Theorem mertens 14824

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.)

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 ⇝ )

Assertion

Ref	Expression
mertens	⊢ (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 mertens

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 11928	. 2 ⊢ ℕ0 = (ℤ≥‘0)
2		0zd 11595	. 2 ⊢ (𝜑 → 0 ∈ ℤ)
3		seqex 13009	. . 3 ⊢ seq0( + , 𝐻) ∈ V
4	3	a1i 11	. 2 ⊢ (𝜑 → seq0( + , 𝐻) ∈ V)
5		mertens.6	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐻‘𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗))))
6		fzfid 12979	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (0...𝑘) ∈ Fin)
7		simpl 468	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝜑)
8		elfznn0 12639	. . . . . . . 8 ⊢ (𝑗 ∈ (0...𝑘) → 𝑗 ∈ ℕ0)
9		mertens.3	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → 𝐴 ∈ ℂ)
10	7, 8, 9	syl2an 583	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → 𝐴 ∈ ℂ)
11		fveq2 6333	. . . . . . . . 9 ⊢ (𝑖 = (𝑘 − 𝑗) → (𝐺‘𝑖) = (𝐺‘(𝑘 − 𝑗)))
12	11	eleq1d 2835	. . . . . . . 8 ⊢ (𝑖 = (𝑘 − 𝑗) → ((𝐺‘𝑖) ∈ ℂ ↔ (𝐺‘(𝑘 − 𝑗)) ∈ ℂ))
13		mertens.4	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) = 𝐵)
14		mertens.5	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)
15	13, 14	eqeltrd 2850	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) ∈ ℂ)
16	15	ralrimiva 3115	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 (𝐺‘𝑘) ∈ ℂ)
17		fveq2 6333	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑖 → (𝐺‘𝑘) = (𝐺‘𝑖))
18	17	eleq1d 2835	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑖 → ((𝐺‘𝑘) ∈ ℂ ↔ (𝐺‘𝑖) ∈ ℂ))
19	18	cbvralv 3320	. . . . . . . . . 10 ⊢ (∀𝑘 ∈ ℕ0 (𝐺‘𝑘) ∈ ℂ ↔ ∀𝑖 ∈ ℕ0 (𝐺‘𝑖) ∈ ℂ)
20	16, 19	sylib 208	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑖 ∈ ℕ0 (𝐺‘𝑖) ∈ ℂ)
21	20	ad2antrr 705	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → ∀𝑖 ∈ ℕ0 (𝐺‘𝑖) ∈ ℂ)
22		fznn0sub 12579	. . . . . . . . 9 ⊢ (𝑗 ∈ (0...𝑘) → (𝑘 − 𝑗) ∈ ℕ0)
23	22	adantl 467	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘 − 𝑗) ∈ ℕ0)
24	12, 21, 23	rspcdva 3466	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → (𝐺‘(𝑘 − 𝑗)) ∈ ℂ)
25	10, 24	mulcld 10265	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → (𝐴 · (𝐺‘(𝑘 − 𝑗))) ∈ ℂ)
26	6, 25	fsumcl 14671	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗))) ∈ ℂ)
27	5, 26	eqeltrd 2850	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐻‘𝑘) ∈ ℂ)
28	1, 2, 27	serf 13035	. . 3 ⊢ (𝜑 → seq0( + , 𝐻):ℕ0⟶ℂ)
29	28	ffvelrnda 6504	. 2 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (seq0( + , 𝐻)‘𝑚) ∈ ℂ)
30		mertens.1	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐹‘𝑗) = 𝐴)
31	30	adantlr 694	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈ ℕ0) → (𝐹‘𝑗) = 𝐴)
32		mertens.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐾‘𝑗) = (abs‘𝐴))
33	32	adantlr 694	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈ ℕ0) → (𝐾‘𝑗) = (abs‘𝐴))
34	9	adantlr 694	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈ ℕ0) → 𝐴 ∈ ℂ)
35	13	adantlr 694	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) = 𝐵)
36	14	adantlr 694	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)
37	5	adantlr 694	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ ℕ0) → (𝐻‘𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗))))
38		mertens.7	. . . . . 6 ⊢ (𝜑 → seq0( + , 𝐾) ∈ dom ⇝ )
39	38	adantr 466	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → seq0( + , 𝐾) ∈ dom ⇝ )
40		mertens.8	. . . . . 6 ⊢ (𝜑 → seq0( + , 𝐺) ∈ dom ⇝ )
41	40	adantr 466	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → seq0( + , 𝐺) ∈ dom ⇝ )
42		simpr 471	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
43		fveq2 6333	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑘 → (𝐺‘𝑙) = (𝐺‘𝑘))
44	43	cbvsumv 14633	. . . . . . . . . . 11 ⊢ Σ𝑙 ∈ (ℤ≥‘(𝑖 + 1))(𝐺‘𝑙) = Σ𝑘 ∈ (ℤ≥‘(𝑖 + 1))(𝐺‘𝑘)
45		fvoveq1 6818	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑛 → (ℤ≥‘(𝑖 + 1)) = (ℤ≥‘(𝑛 + 1)))
46	45	sumeq1d 14638	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑛 → Σ𝑘 ∈ (ℤ≥‘(𝑖 + 1))(𝐺‘𝑘) = Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))
47	44, 46	syl5eq 2817	. . . . . . . . . 10 ⊢ (𝑖 = 𝑛 → Σ𝑙 ∈ (ℤ≥‘(𝑖 + 1))(𝐺‘𝑙) = Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))
48	47	fveq2d 6337	. . . . . . . . 9 ⊢ (𝑖 = 𝑛 → (abs‘Σ𝑙 ∈ (ℤ≥‘(𝑖 + 1))(𝐺‘𝑙)) = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)))
49	48	eqeq2d 2781	. . . . . . . 8 ⊢ (𝑖 = 𝑛 → (𝑢 = (abs‘Σ𝑙 ∈ (ℤ≥‘(𝑖 + 1))(𝐺‘𝑙)) ↔ 𝑢 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))))
50	49	cbvrexv 3321	. . . . . . 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)))
51		eqeq1 2775	. . . . . . . 8 ⊢ (𝑢 = 𝑧 → (𝑢 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) ↔ 𝑧 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))))
52	51	rexbidv 3200	. . . . . . 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))))
53	50, 52	syl5bb 272	. . . . . 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))))
54	53	cbvabv 2896	. . . . 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))}
55		fveq2 6333	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → (𝐾‘𝑖) = (𝐾‘𝑗))
56	55	cbvsumv 14633	. . . . . . . . . . 11 ⊢ Σ𝑖 ∈ ℕ0 (𝐾‘𝑖) = Σ𝑗 ∈ ℕ0 (𝐾‘𝑗)
57	56	oveq1i 6805	. . . . . . . . . 10 ⊢ (Σ𝑖 ∈ ℕ0 (𝐾‘𝑖) + 1) = (Σ𝑗 ∈ ℕ0 (𝐾‘𝑗) + 1)
58	57	oveq2i 6806	. . . . . . . . 9 ⊢ ((𝑥 / 2) / (Σ𝑖 ∈ ℕ0 (𝐾‘𝑖) + 1)) = ((𝑥 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾‘𝑗) + 1))
59	58	breq2i 4795	. . . . . . . 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)))
60		fveq2 6333	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑘 → (𝐺‘𝑖) = (𝐺‘𝑘))
61	60	cbvsumv 14633	. . . . . . . . . . 11 ⊢ Σ𝑖 ∈ (ℤ≥‘(𝑢 + 1))(𝐺‘𝑖) = Σ𝑘 ∈ (ℤ≥‘(𝑢 + 1))(𝐺‘𝑘)
62		fvoveq1 6818	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑛 → (ℤ≥‘(𝑢 + 1)) = (ℤ≥‘(𝑛 + 1)))
63	62	sumeq1d 14638	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑛 → Σ𝑘 ∈ (ℤ≥‘(𝑢 + 1))(𝐺‘𝑘) = Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))
64	61, 63	syl5eq 2817	. . . . . . . . . 10 ⊢ (𝑢 = 𝑛 → Σ𝑖 ∈ (ℤ≥‘(𝑢 + 1))(𝐺‘𝑖) = Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))
65	64	fveq2d 6337	. . . . . . . . 9 ⊢ (𝑢 = 𝑛 → (abs‘Σ𝑖 ∈ (ℤ≥‘(𝑢 + 1))(𝐺‘𝑖)) = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)))
66	65	breq1d 4797	. . . . . . . 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))))
67	59, 66	syl5bb 272	. . . . . . 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))))
68	67	cbvralv 3320	. . . . . 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)))
69	68	anbi2i 609	. . . . 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))))
70	31, 33, 34, 35, 36, 37, 39, 41, 42, 54, 69	mertenslem2 14823	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℕ0 ∀𝑚 ∈ (ℤ≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝑥)
71		eluznn0 11964	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑚 ∈ (ℤ≥‘𝑦)) → 𝑚 ∈ ℕ0)
72		fzfid 12979	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (0...𝑚) ∈ Fin)
73		simpll 750	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → 𝜑)
74		elfznn0 12639	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (0...𝑚) → 𝑗 ∈ ℕ0)
75	74	adantl 467	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → 𝑗 ∈ ℕ0)
76	1, 2, 13, 14, 40	isumcl 14699	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → Σ𝑘 ∈ ℕ0 𝐵 ∈ ℂ)
77	76	adantr 466	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → Σ𝑘 ∈ ℕ0 𝐵 ∈ ℂ)
78	30, 9	eqeltrd 2850	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐹‘𝑗) ∈ ℂ)
79	77, 78	mulcld 10265	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)) ∈ ℂ)
80	73, 75, 79	syl2anc 573	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)) ∈ ℂ)
81		fzfid 12979	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (0...(𝑚 − 𝑗)) ∈ Fin)
82		simplll 758	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚 − 𝑗))) → 𝜑)
83	74	ad2antlr 706	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚 − 𝑗))) → 𝑗 ∈ ℕ0)
84	82, 83, 9	syl2anc 573	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚 − 𝑗))) → 𝐴 ∈ ℂ)
85		elfznn0 12639	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (0...(𝑚 − 𝑗)) → 𝑘 ∈ ℕ0)
86	85	adantl 467	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚 − 𝑗))) → 𝑘 ∈ ℕ0)
87	82, 86, 15	syl2anc 573	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚 − 𝑗))) → (𝐺‘𝑘) ∈ ℂ)
88	84, 87	mulcld 10265	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚 − 𝑗))) → (𝐴 · (𝐺‘𝑘)) ∈ ℂ)
89	81, 88	fsumcl 14671	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐴 · (𝐺‘𝑘)) ∈ ℂ)
90	72, 80, 89	fsumsub 14726	. . . . . . . . . . . 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))))
91	73, 75, 9	syl2anc 573	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → 𝐴 ∈ ℂ)
92	76	ad2antrr 705	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ ℕ0 𝐵 ∈ ℂ)
93	81, 87	fsumcl 14671	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘) ∈ ℂ)
94	91, 92, 93	subdid 10691	. . . . . . . . . . . . . 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))))
95		eqid 2771	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℤ≥‘((𝑚 − 𝑗) + 1)) = (ℤ≥‘((𝑚 − 𝑗) + 1))
96		fznn0sub 12579	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ (0...𝑚) → (𝑚 − 𝑗) ∈ ℕ0)
97	96	adantl 467	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝑚 − 𝑗) ∈ ℕ0)
98		peano2nn0 11539	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 − 𝑗) ∈ ℕ0 → ((𝑚 − 𝑗) + 1) ∈ ℕ0)
99	97, 98	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → ((𝑚 − 𝑗) + 1) ∈ ℕ0)
100	73, 13	sylan 569	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) = 𝐵)
101	73, 14	sylan 569	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)
102	40	ad2antrr 705	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → seq0( + , 𝐺) ∈ dom ⇝ )
103	1, 95, 99, 100, 101, 102	isumsplit 14778	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ ℕ0 𝐵 = (Σ𝑘 ∈ (0...(((𝑚 − 𝑗) + 1) − 1))𝐵 + Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵))
104	97	nn0cnd 11559	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝑚 − 𝑗) ∈ ℂ)
105		ax-1cn 10199	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ∈ ℂ
106		pncan 10492	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑚 − 𝑗) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑚 − 𝑗) + 1) − 1) = (𝑚 − 𝑗))
107	104, 105, 106	sylancl 574	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (((𝑚 − 𝑗) + 1) − 1) = (𝑚 − 𝑗))
108	107	oveq2d 6811	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (0...(((𝑚 − 𝑗) + 1) − 1)) = (0...(𝑚 − 𝑗)))
109	108	sumeq1d 14638	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(((𝑚 − 𝑗) + 1) − 1))𝐵 = Σ𝑘 ∈ (0...(𝑚 − 𝑗))𝐵)
110	82, 86, 13	syl2anc 573	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚 − 𝑗))) → (𝐺‘𝑘) = 𝐵)
111	110	sumeq2dv 14640	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘) = Σ𝑘 ∈ (0...(𝑚 − 𝑗))𝐵)
112	109, 111	eqtr4d 2808	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(((𝑚 − 𝑗) + 1) − 1))𝐵 = Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘))
113	112	oveq1d 6810	. . . . . . . . . . . . . . . . . 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))
114	103, 113	eqtrd 2805	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ ℕ0 𝐵 = (Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘) + Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵))
115	114	oveq1d 6810	. . . . . . . . . . . . . . . 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)))
116	99	nn0zd 11686	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → ((𝑚 − 𝑗) + 1) ∈ ℤ)
117		simplll 758	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))) → 𝜑)
118		eluznn0 11964	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑚 − 𝑗) + 1) ∈ ℕ0 ∧ 𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))) → 𝑘 ∈ ℕ0)
119	99, 118	sylan 569	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))) → 𝑘 ∈ ℕ0)
120	117, 119, 13	syl2anc 573	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))) → (𝐺‘𝑘) = 𝐵)
121	117, 119, 14	syl2anc 573	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))) → 𝐵 ∈ ℂ)
122	100, 101	eqeltrd 2850	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) ∈ ℂ)
123	1, 99, 122	iserex 14594	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (seq0( + , 𝐺) ∈ dom ⇝ ↔ seq((𝑚 − 𝑗) + 1)( + , 𝐺) ∈ dom ⇝ ))
124	102, 123	mpbid 222	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → seq((𝑚 − 𝑗) + 1)( + , 𝐺) ∈ dom ⇝ )
125	95, 116, 120, 121, 124	isumcl 14699	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵 ∈ ℂ)
126	93, 125	pncan2d 10599	. . . . . . . . . . . . . . . 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)
127	115, 126	eqtrd 2805	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (Σ𝑘 ∈ ℕ0 𝐵 − Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘)) = Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)
128	127	oveq2d 6811	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝐴 · (Σ𝑘 ∈ ℕ0 𝐵 − Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘))) = (𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵))
129	9, 77	mulcomd 10266	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐴 · Σ𝑘 ∈ ℕ0 𝐵) = (Σ𝑘 ∈ ℕ0 𝐵 · 𝐴))
130	30	oveq2d 6811	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)) = (Σ𝑘 ∈ ℕ0 𝐵 · 𝐴))
131	129, 130	eqtr4d 2808	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐴 · Σ𝑘 ∈ ℕ0 𝐵) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)))
132	73, 75, 131	syl2anc 573	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝐴 · Σ𝑘 ∈ ℕ0 𝐵) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)))
133	81, 91, 87	fsummulc2 14722	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝐴 · Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘)) = Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐴 · (𝐺‘𝑘)))
134	132, 133	oveq12d 6813	. . . . . . . . . . . . . 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))))
135	94, 128, 134	3eqtr3rd 2814	. . . . . . . . . . . . 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))
136	135	sumeq2dv 14640	. . . . . . . . . . . 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))
137		fveq2 6333	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑗 → (𝐹‘𝑛) = (𝐹‘𝑗))
138	137	oveq2d 6811	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑗 → (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛)) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)))
139		eqid 2771	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛))) = (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛)))
140		ovex 6826	. . . . . . . . . . . . . . . 16 ⊢ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)) ∈ V
141	138, 139, 140	fvmpt 6426	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛)))‘𝑗) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)))
142	75, 141	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → ((𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛)))‘𝑗) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)))
143		simpr 471	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℕ0)
144	143, 1	syl6eleq 2860	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ (ℤ≥‘0))
145	142, 144, 80	fsumser 14668	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → Σ𝑗 ∈ (0...𝑚)(Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)) = (seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛))))‘𝑚))
146		fveq2 6333	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑘 → (𝐺‘𝑛) = (𝐺‘𝑘))
147	146	oveq2d 6811	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑘 → (𝐴 · (𝐺‘𝑛)) = (𝐴 · (𝐺‘𝑘)))
148		fveq2 6333	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝑘 − 𝑗) → (𝐺‘𝑛) = (𝐺‘(𝑘 − 𝑗)))
149	148	oveq2d 6811	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑘 − 𝑗) → (𝐴 · (𝐺‘𝑛)) = (𝐴 · (𝐺‘(𝑘 − 𝑗))))
150	88	anasss 452	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑗 ∈ (0...𝑚) ∧ 𝑘 ∈ (0...(𝑚 − 𝑗)))) → (𝐴 · (𝐺‘𝑘)) ∈ ℂ)
151	147, 149, 150	fsum0diag2 14721	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → Σ𝑗 ∈ (0...𝑚)Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐴 · (𝐺‘𝑘)) = Σ𝑘 ∈ (0...𝑚)Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗))))
152		simpll 750	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑚)) → 𝜑)
153		elfznn0 12639	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (0...𝑚) → 𝑘 ∈ ℕ0)
154	153	adantl 467	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑚)) → 𝑘 ∈ ℕ0)
155	152, 154, 5	syl2anc 573	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑚)) → (𝐻‘𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗))))
156	152, 154, 26	syl2anc 573	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑚)) → Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗))) ∈ ℂ)
157	155, 144, 156	fsumser 14668	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → Σ𝑘 ∈ (0...𝑚)Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗))) = (seq0( + , 𝐻)‘𝑚))
158	151, 157	eqtrd 2805	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → Σ𝑗 ∈ (0...𝑚)Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐴 · (𝐺‘𝑘)) = (seq0( + , 𝐻)‘𝑚))
159	145, 158	oveq12d 6813	. . . . . . . . . . . 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)))
160	90, 136, 159	3eqtr3rd 2814	. . . . . . . . . . 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))
161	160	fveq2d 6337	. . . . . . . . . 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)))
162	161	breq1d 4797	. . . . . . . . 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))
163	71, 162	sylan2 580	. . . . . . . 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))
164	163	anassrs 453	. . . . . . 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))
165	164	ralbidva 3134	. . . . . 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))
166	165	rexbidva 3197	. . . . 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))
167	166	adantr 466	. . . 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))
168	70, 167	mpbird 247	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℕ0 ∀𝑚 ∈ (ℤ≥‘𝑦)(abs‘((seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) < 𝑥)
169	168	ralrimiva 3115	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ0 ∀𝑚 ∈ (ℤ≥‘𝑦)(abs‘((seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) < 𝑥)
170	30	fveq2d 6337	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (abs‘(𝐹‘𝑗)) = (abs‘𝐴))
171	32, 170	eqtr4d 2808	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐾‘𝑗) = (abs‘(𝐹‘𝑗)))
172	1, 2, 171, 78, 38	abscvgcvg 14757	. . . . 5 ⊢ (𝜑 → seq0( + , 𝐹) ∈ dom ⇝ )
173	1, 2, 30, 9, 172	isumclim2 14696	. . . 4 ⊢ (𝜑 → seq0( + , 𝐹) ⇝ Σ𝑗 ∈ ℕ0 𝐴)
174	78	ralrimiva 3115	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ ℕ0 (𝐹‘𝑗) ∈ ℂ)
175		fveq2 6333	. . . . . . 7 ⊢ (𝑗 = 𝑚 → (𝐹‘𝑗) = (𝐹‘𝑚))
176	175	eleq1d 2835	. . . . . 6 ⊢ (𝑗 = 𝑚 → ((𝐹‘𝑗) ∈ ℂ ↔ (𝐹‘𝑚) ∈ ℂ))
177	176	rspccva 3459	. . . . 5 ⊢ ((∀𝑗 ∈ ℕ0 (𝐹‘𝑗) ∈ ℂ ∧ 𝑚 ∈ ℕ0) → (𝐹‘𝑚) ∈ ℂ)
178	174, 177	sylan 569	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (𝐹‘𝑚) ∈ ℂ)
179		fveq2 6333	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝐹‘𝑛) = (𝐹‘𝑚))
180	179	oveq2d 6811	. . . . . 6 ⊢ (𝑛 = 𝑚 → (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛)) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑚)))
181		ovex 6826	. . . . . 6 ⊢ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑚)) ∈ V
182	180, 139, 181	fvmpt 6426	. . . . 5 ⊢ (𝑚 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛)))‘𝑚) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑚)))
183	182	adantl 467	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛)))‘𝑚) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑚)))
184	1, 2, 76, 173, 178, 183	isermulc2 14595	. . 3 ⊢ (𝜑 → seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛)))) ⇝ (Σ𝑘 ∈ ℕ0 𝐵 · Σ𝑗 ∈ ℕ0 𝐴))
185	1, 2, 30, 9, 172	isumcl 14699	. . . 4 ⊢ (𝜑 → Σ𝑗 ∈ ℕ0 𝐴 ∈ ℂ)
186	76, 185	mulcomd 10266	. . 3 ⊢ (𝜑 → (Σ𝑘 ∈ ℕ0 𝐵 · Σ𝑗 ∈ ℕ0 𝐴) = (Σ𝑗 ∈ ℕ0 𝐴 · Σ𝑘 ∈ ℕ0 𝐵))
187	184, 186	breqtrd 4813	. 2 ⊢ (𝜑 → seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛)))) ⇝ (Σ𝑗 ∈ ℕ0 𝐴 · Σ𝑘 ∈ ℕ0 𝐵))
188	1, 2, 4, 29, 169, 187	2clim 14510	1 ⊢ (𝜑 → seq0( + , 𝐻) ⇝ (Σ𝑗 ∈ ℕ0 𝐴 · Σ𝑘 ∈ ℕ0 𝐵))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1631   ∈ wcel 2145  {cab 2757  ∀wral 3061  ∃wrex 3062  Vcvv 3351   class class class wbr 4787   ↦ cmpt 4864  dom cdm 5250  ‘cfv 6030  (class class class)co 6795  ℂcc 10139  0cc0 10141  1c1 10142   + caddc 10144   · cmul 10146   < clt 10279   − cmin 10471   / cdiv 10889  ℕcn 11225  2c2 11275  ℕ₀cn0 11498  ℤ_≥cuz 11892  ℝ⁺crp 12034  ...cfz 12532  seqcseq 13007  abscabs 14181   ⇝ cli 14422  Σcsu 14623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7099  ax-inf2 8705  ax-cnex 10197  ax-resscn 10198  ax-1cn 10199  ax-icn 10200  ax-addcl 10201  ax-addrcl 10202  ax-mulcl 10203  ax-mulrcl 10204  ax-mulcom 10205  ax-addass 10206  ax-mulass 10207  ax-distr 10208  ax-i2m1 10209  ax-1ne0 10210  ax-1rid 10211  ax-rnegex 10212  ax-rrecex 10213  ax-cnre 10214  ax-pre-lttri 10215  ax-pre-lttrn 10216  ax-pre-ltadd 10217  ax-pre-mulgt0 10218  ax-pre-sup 10219  ax-addf 10220  ax-mulf 10221

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-se 5210  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-isom 6039  df-riota 6756  df-ov 6798  df-oprab 6799  df-mpt2 6800  df-om 7216  df-1st 7318  df-2nd 7319  df-wrecs 7562  df-recs 7624  df-rdg 7662  df-1o 7716  df-oadd 7720  df-er 7899  df-pm 8015  df-en 8113  df-dom 8114  df-sdom 8115  df-fin 8116  df-sup 8507  df-inf 8508  df-oi 8574  df-card 8968  df-pnf 10281  df-mnf 10282  df-xr 10283  df-ltxr 10284  df-le 10285  df-sub 10473  df-neg 10474  df-div 10890  df-nn 11226  df-2 11284  df-3 11285  df-n0 11499  df-z 11584  df-uz 11893  df-rp 12035  df-ico 12385  df-fz 12533  df-fzo 12673  df-fl 12800  df-seq 13008  df-exp 13067  df-hash 13321  df-cj 14046  df-re 14047  df-im 14048  df-sqrt 14182  df-abs 14183  df-limsup 14409  df-clim 14426  df-rlim 14427  df-sum 14624

This theorem is referenced by:  efaddlem  15028