈ ℤ) → ((𝑚 − 𝑠) ∈ (ℤ_≥‘𝑡) ↔ 𝑡 ≤ (𝑚 − 𝑠)))
240	235, 69, 239	syl2anc 691	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → ((𝑚 − 𝑠) ∈ (ℤ_≥‘𝑡) ↔ 𝑡 ≤ (𝑚 − 𝑠)))
241	238, 240	mpbird 246	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (𝑚 − 𝑠) ∈ (ℤ_≥‘𝑡))
242		peano2uz 11617	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 − 𝑠) ∈ (ℤ_≥‘𝑡) → ((𝑚 − 𝑠) + 1) ∈ (ℤ_≥‘𝑡))
243	241, 242	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → ((𝑚 − 𝑠) + 1) ∈ (ℤ_≥‘𝑡))
244		uztrn 11580	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ (ℤ_≥‘((𝑚 − 𝑠) + 1)) ∧ ((𝑚 − 𝑠) + 1) ∈ (ℤ_≥‘𝑡)) → 𝑗 ∈ (ℤ_≥‘𝑡))
245	231, 243, 244	syl2anr 494	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → 𝑗 ∈ (ℤ_≥‘𝑡))
246	7	simprd 478	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑚 ∈ (ℤ_≥‘𝑡)(𝐾‘𝑚) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)))
247	246	ad2antrr 758	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → ∀𝑚 ∈ (ℤ_≥‘𝑡)(𝐾‘𝑚) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)))
248		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑗 → (𝐾‘𝑚) = (𝐾‘𝑗))
249	248	breq1d 4593	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑗 → ((𝐾‘𝑚) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) ↔ (𝐾‘𝑗) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1))))
250	249	rspcv 3278	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (ℤ_≥‘𝑡) → (∀𝑚 ∈ (ℤ_≥‘𝑡)(𝐾‘𝑚) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) → (𝐾‘𝑗) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1))))
251	245, 247, 250	sylc 63	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → (𝐾‘𝑗) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)))
252	230, 251	eqbrtrrd 4607	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → (abs‘𝐴) < (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)))
253	228, 229, 252	ltled 10064	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → (abs‘𝐴) ≤ (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)))
254	212	ad2antrr 758	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → (𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤 ∈ 𝑇 𝑤 ≤ 𝑧))
255	51	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → 𝑚 ∈ ℝ)
256		peano2zm 11297	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ ℤ → (𝑠 − 1) ∈ ℤ)
257	56, 256	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑠 − 1) ∈ ℤ)
258	257	zred 11358	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑠 − 1) ∈ ℝ)
259	258	ad2antrr 758	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → (𝑠 − 1) ∈ ℝ)
260	227	nn0red 11229	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → 𝑗 ∈ ℝ)
261	50	zcnd 11359	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → 𝑚 ∈ ℂ)
262	52	recnd 9947	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → 𝑠 ∈ ℂ)
263		1cnd 9935	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → 1 ∈ ℂ)
264	261, 262, 263	subsubd 10299	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (𝑚 − (𝑠 − 1)) = ((𝑚 − 𝑠) + 1))
265	264	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → (𝑚 − (𝑠 − 1)) = ((𝑚 − 𝑠) + 1))
266		elfzle1 12215	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚) → ((𝑚 − 𝑠) + 1) ≤ 𝑗)
267	266	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → ((𝑚 − 𝑠) + 1) ≤ 𝑗)
268	265, 267	eqbrtrd 4605	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → (𝑚 − (𝑠 − 1)) ≤ 𝑗)
269	255, 259, 260, 268	subled 10509	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → (𝑚 − 𝑗) ≤ (𝑠 − 1))
270	90, 16	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → (𝑚 − 𝑗) ∈ ℕ₀)
271	270, 30	syl6eleq 2698	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → (𝑚 − 𝑗) ∈ (ℤ_≥‘0))
272	257	ad2antrr 758	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → (𝑠 − 1) ∈ ℤ)
273		elfz5 12205	. . . . . . . . . . . . . . 15 ⊢ (((𝑚 − 𝑗) ∈ (ℤ_≥‘0) ∧ (𝑠 − 1) ∈ ℤ) → ((𝑚 − 𝑗) ∈ (0...(𝑠 − 1)) ↔ (𝑚 − 𝑗) ≤ (𝑠 − 1)))
274	271, 272, 273	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → ((𝑚 − 𝑗) ∈ (0...(𝑠 − 1)) ↔ (𝑚 − 𝑗) ≤ (𝑠 − 1)))
275	269, 274	mpbird 246	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → (𝑚 − 𝑗) ∈ (0...(𝑠 − 1)))
276		simplll 794	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) ∧ 𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))) → 𝜑)
277	90, 19	syldan 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → ((𝑚 − 𝑗) + 1) ∈ ℕ₀)
278	277, 22	sylan 487	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) ∧ 𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))) → 𝑘 ∈ ℕ₀)
279	276, 278, 24	syl2anc 691	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) ∧ 𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))) → (𝐺‘𝑘) = 𝐵)
280	279	sumeq2dv 14281	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))(𝐺‘𝑘) = Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)
281	280	eqcomd 2616	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵 = Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))(𝐺‘𝑘))
282	281	fveq2d 6107	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵) = (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))(𝐺‘𝑘)))
283	148	eqeq2d 2620	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑚 − 𝑗) → ((abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵) = (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘)) ↔ (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵) = (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))(𝐺‘𝑘))))
284	283	rspcev 3282	. . . . . . . . . . . . 13 ⊢ (((𝑚 − 𝑗) ∈ (0...(𝑠 − 1)) ∧ (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵) = (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))(𝐺‘𝑘))) → ∃𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵) = (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘)))
285	275, 282, 284	syl2anc 691	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → ∃𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵) = (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘)))
286		fvex 6113	. . . . . . . . . . . . 13 ⊢ (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵) ∈ V
287		eqeq1 2614	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵) → (𝑧 = (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘)) ↔ (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵) = (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘))))
288	287	rexbidv 3034	. . . . . . . . . . . . 13 ⊢ (𝑧 = (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵) → (∃𝑛 ∈ (0...(𝑠 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘)) ↔ ∃𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵) = (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘))))
289		mertens.10	. . . . . . . . . . . . 13 ⊢ 𝑇 = {𝑧 ∣ ∃𝑛 ∈ (0...(𝑠 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘))}
290	286, 288, 289	elab2 3323	. . . . . . . . . . . 12 ⊢ ((abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵) ∈ 𝑇 ↔ ∃𝑛 ∈ (0...(𝑠 − 1))(abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵) = (abs‘Σ𝑘 ∈ (ℤ_≥‘(𝑛 + 1))(𝐺‘𝑘)))
291	285, 290	sylibr 223	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵) ∈ 𝑇)
292		suprub 10863	. . . . . . . . . . 11 ⊢ (((𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤 ∈ 𝑇 𝑤 ≤ 𝑧) ∧ (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵) ∈ 𝑇) → (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵) ≤ sup(𝑇, ℝ, < ))
293	254, 291, 292	syl2anc 691	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵) ≤ sup(𝑇, ℝ, < ))
294	226, 227, 125	syl2anc 691	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → ((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)))
295	90, 78	syldan 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵) ∈ ℝ)
296	36	absge0d 14031	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...𝑚)) → 0 ≤ (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵))
297	90, 296	syldan 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → 0 ≤ (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵))
298	295, 297	jca 553	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → ((abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵) ∈ ℝ ∧ 0 ≤ (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)))
299	214	ad2antrr 758	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → sup(𝑇, ℝ, < ) ∈ ℝ)
300		lemul12a 10760	. . . . . . . . . . 11 ⊢ (((((abs‘𝐴) ∈ ℝ ∧ 0 ≤ (abs‘𝐴)) ∧ (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) ∈ ℝ) ∧ (((abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵) ∈ ℝ ∧ 0 ≤ (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) ∧ sup(𝑇, ℝ, < ) ∈ ℝ)) → (((abs‘𝐴) ≤ (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) ∧ (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵) ≤ sup(𝑇, ℝ, < )) → ((abs‘𝐴) · (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) ≤ ((((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) · sup(𝑇, ℝ, < ))))
301	294, 229, 298, 299, 300	syl22anc 1319	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → (((abs‘𝐴) ≤ (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) ∧ (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵) ≤ sup(𝑇, ℝ, < )) → ((abs‘𝐴) · (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) ≤ ((((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) · sup(𝑇, ℝ, < ))))
302	253, 293, 301	mp2and 711	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)) → ((abs‘𝐴) · (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) ≤ ((((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) · sup(𝑇, ℝ, < )))
303	82, 91, 225, 302	fsumle 14372	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → Σ𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)((abs‘𝐴) · (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) ≤ Σ𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)((((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) · sup(𝑇, ℝ, < )))
304	224	recnd 9947	. . . . . . . . . . 11 ⊢ (𝜑 → ((((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) · sup(𝑇, ℝ, < )) ∈ ℂ)
305	304	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → ((((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) · sup(𝑇, ℝ, < )) ∈ ℂ)
306		fsumconst 14364	. . . . . . . . . 10 ⊢ (((((𝑚 − 𝑠) + 1)...𝑚) ∈ Fin ∧ ((((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) · sup(𝑇, ℝ, < )) ∈ ℂ) → Σ𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)((((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) · sup(𝑇, ℝ, < )) = ((#‘(((𝑚 − 𝑠) + 1)...𝑚)) · ((((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) · sup(𝑇, ℝ, < ))))
307	82, 305, 306	syl2anc 691	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → Σ𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)((((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) · sup(𝑇, ℝ, < )) = ((#‘(((𝑚 − 𝑠) + 1)...𝑚)) · ((((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) · sup(𝑇, ℝ, < ))))
308		1zzd 11285	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → 1 ∈ ℤ)
309	56	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → 𝑠 ∈ ℤ)
310		fzen 12229	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ (𝑚 − 𝑠) ∈ ℤ) → (1...𝑠) ≈ ((1 + (𝑚 − 𝑠))...(𝑠 + (𝑚 − 𝑠))))
311	308, 309, 69, 310	syl3anc 1318	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (1...𝑠) ≈ ((1 + (𝑚 − 𝑠))...(𝑠 + (𝑚 − 𝑠))))
312		ax-1cn 9873	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
313	69	zcnd 11359	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (𝑚 − 𝑠) ∈ ℂ)
314		addcom 10101	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℂ ∧ (𝑚 − 𝑠) ∈ ℂ) → (1 + (𝑚 − 𝑠)) = ((𝑚 − 𝑠) + 1))
315	312, 313, 314	sylancr 694	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (1 + (𝑚 − 𝑠)) = ((𝑚 − 𝑠) + 1))
316	262, 261	pncan3d 10274	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (𝑠 + (𝑚 − 𝑠)) = 𝑚)
317	315, 316	oveq12d 6567	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → ((1 + (𝑚 − 𝑠))...(𝑠 + (𝑚 − 𝑠))) = (((𝑚 − 𝑠) + 1)...𝑚))
318	311, 317	breqtrd 4609	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (1...𝑠) ≈ (((𝑚 − 𝑠) + 1)...𝑚))
319		fzfid 12634	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (1...𝑠) ∈ Fin)
320		hashen 12997	. . . . . . . . . . . . 13 ⊢ (((1...𝑠) ∈ Fin ∧ (((𝑚 − 𝑠) + 1)...𝑚) ∈ Fin) → ((#‘(1...𝑠)) = (#‘(((𝑚 − 𝑠) + 1)...𝑚)) ↔ (1...𝑠) ≈ (((𝑚 − 𝑠) + 1)...𝑚)))
321	319, 82, 320	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → ((#‘(1...𝑠)) = (#‘(((𝑚 − 𝑠) + 1)...𝑚)) ↔ (1...𝑠) ≈ (((𝑚 − 𝑠) + 1)...𝑚)))
322	318, 321	mpbird 246	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (#‘(1...𝑠)) = (#‘(((𝑚 − 𝑠) + 1)...𝑚)))
323		hashfz1 12996	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ℕ₀ → (#‘(1...𝑠)) = 𝑠)
324	47, 323	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (#‘(1...𝑠)) = 𝑠)
325	322, 324	eqtr3d 2646	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (#‘(((𝑚 − 𝑠) + 1)...𝑚)) = 𝑠)
326	325	oveq1d 6564	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → ((#‘(((𝑚 − 𝑠) + 1)...𝑚)) · ((((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) · sup(𝑇, ℝ, < ))) = (𝑠 · ((((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) · sup(𝑇, ℝ, < ))))
327	214	recnd 9947	. . . . . . . . . . . 12 ⊢ (𝜑 → sup(𝑇, ℝ, < ) ∈ ℂ)
328	217	rpcnne0d 11757	. . . . . . . . . . . 12 ⊢ (𝜑 → ((sup(𝑇, ℝ, < ) + 1) ∈ ℂ ∧ (sup(𝑇, ℝ, < ) + 1) ≠ 0))
329		div23 10583	. . . . . . . . . . . 12 ⊢ (((𝐸 / 2) ∈ ℂ ∧ sup(𝑇, ℝ, < ) ∈ ℂ ∧ ((sup(𝑇, ℝ, < ) + 1) ∈ ℂ ∧ (sup(𝑇, ℝ, < ) + 1) ≠ 0)) → (((𝐸 / 2) · sup(𝑇, ℝ, < )) / (sup(𝑇, ℝ, < ) + 1)) = (((𝐸 / 2) / (sup(𝑇, ℝ, < ) + 1)) · sup(𝑇, ℝ, < )))
330	158, 327, 328, 329	syl3anc 1318	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐸 / 2) · sup(𝑇, ℝ, < )) / (sup(𝑇, ℝ, < ) + 1)) = (((𝐸 / 2) / (sup(𝑇, ℝ, < ) + 1)) · sup(𝑇, ℝ, < )))
331	56	zcnd 11359	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑠 ∈ ℂ)
332	221	rpcnd 11750	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐸 / 2) / 𝑠) ∈ ℂ)
333		divass 10582	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ ℂ ∧ ((𝐸 / 2) / 𝑠) ∈ ℂ ∧ ((sup(𝑇, ℝ, < ) + 1) ∈ ℂ ∧ (sup(𝑇, ℝ, < ) + 1) ≠ 0)) → ((𝑠 · ((𝐸 / 2) / 𝑠)) / (sup(𝑇, ℝ, < ) + 1)) = (𝑠 · (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1))))
334	331, 332, 328, 333	syl3anc 1318	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑠 · ((𝐸 / 2) / 𝑠)) / (sup(𝑇, ℝ, < ) + 1)) = (𝑠 · (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1))))
335	5	nnne0d 10942	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑠 ≠ 0)
336	158, 331, 335	divcan2d 10682	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑠 · ((𝐸 / 2) / 𝑠)) = (𝐸 / 2))
337	336	oveq1d 6564	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑠 · ((𝐸 / 2) / 𝑠)) / (sup(𝑇, ℝ, < ) + 1)) = ((𝐸 / 2) / (sup(𝑇, ℝ, < ) + 1)))
338	334, 337	eqtr3d 2646	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑠 · (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1))) = ((𝐸 / 2) / (sup(𝑇, ℝ, < ) + 1)))
339	338	oveq1d 6564	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑠 · (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1))) · sup(𝑇, ℝ, < )) = (((𝐸 / 2) / (sup(𝑇, ℝ, < ) + 1)) · sup(𝑇, ℝ, < )))
340	222	rpcnd 11750	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) ∈ ℂ)
341	331, 340, 327	mulassd 9942	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑠 · (((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1))) · sup(𝑇, ℝ, < )) = (𝑠 · ((((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) · sup(𝑇, ℝ, < ))))
342	330, 339, 341	3eqtr2rd 2651	. . . . . . . . . 10 ⊢ (𝜑 → (𝑠 · ((((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) · sup(𝑇, ℝ, < ))) = (((𝐸 / 2) · sup(𝑇, ℝ, < )) / (sup(𝑇, ℝ, < ) + 1)))
343	342	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (𝑠 · ((((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) · sup(𝑇, ℝ, < ))) = (((𝐸 / 2) · sup(𝑇, ℝ, < )) / (sup(𝑇, ℝ, < ) + 1)))
344	307, 326, 343	3eqtrd 2648	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → Σ𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)((((𝐸 / 2) / 𝑠) / (sup(𝑇, ℝ, < ) + 1)) · sup(𝑇, ℝ, < )) = (((𝐸 / 2) · sup(𝑇, ℝ, < )) / (sup(𝑇, ℝ, < ) + 1)))
345	303, 344	breqtrd 4609	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → Σ𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)((abs‘𝐴) · (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) ≤ (((𝐸 / 2) · sup(𝑇, ℝ, < )) / (sup(𝑇, ℝ, < ) + 1)))
346		peano2re 10088	. . . . . . . . . . 11 ⊢ (sup(𝑇, ℝ, < ) ∈ ℝ → (sup(𝑇, ℝ, < ) + 1) ∈ ℝ)
347	214, 346	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (sup(𝑇, ℝ, < ) + 1) ∈ ℝ)
348	214	ltp1d 10833	. . . . . . . . . 10 ⊢ (𝜑 → sup(𝑇, ℝ, < ) < (sup(𝑇, ℝ, < ) + 1))
349	214, 347, 93, 348	ltmul2dd 11804	. . . . . . . . 9 ⊢ (𝜑 → ((𝐸 / 2) · sup(𝑇, ℝ, < )) < ((𝐸 / 2) · (sup(𝑇, ℝ, < ) + 1)))
350	215, 94, 217	ltdivmul2d 11800	. . . . . . . . 9 ⊢ (𝜑 → ((((𝐸 / 2) · sup(𝑇, ℝ, < )) / (sup(𝑇, ℝ, < ) + 1)) < (𝐸 / 2) ↔ ((𝐸 / 2) · sup(𝑇, ℝ, < )) < ((𝐸 / 2) · (sup(𝑇, ℝ, < ) + 1))))
351	349, 350	mpbird 246	. . . . . . . 8 ⊢ (𝜑 → (((𝐸 / 2) · sup(𝑇, ℝ, < )) / (sup(𝑇, ℝ, < ) + 1)) < (𝐸 / 2))
352	351	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (((𝐸 / 2) · sup(𝑇, ℝ, < )) / (sup(𝑇, ℝ, < ) + 1)) < (𝐸 / 2))
353	92, 219, 95, 345, 352	lelttrd 10074	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → Σ𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)((abs‘𝐴) · (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) < (𝐸 / 2))
354	81, 92, 95, 95, 210, 353	lt2addd 10529	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (Σ𝑗 ∈ (0...(𝑚 − 𝑠))((abs‘𝐴) · (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) + Σ𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)((abs‘𝐴) · (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵))) < ((𝐸 / 2) + (𝐸 / 2)))
355	14, 36	absmuld 14041	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...𝑚)) → (abs‘(𝐴 · Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) = ((abs‘𝐴) · (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)))
356	355	sumeq2dv 14281	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → Σ𝑗 ∈ (0...𝑚)(abs‘(𝐴 · Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) = Σ𝑗 ∈ (0...𝑚)((abs‘𝐴) · (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)))
357	69	zred 11358	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (𝑚 − 𝑠) ∈ ℝ)
358	357	ltp1d 10833	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (𝑚 − 𝑠) < ((𝑚 − 𝑠) + 1))
359		fzdisj 12239	. . . . . . . 8 ⊢ ((𝑚 − 𝑠) < ((𝑚 − 𝑠) + 1) → ((0...(𝑚 − 𝑠)) ∩ (((𝑚 − 𝑠) + 1)...𝑚)) = ∅)
360	358, 359	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → ((0...(𝑚 − 𝑠)) ∩ (((𝑚 − 𝑠) + 1)...𝑚)) = ∅)
361		fzsplit 12238	. . . . . . . 8 ⊢ ((𝑚 − 𝑠) ∈ (0...𝑚) → (0...𝑚) = ((0...(𝑚 − 𝑠)) ∪ (((𝑚 − 𝑠) + 1)...𝑚)))
362	67, 361	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (0...𝑚) = ((0...(𝑚 − 𝑠)) ∪ (((𝑚 − 𝑠) + 1)...𝑚)))
363	79	recnd 9947	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) ∧ 𝑗 ∈ (0...𝑚)) → ((abs‘𝐴) · (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) ∈ ℂ)
364	360, 362, 10, 363	fsumsplit 14318	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → Σ𝑗 ∈ (0...𝑚)((abs‘𝐴) · (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) = (Σ𝑗 ∈ (0...(𝑚 − 𝑠))((abs‘𝐴) · (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) + Σ𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)((abs‘𝐴) · (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵))))
365	356, 364	eqtr2d 2645	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (Σ𝑗 ∈ (0...(𝑚 − 𝑠))((abs‘𝐴) · (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) + Σ𝑗 ∈ (((𝑚 − 𝑠) + 1)...𝑚)((abs‘𝐴) · (abs‘Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵))) = Σ𝑗 ∈ (0...𝑚)(abs‘(𝐴 · Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)))
366	42	rpcnd 11750	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℂ)
367	366	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → 𝐸 ∈ ℂ)
368	367	2halvesd 11155	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → ((𝐸 / 2) + (𝐸 / 2)) = 𝐸)
369	354, 365, 368	3brtr3d 4614	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → Σ𝑗 ∈ (0...𝑚)(abs‘(𝐴 · Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝐸)
370	39, 41, 44, 45, 369	lelttrd 10074	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))) → (abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝐸)
371	370	ralrimiva 2949	. 2 ⊢ (𝜑 → ∀𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝐸)
372		fveq2 6103	. . . 4 ⊢ (𝑦 = (𝑠 + 𝑡) → (ℤ_≥‘𝑦) = (ℤ_≥‘(𝑠 + 𝑡)))
373	372	raleqdv 3121	. . 3 ⊢ (𝑦 = (𝑠 + 𝑡) → (∀𝑚 ∈ (ℤ_≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝐸 ↔ ∀𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝐸))
374	373	rspcev 3282	. 2 ⊢ (((𝑠 + 𝑡) ∈ ℕ₀ ∧ ∀𝑚 ∈ (ℤ_≥‘(𝑠 + 𝑡))(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝐸) → ∃𝑦 ∈ ℕ₀ ∀𝑚 ∈ (ℤ_≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ_≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝐸)
375	9, 371, 374	syl2anc 691	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 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 {cab 2596 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 class class class wbr 4583 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 ≈ cen 7838 Fincfn 7841 supcsup 8229 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 ≤ cle 9954 − cmin 10145 / cdiv 10563 ℕcn 10897 2c2 10947 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 ℝ⁺crp 11708 ...cfz 12197 seqcseq 12663 #chash 12979 abscabs 13822 ⇝ cli 14063 Σcsu 14264

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-ico 12052 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-rlim 14068 df-sum 14265

This theorem is referenced by: mertenslem2 14456

W3C validator