Theorem mertensabs 12251

Description: Mertens' theorem. If 𝐴(𝑗) is an absolutely convergent series and 𝐵(𝑘) is convergent, then (Σ𝑗 ∈ ℕ₀𝐴(𝑗) · Σ𝑘 ∈ ℕ₀𝐵(𝑘)) = Σ𝑘 ∈ ℕ₀Σ𝑗 ∈ (0...𝑘)(𝐴(𝑗) · 𝐵(𝑘 − 𝑗)) (and this latter series is convergent). This latter sum is commonly known as the Cauchy product of the sequences. The proof follows the outline at http://en.wikipedia.org/wiki/Cauchy_product#Proof_of_Mertens.27_theorem. (Contributed by Mario Carneiro, 29-Apr-2014.) (Revised by Jim Kingdon, 8-Dec-2022.)

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.f	⊢ (𝜑 → seq0( + , 𝐹) ∈ dom ⇝ )

Assertion

Ref	Expression
mertensabs	⊢ (𝜑 → seq0( + , 𝐻) ⇝ (Σ𝑗 ∈ ℕ0 𝐴 · Σ𝑘 ∈ ℕ0 𝐵))

Distinct variable groups:   𝐵,𝑗   𝑗,𝑘,𝐺   𝜑,𝑗,𝑘   𝐴,𝑘   𝑗,𝐾,𝑘   𝑗,𝐹   𝑘,𝐻

Allowed substitution hints:   𝐴(𝑗)   𝐵(𝑘)   𝐹(𝑘)   𝐻(𝑗)

Proof of Theorem mertensabs

Dummy variables 𝑚 𝑛 𝑠 𝑥 𝑦 𝑧 𝑖 𝑙 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0uz 9910	. 2 ⊢ ℕ0 = (ℤ≥‘0)
2		0zd 9609	. 2 ⊢ (𝜑 → 0 ∈ ℤ)
3		seqex 10838	. . 3 ⊢ seq0( + , 𝐻) ∈ V
4	3	a1i 9	. 2 ⊢ (𝜑 → seq0( + , 𝐻) ∈ V)
5		mertens.6	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐻‘𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗))))
6		0zd 9609	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 0 ∈ ℤ)
7		nn0z 9617	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℤ)
8	7	adantl 277	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℤ)
9	6, 8	fzfigd 10820	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (0...𝑘) ∈ Fin)
10		simpl 109	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝜑)
11		elfznn0 10473	. . . . . . . 8 ⊢ (𝑗 ∈ (0...𝑘) → 𝑗 ∈ ℕ0)
12		mertens.3	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → 𝐴 ∈ ℂ)
13	10, 11, 12	syl2an 289	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → 𝐴 ∈ ℂ)
14		fveq2 5675	. . . . . . . . 9 ⊢ (𝑖 = (𝑘 − 𝑗) → (𝐺‘𝑖) = (𝐺‘(𝑘 − 𝑗)))
15	14	eleq1d 2303	. . . . . . . 8 ⊢ (𝑖 = (𝑘 − 𝑗) → ((𝐺‘𝑖) ∈ ℂ ↔ (𝐺‘(𝑘 − 𝑗)) ∈ ℂ))
16		mertens.4	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) = 𝐵)
17		mertens.5	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)
18	16, 17	eqeltrd 2311	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) ∈ ℂ)
19	18	ralrimiva 2617	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 (𝐺‘𝑘) ∈ ℂ)
20		fveq2 5675	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑖 → (𝐺‘𝑘) = (𝐺‘𝑖))
21	20	eleq1d 2303	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑖 → ((𝐺‘𝑘) ∈ ℂ ↔ (𝐺‘𝑖) ∈ ℂ))
22	21	cbvralv 2780	. . . . . . . . . 10 ⊢ (∀𝑘 ∈ ℕ0 (𝐺‘𝑘) ∈ ℂ ↔ ∀𝑖 ∈ ℕ0 (𝐺‘𝑖) ∈ ℂ)
23	19, 22	sylib 122	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑖 ∈ ℕ0 (𝐺‘𝑖) ∈ ℂ)
24	23	ad2antrr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → ∀𝑖 ∈ ℕ0 (𝐺‘𝑖) ∈ ℂ)
25		fznn0sub 10415	. . . . . . . . 9 ⊢ (𝑗 ∈ (0...𝑘) → (𝑘 − 𝑗) ∈ ℕ0)
26	25	adantl 277	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘 − 𝑗) ∈ ℕ0)
27	15, 24, 26	rspcdva 2928	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → (𝐺‘(𝑘 − 𝑗)) ∈ ℂ)
28	13, 27	mulcld 8310	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → (𝐴 · (𝐺‘(𝑘 − 𝑗))) ∈ ℂ)
29	9, 28	fsumcl 12114	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗))) ∈ ℂ)
30	5, 29	eqeltrd 2311	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐻‘𝑘) ∈ ℂ)
31	1, 2, 30	serf 10872	. . 3 ⊢ (𝜑 → seq0( + , 𝐻):ℕ0⟶ℂ)
32	31	ffvelcdmda 5817	. 2 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (seq0( + , 𝐻)‘𝑚) ∈ ℂ)
33		mertens.1	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐹‘𝑗) = 𝐴)
34	33	adantlr 477	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈ ℕ0) → (𝐹‘𝑗) = 𝐴)
35		mertens.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐾‘𝑗) = (abs‘𝐴))
36	35	adantlr 477	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈ ℕ0) → (𝐾‘𝑗) = (abs‘𝐴))
37	12	adantlr 477	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈ ℕ0) → 𝐴 ∈ ℂ)
38	16	adantlr 477	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) = 𝐵)
39	17	adantlr 477	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)
40	5	adantlr 477	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ ℕ0) → (𝐻‘𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗))))
41		mertens.7	. . . . . 6 ⊢ (𝜑 → seq0( + , 𝐾) ∈ dom ⇝ )
42	41	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → seq0( + , 𝐾) ∈ dom ⇝ )
43		mertens.8	. . . . . 6 ⊢ (𝜑 → seq0( + , 𝐺) ∈ dom ⇝ )
44	43	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → seq0( + , 𝐺) ∈ dom ⇝ )
45		simpr 110	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
46		fveq2 5675	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑘 → (𝐺‘𝑙) = (𝐺‘𝑘))
47	46	cbvsumv 12074	. . . . . . . . . . 11 ⊢ Σ𝑙 ∈ (ℤ≥‘(𝑖 + 1))(𝐺‘𝑙) = Σ𝑘 ∈ (ℤ≥‘(𝑖 + 1))(𝐺‘𝑘)
48		fvoveq1 6081	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑛 → (ℤ≥‘(𝑖 + 1)) = (ℤ≥‘(𝑛 + 1)))
49	48	sumeq1d 12079	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑛 → Σ𝑘 ∈ (ℤ≥‘(𝑖 + 1))(𝐺‘𝑘) = Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))
50	47, 49	eqtrid 2279	. . . . . . . . . 10 ⊢ (𝑖 = 𝑛 → Σ𝑙 ∈ (ℤ≥‘(𝑖 + 1))(𝐺‘𝑙) = Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))
51	50	fveq2d 5679	. . . . . . . . 9 ⊢ (𝑖 = 𝑛 → (abs‘Σ𝑙 ∈ (ℤ≥‘(𝑖 + 1))(𝐺‘𝑙)) = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)))
52	51	eqeq2d 2246	. . . . . . . 8 ⊢ (𝑖 = 𝑛 → (𝑢 = (abs‘Σ𝑙 ∈ (ℤ≥‘(𝑖 + 1))(𝐺‘𝑙)) ↔ 𝑢 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))))
53	52	cbvrexv 2781	. . . . . . 7 ⊢ (∃𝑖 ∈ (0...(𝑠 − 1))𝑢 = (abs‘Σ𝑙 ∈ (ℤ≥‘(𝑖 + 1))(𝐺‘𝑙)) ↔ ∃𝑛 ∈ (0...(𝑠 − 1))𝑢 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)))
54		eqeq1 2241	. . . . . . . 8 ⊢ (𝑢 = 𝑧 → (𝑢 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) ↔ 𝑧 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))))
55	54	rexbidv 2545	. . . . . . 7 ⊢ (𝑢 = 𝑧 → (∃𝑛 ∈ (0...(𝑠 − 1))𝑢 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) ↔ ∃𝑛 ∈ (0...(𝑠 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))))
56	53, 55	bitrid 192	. . . . . 6 ⊢ (𝑢 = 𝑧 → (∃𝑖 ∈ (0...(𝑠 − 1))𝑢 = (abs‘Σ𝑙 ∈ (ℤ≥‘(𝑖 + 1))(𝐺‘𝑙)) ↔ ∃𝑛 ∈ (0...(𝑠 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))))
57	56	cbvabv 2361	. . . . 5 ⊢ {𝑢 ∣ ∃𝑖 ∈ (0...(𝑠 − 1))𝑢 = (abs‘Σ𝑙 ∈ (ℤ≥‘(𝑖 + 1))(𝐺‘𝑙))} = {𝑧 ∣ ∃𝑛 ∈ (0...(𝑠 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))}
58		fveq2 5675	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → (𝐾‘𝑖) = (𝐾‘𝑗))
59	58	cbvsumv 12074	. . . . . . . . . . 11 ⊢ Σ𝑖 ∈ ℕ0 (𝐾‘𝑖) = Σ𝑗 ∈ ℕ0 (𝐾‘𝑗)
60	59	oveq1i 6068	. . . . . . . . . 10 ⊢ (Σ𝑖 ∈ ℕ0 (𝐾‘𝑖) + 1) = (Σ𝑗 ∈ ℕ0 (𝐾‘𝑗) + 1)
61	60	oveq2i 6069	. . . . . . . . 9 ⊢ ((𝑥 / 2) / (Σ𝑖 ∈ ℕ0 (𝐾‘𝑖) + 1)) = ((𝑥 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾‘𝑗) + 1))
62	61	breq2i 4122	. . . . . . . 8 ⊢ ((abs‘Σ𝑖 ∈ (ℤ≥‘(𝑢 + 1))(𝐺‘𝑖)) < ((𝑥 / 2) / (Σ𝑖 ∈ ℕ0 (𝐾‘𝑖) + 1)) ↔ (abs‘Σ𝑖 ∈ (ℤ≥‘(𝑢 + 1))(𝐺‘𝑖)) < ((𝑥 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾‘𝑗) + 1)))
63		fveq2 5675	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑘 → (𝐺‘𝑖) = (𝐺‘𝑘))
64	63	cbvsumv 12074	. . . . . . . . . . 11 ⊢ Σ𝑖 ∈ (ℤ≥‘(𝑢 + 1))(𝐺‘𝑖) = Σ𝑘 ∈ (ℤ≥‘(𝑢 + 1))(𝐺‘𝑘)
65		fvoveq1 6081	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑛 → (ℤ≥‘(𝑢 + 1)) = (ℤ≥‘(𝑛 + 1)))
66	65	sumeq1d 12079	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑛 → Σ𝑘 ∈ (ℤ≥‘(𝑢 + 1))(𝐺‘𝑘) = Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))
67	64, 66	eqtrid 2279	. . . . . . . . . 10 ⊢ (𝑢 = 𝑛 → Σ𝑖 ∈ (ℤ≥‘(𝑢 + 1))(𝐺‘𝑖) = Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))
68	67	fveq2d 5679	. . . . . . . . 9 ⊢ (𝑢 = 𝑛 → (abs‘Σ𝑖 ∈ (ℤ≥‘(𝑢 + 1))(𝐺‘𝑖)) = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)))
69	68	breq1d 4124	. . . . . . . 8 ⊢ (𝑢 = 𝑛 → ((abs‘Σ𝑖 ∈ (ℤ≥‘(𝑢 + 1))(𝐺‘𝑖)) < ((𝑥 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾‘𝑗) + 1)) ↔ (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) < ((𝑥 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾‘𝑗) + 1))))
70	62, 69	bitrid 192	. . . . . . 7 ⊢ (𝑢 = 𝑛 → ((abs‘Σ𝑖 ∈ (ℤ≥‘(𝑢 + 1))(𝐺‘𝑖)) < ((𝑥 / 2) / (Σ𝑖 ∈ ℕ0 (𝐾‘𝑖) + 1)) ↔ (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) < ((𝑥 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾‘𝑗) + 1))))
71	70	cbvralv 2780	. . . . . 6 ⊢ (∀𝑢 ∈ (ℤ≥‘𝑠)(abs‘Σ𝑖 ∈ (ℤ≥‘(𝑢 + 1))(𝐺‘𝑖)) < ((𝑥 / 2) / (Σ𝑖 ∈ ℕ0 (𝐾‘𝑖) + 1)) ↔ ∀𝑛 ∈ (ℤ≥‘𝑠)(abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) < ((𝑥 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾‘𝑗) + 1)))
72	71	anbi2i 457	. . . . 5 ⊢ ((𝑠 ∈ ℕ ∧ ∀𝑢 ∈ (ℤ≥‘𝑠)(abs‘Σ𝑖 ∈ (ℤ≥‘(𝑢 + 1))(𝐺‘𝑖)) < ((𝑥 / 2) / (Σ𝑖 ∈ ℕ0 (𝐾‘𝑖) + 1))) ↔ (𝑠 ∈ ℕ ∧ ∀𝑛 ∈ (ℤ≥‘𝑠)(abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) < ((𝑥 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾‘𝑗) + 1))))
73	34, 36, 37, 38, 39, 40, 42, 44, 45, 57, 72	mertenslem2 12250	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℕ0 ∀𝑚 ∈ (ℤ≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝑥)
74		eluznn0 9952	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑚 ∈ (ℤ≥‘𝑦)) → 𝑚 ∈ ℕ0)
75		0zd 9609	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → 0 ∈ ℤ)
76		nn0z 9617	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ ℤ)
77	76	adantl 277	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℤ)
78	75, 77	fzfigd 10820	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (0...𝑚) ∈ Fin)
79		simpll 527	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → 𝜑)
80		elfznn0 10473	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (0...𝑚) → 𝑗 ∈ ℕ0)
81	80	adantl 277	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → 𝑗 ∈ ℕ0)
82	1, 2, 16, 17, 43	isumcl 12139	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → Σ𝑘 ∈ ℕ0 𝐵 ∈ ℂ)
83	82	adantr 276	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → Σ𝑘 ∈ ℕ0 𝐵 ∈ ℂ)
84	33, 12	eqeltrd 2311	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐹‘𝑗) ∈ ℂ)
85	83, 84	mulcld 8310	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)) ∈ ℂ)
86	79, 81, 85	syl2anc 411	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)) ∈ ℂ)
87		0zd 9609	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → 0 ∈ ℤ)
88	77	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → 𝑚 ∈ ℤ)
89	81	nn0zd 9719	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → 𝑗 ∈ ℤ)
90	88, 89	zsubcld 9726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝑚 − 𝑗) ∈ ℤ)
91	87, 90	fzfigd 10820	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (0...(𝑚 − 𝑗)) ∈ Fin)
92		simplll 535	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚 − 𝑗))) → 𝜑)
93	80	ad2antlr 489	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚 − 𝑗))) → 𝑗 ∈ ℕ0)
94	92, 93, 12	syl2anc 411	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚 − 𝑗))) → 𝐴 ∈ ℂ)
95		elfznn0 10473	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (0...(𝑚 − 𝑗)) → 𝑘 ∈ ℕ0)
96	95	adantl 277	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚 − 𝑗))) → 𝑘 ∈ ℕ0)
97	92, 96, 18	syl2anc 411	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚 − 𝑗))) → (𝐺‘𝑘) ∈ ℂ)
98	94, 97	mulcld 8310	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚 − 𝑗))) → (𝐴 · (𝐺‘𝑘)) ∈ ℂ)
99	91, 98	fsumcl 12114	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐴 · (𝐺‘𝑘)) ∈ ℂ)
100	78, 86, 99	fsumsub 12166	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → Σ𝑗 ∈ (0...𝑚)((Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)) − Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐴 · (𝐺‘𝑘))) = (Σ𝑗 ∈ (0...𝑚)(Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)) − Σ𝑗 ∈ (0...𝑚)Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐴 · (𝐺‘𝑘))))
101	79, 81, 12	syl2anc 411	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → 𝐴 ∈ ℂ)
102	82	ad2antrr 488	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ ℕ0 𝐵 ∈ ℂ)
103	91, 97	fsumcl 12114	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘) ∈ ℂ)
104	101, 102, 103	subdid 8705	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝐴 · (Σ𝑘 ∈ ℕ0 𝐵 − Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘))) = ((𝐴 · Σ𝑘 ∈ ℕ0 𝐵) − (𝐴 · Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘))))
105		eqid 2234	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℤ≥‘((𝑚 − 𝑗) + 1)) = (ℤ≥‘((𝑚 − 𝑗) + 1))
106		fznn0sub 10415	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ (0...𝑚) → (𝑚 − 𝑗) ∈ ℕ0)
107	106	adantl 277	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝑚 − 𝑗) ∈ ℕ0)
108		peano2nn0 9556	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 − 𝑗) ∈ ℕ0 → ((𝑚 − 𝑗) + 1) ∈ ℕ0)
109	107, 108	syl 14	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → ((𝑚 − 𝑗) + 1) ∈ ℕ0)
110	79, 16	sylan 283	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) = 𝐵)
111	79, 17	sylan 283	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)
112	43	ad2antrr 488	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → seq0( + , 𝐺) ∈ dom ⇝ )
113	1, 105, 109, 110, 111, 112	isumsplit 12205	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ ℕ0 𝐵 = (Σ𝑘 ∈ (0...(((𝑚 − 𝑗) + 1) − 1))𝐵 + Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵))
114	107	nn0cnd 9575	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝑚 − 𝑗) ∈ ℂ)
115		ax-1cn 8236	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ∈ ℂ
116		pncan 8496	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑚 − 𝑗) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑚 − 𝑗) + 1) − 1) = (𝑚 − 𝑗))
117	114, 115, 116	sylancl 413	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (((𝑚 − 𝑗) + 1) − 1) = (𝑚 − 𝑗))
118	117	oveq2d 6074	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (0...(((𝑚 − 𝑗) + 1) − 1)) = (0...(𝑚 − 𝑗)))
119	118	sumeq1d 12079	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(((𝑚 − 𝑗) + 1) − 1))𝐵 = Σ𝑘 ∈ (0...(𝑚 − 𝑗))𝐵)
120	92, 96, 16	syl2anc 411	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚 − 𝑗))) → (𝐺‘𝑘) = 𝐵)
121	120	sumeq2dv 12081	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘) = Σ𝑘 ∈ (0...(𝑚 − 𝑗))𝐵)
122	119, 121	eqtr4d 2270	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(((𝑚 − 𝑗) + 1) − 1))𝐵 = Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘))
123	122	oveq1d 6073	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (Σ𝑘 ∈ (0...(((𝑚 − 𝑗) + 1) − 1))𝐵 + Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵) = (Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘) + Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵))
124	113, 123	eqtrd 2267	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ ℕ0 𝐵 = (Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘) + Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵))
125	124	oveq1d 6073	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (Σ𝑘 ∈ ℕ0 𝐵 − Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘)) = ((Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘) + Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵) − Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘)))
126	109	nn0zd 9719	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → ((𝑚 − 𝑗) + 1) ∈ ℤ)
127		simplll 535	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))) → 𝜑)
128		eluznn0 9952	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑚 − 𝑗) + 1) ∈ ℕ0 ∧ 𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))) → 𝑘 ∈ ℕ0)
129	109, 128	sylan 283	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))) → 𝑘 ∈ ℕ0)
130	127, 129, 16	syl2anc 411	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))) → (𝐺‘𝑘) = 𝐵)
131	127, 129, 17	syl2anc 411	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))) → 𝐵 ∈ ℂ)
132	110, 111	eqeltrd 2311	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) ∈ ℂ)
133	1, 109, 132	iserex 12052	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (seq0( + , 𝐺) ∈ dom ⇝ ↔ seq((𝑚 − 𝑗) + 1)( + , 𝐺) ∈ dom ⇝ ))
134	112, 133	mpbid 147	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → seq((𝑚 − 𝑗) + 1)( + , 𝐺) ∈ dom ⇝ )
135	105, 126, 130, 131, 134	isumcl 12139	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵 ∈ ℂ)
136	103, 135	pncan2d 8603	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → ((Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘) + Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵) − Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘)) = Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)
137	125, 136	eqtrd 2267	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (Σ𝑘 ∈ ℕ0 𝐵 − Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘)) = Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)
138	137	oveq2d 6074	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝐴 · (Σ𝑘 ∈ ℕ0 𝐵 − Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘))) = (𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵))
139	12, 83	mulcomd 8311	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐴 · Σ𝑘 ∈ ℕ0 𝐵) = (Σ𝑘 ∈ ℕ0 𝐵 · 𝐴))
140	33	oveq2d 6074	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)) = (Σ𝑘 ∈ ℕ0 𝐵 · 𝐴))
141	139, 140	eqtr4d 2270	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐴 · Σ𝑘 ∈ ℕ0 𝐵) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)))
142	79, 81, 141	syl2anc 411	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝐴 · Σ𝑘 ∈ ℕ0 𝐵) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)))
143	91, 101, 97	fsummulc2 12162	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝐴 · Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘)) = Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐴 · (𝐺‘𝑘)))
144	142, 143	oveq12d 6076	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → ((𝐴 · Σ𝑘 ∈ ℕ0 𝐵) − (𝐴 · Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘))) = ((Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)) − Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐴 · (𝐺‘𝑘))))
145	104, 138, 144	3eqtr3rd 2276	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → ((Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)) − Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐴 · (𝐺‘𝑘))) = (𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵))
146	145	sumeq2dv 12081	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → Σ𝑗 ∈ (0...𝑚)((Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)) − Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐴 · (𝐺‘𝑘))) = Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵))
147		elnn0uz 9913	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ ℕ0 ↔ 𝑗 ∈ (ℤ≥‘0))
148	147	biimpri 133	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (ℤ≥‘0) → 𝑗 ∈ ℕ0)
149	82	ad2antrr 488	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (ℤ≥‘0)) → Σ𝑘 ∈ ℕ0 𝐵 ∈ ℂ)
150	148, 84	sylan2 286	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘0)) → (𝐹‘𝑗) ∈ ℂ)
151	150	adantlr 477	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (ℤ≥‘0)) → (𝐹‘𝑗) ∈ ℂ)
152	149, 151	mulcld 8310	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (ℤ≥‘0)) → (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)) ∈ ℂ)
153		fveq2 5675	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑗 → (𝐹‘𝑛) = (𝐹‘𝑗))
154	153	oveq2d 6074	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑗 → (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛)) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)))
155		eqid 2234	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛))) = (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛)))
156	154, 155	fvmptg 5758	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ ℕ0 ∧ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)) ∈ ℂ) → ((𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛)))‘𝑗) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)))
157	148, 152, 156	syl2an2 598	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (ℤ≥‘0)) → ((𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛)))‘𝑗) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)))
158		simpr 110	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℕ0)
159	158, 1	eleqtrdi 2327	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ (ℤ≥‘0))
160	157, 159, 152	fsum3ser 12111	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → Σ𝑗 ∈ (0...𝑚)(Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)) = (seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛))))‘𝑚))
161		fveq2 5675	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑘 → (𝐺‘𝑛) = (𝐺‘𝑘))
162	161	oveq2d 6074	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑘 → (𝐴 · (𝐺‘𝑛)) = (𝐴 · (𝐺‘𝑘)))
163		fveq2 5675	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝑘 − 𝑗) → (𝐺‘𝑛) = (𝐺‘(𝑘 − 𝑗)))
164	163	oveq2d 6074	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑘 − 𝑗) → (𝐴 · (𝐺‘𝑛)) = (𝐴 · (𝐺‘(𝑘 − 𝑗))))
165	98	anasss 399	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑗 ∈ (0...𝑚) ∧ 𝑘 ∈ (0...(𝑚 − 𝑗)))) → (𝐴 · (𝐺‘𝑘)) ∈ ℂ)
166	162, 164, 165, 77	fisum0diag2 12161	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → Σ𝑗 ∈ (0...𝑚)Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐴 · (𝐺‘𝑘)) = Σ𝑘 ∈ (0...𝑚)Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗))))
167		simpll 527	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑘 ∈ (ℤ≥‘0)) → 𝜑)
168		elnn0uz 9913	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℕ0 ↔ 𝑘 ∈ (ℤ≥‘0))
169	168	biimpri 133	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (ℤ≥‘0) → 𝑘 ∈ ℕ0)
170	169	adantl 277	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑘 ∈ (ℤ≥‘0)) → 𝑘 ∈ ℕ0)
171	167, 170, 5	syl2anc 411	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑘 ∈ (ℤ≥‘0)) → (𝐻‘𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗))))
172	167, 170, 29	syl2anc 411	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑘 ∈ (ℤ≥‘0)) → Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗))) ∈ ℂ)
173	171, 159, 172	fsum3ser 12111	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → Σ𝑘 ∈ (0...𝑚)Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗))) = (seq0( + , 𝐻)‘𝑚))
174	166, 173	eqtrd 2267	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → Σ𝑗 ∈ (0...𝑚)Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐴 · (𝐺‘𝑘)) = (seq0( + , 𝐻)‘𝑚))
175	160, 174	oveq12d 6076	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (Σ𝑗 ∈ (0...𝑚)(Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)) − Σ𝑗 ∈ (0...𝑚)Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐴 · (𝐺‘𝑘))) = ((seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚)))
176	100, 146, 175	3eqtr3rd 2276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → ((seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚)) = Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵))
177	176	fveq2d 5679	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (abs‘((seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) = (abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)))
178	177	breq1d 4124	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → ((abs‘((seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) < 𝑥 ↔ (abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝑥))
179	74, 178	sylan2 286	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ ℕ0 ∧ 𝑚 ∈ (ℤ≥‘𝑦))) → ((abs‘((seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) < 𝑥 ↔ (abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝑥))
180	179	anassrs 400	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ 𝑚 ∈ (ℤ≥‘𝑦)) → ((abs‘((seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) < 𝑥 ↔ (abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝑥))
181	180	ralbidva 2540	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ0) → (∀𝑚 ∈ (ℤ≥‘𝑦)(abs‘((seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) < 𝑥 ↔ ∀𝑚 ∈ (ℤ≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝑥))
182	181	rexbidva 2541	. . . . 5 ⊢ (𝜑 → (∃𝑦 ∈ ℕ0 ∀𝑚 ∈ (ℤ≥‘𝑦)(abs‘((seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) < 𝑥 ↔ ∃𝑦 ∈ ℕ0 ∀𝑚 ∈ (ℤ≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝑥))
183	182	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (∃𝑦 ∈ ℕ0 ∀𝑚 ∈ (ℤ≥‘𝑦)(abs‘((seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) < 𝑥 ↔ ∃𝑦 ∈ ℕ0 ∀𝑚 ∈ (ℤ≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝑥))
184	73, 183	mpbird 167	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℕ0 ∀𝑚 ∈ (ℤ≥‘𝑦)(abs‘((seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) < 𝑥)
185	184	ralrimiva 2617	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ0 ∀𝑚 ∈ (ℤ≥‘𝑦)(abs‘((seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) < 𝑥)
186		mertens.f	. . . . 5 ⊢ (𝜑 → seq0( + , 𝐹) ∈ dom ⇝ )
187	1, 2, 33, 12, 186	isumclim2 12136	. . . 4 ⊢ (𝜑 → seq0( + , 𝐹) ⇝ Σ𝑗 ∈ ℕ0 𝐴)
188	84	ralrimiva 2617	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ ℕ0 (𝐹‘𝑗) ∈ ℂ)
189		fveq2 5675	. . . . . . 7 ⊢ (𝑗 = 𝑚 → (𝐹‘𝑗) = (𝐹‘𝑚))
190	189	eleq1d 2303	. . . . . 6 ⊢ (𝑗 = 𝑚 → ((𝐹‘𝑗) ∈ ℂ ↔ (𝐹‘𝑚) ∈ ℂ))
191	190	rspccva 2922	. . . . 5 ⊢ ((∀𝑗 ∈ ℕ0 (𝐹‘𝑗) ∈ ℂ ∧ 𝑚 ∈ ℕ0) → (𝐹‘𝑚) ∈ ℂ)
192	188, 191	sylan 283	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (𝐹‘𝑚) ∈ ℂ)
193	82	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → Σ𝑘 ∈ ℕ0 𝐵 ∈ ℂ)
194	193, 192	mulcld 8310	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑚)) ∈ ℂ)
195		fveq2 5675	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝐹‘𝑛) = (𝐹‘𝑚))
196	195	oveq2d 6074	. . . . . 6 ⊢ (𝑛 = 𝑚 → (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛)) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑚)))
197	196, 155	fvmptg 5758	. . . . 5 ⊢ ((𝑚 ∈ ℕ0 ∧ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑚)) ∈ ℂ) → ((𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛)))‘𝑚) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑚)))
198	158, 194, 197	syl2anc 411	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛)))‘𝑚) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑚)))
199	1, 2, 82, 187, 192, 198	isermulc2 12053	. . 3 ⊢ (𝜑 → seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛)))) ⇝ (Σ𝑘 ∈ ℕ0 𝐵 · Σ𝑗 ∈ ℕ0 𝐴))
200	1, 2, 33, 12, 186	isumcl 12139	. . . 4 ⊢ (𝜑 → Σ𝑗 ∈ ℕ0 𝐴 ∈ ℂ)
201	82, 200	mulcomd 8311	. . 3 ⊢ (𝜑 → (Σ𝑘 ∈ ℕ0 𝐵 · Σ𝑗 ∈ ℕ0 𝐴) = (Σ𝑗 ∈ ℕ0 𝐴 · Σ𝑘 ∈ ℕ0 𝐵))
202	199, 201	breqtrd 4140	. 2 ⊢ (𝜑 → seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛)))) ⇝ (Σ𝑗 ∈ ℕ0 𝐴 · Σ𝑘 ∈ ℕ0 𝐵))
203	1, 2, 4, 32, 185, 202	2clim 12014	1 ⊢ (𝜑 → seq0( + , 𝐻) ⇝ (Σ𝑗 ∈ ℕ0 𝐴 · Σ𝑘 ∈ ℕ0 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2205  {cab 2220  ∀wral 2522  ∃wrex 2523  Vcvv 2815   class class class wbr 4114   ↦ cmpt 4176  dom cdm 4754  ‘cfv 5357  (class class class)co 6058  ℂcc 8141  0cc0 8143  1c1 8144   + caddc 8146   · cmul 8148   < clt 8324   − cmin 8461   / cdiv 8966  ℕcn 9257  2c2 9308  ℕ₀cn0 9516  ℤcz 9597  ℤ_≥cuz 9874  ℝ⁺crp 10007  ...cfz 10364  seqcseq 10836  abscabs 11710   ⇝ cli 11991  Σcsu 12066

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-disj 4091  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-oadd 6664  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-sup 7288  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8463  df-neg 8464  df-reap 8867  df-ap 8874  df-div 8967  df-inn 9258  df-2 9316  df-3 9317  df-4 9318  df-n0 9517  df-z 9598  df-uz 9875  df-q 9973  df-rp 10008  df-ico 10249  df-fz 10365  df-fzo 10502  df-seqfrec 10837  df-exp 10928  df-ihash 11167  df-cj 11555  df-re 11556  df-im 11557  df-rsqrt 11711  df-abs 11712  df-clim 11992  df-sumdc 12067

This theorem is referenced by:  efaddlem  12388