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Theorem iseraltlem3 15626

Description: Lemma for iseralt 15627. From iseraltlem2 15625, we have (-1↑𝑛) · 𝑆(𝑛 + 2𝑘) ≤ (-1↑𝑛) · 𝑆(𝑛) and (-1↑𝑛) · 𝑆(𝑛 + 1) ≤ (-1↑𝑛) · 𝑆(𝑛 + 2𝑘 + 1), and we also have (-1↑𝑛) · 𝑆(𝑛 + 1) = (-1↑𝑛) · 𝑆(𝑛) − 𝐺(𝑛 + 1) for each 𝑛 by the definition of the partial sum 𝑆, so combining the inequalities we get (-1↑𝑛) · 𝑆(𝑛) − 𝐺(𝑛 + 1) = (-1↑𝑛) · 𝑆(𝑛 + 1) ≤ (-1↑𝑛) · 𝑆(𝑛 + 2𝑘 + 1) = (-1↑𝑛) · 𝑆(𝑛 + 2𝑘) − 𝐺(𝑛 + 2𝑘 + 1) ≤ (-1↑𝑛) · 𝑆(𝑛 + 2𝑘) ≤ (-1↑𝑛) · 𝑆(𝑛) ≤ (-1↑𝑛) · 𝑆(𝑛) + 𝐺(𝑛 + 1), so ∣ (-1↑𝑛) · 𝑆(𝑛 + 2𝑘 + 1) − (-1↑𝑛) · 𝑆(𝑛) ∣ = ∣ 𝑆(𝑛 + 2𝑘 + 1) − 𝑆(𝑛) ∣ ≤ 𝐺(𝑛 + 1) and ∣ (-1↑𝑛) · 𝑆(𝑛 + 2𝑘) − (-1↑𝑛) · 𝑆(𝑛) ∣ = ∣ 𝑆(𝑛 + 2𝑘) − 𝑆(𝑛) ∣ ≤ 𝐺(𝑛 + 1). Thus, both even and odd partial sums are Cauchy if 𝐺 converges to 0. (Contributed by Mario Carneiro, 6-Apr-2015.)

**Hypotheses**
Ref	Expression
iseralt.1	⊢ 𝑍 = (ℤ_≥‘𝑀)
iseralt.2	⊢ (𝜑 → 𝑀 ∈ ℤ)
iseralt.3	⊢ (𝜑 → 𝐺:𝑍⟶ℝ)
iseralt.4	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘(𝑘 + 1)) ≤ (𝐺‘𝑘))
iseralt.5	⊢ (𝜑 → 𝐺 ⇝ 0)
iseralt.6	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = ((-1↑𝑘) · (𝐺‘𝑘)))

**Assertion**
Ref	Expression
iseraltlem3	⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((abs‘((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) − (seq𝑀( + , 𝐹)‘𝑁))) ≤ (𝐺‘(𝑁 + 1)) ∧ (abs‘((seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)) − (seq𝑀( + , 𝐹)‘𝑁))) ≤ (𝐺‘(𝑁 + 1))))

Distinct variable groups: 𝑘,𝐹 𝑘,𝐺 𝑘,𝑀 𝜑,𝑘 𝑘,𝐾 𝑘,𝑁 𝑘,𝑍

**Proof of Theorem iseraltlem3**
Step	Hyp	Ref	Expression
1		neg1rr 12323	. . . . . . . . . 10 ⊢ -1 ∈ ℝ
2	1	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → -1 ∈ ℝ)
3		neg1ne0 12324	. . . . . . . . . 10 ⊢ -1 ≠ 0
4	3	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → -1 ≠ 0)
5		iseralt.1	. . . . . . . . . . 11 ⊢ 𝑍 = (ℤ_≥‘𝑀)
6		uzssz 12839	. . . . . . . . . . 11 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
7	5, 6	eqsstri 4015	. . . . . . . . . 10 ⊢ 𝑍 ⊆ ℤ
8		simp2 1137	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → 𝑁 ∈ 𝑍)
9	7, 8	sselid 3979	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → 𝑁 ∈ ℤ)
10	2, 4, 9	reexpclzd 14208	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (-1↑𝑁) ∈ ℝ)
11	10	recnd 11238	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (-1↑𝑁) ∈ ℂ)
12		iseralt.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
13		iseralt.6	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = ((-1↑𝑘) · (𝐺‘𝑘)))
14	1	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -1 ∈ ℝ)
15	3	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -1 ≠ 0)
16		simpr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍)
17	7, 16	sselid 3979	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ ℤ)
18	14, 15, 17	reexpclzd 14208	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (-1↑𝑘) ∈ ℝ)
19		iseralt.3	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺:𝑍⟶ℝ)
20	19	ffvelcdmda 7083	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ)
21	18, 20	remulcld 11240	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((-1↑𝑘) · (𝐺‘𝑘)) ∈ ℝ)
22	13, 21	eqeltrd 2833	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
23	5, 12, 22	serfre 13993	. . . . . . . . . 10 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℝ)
24	23	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → seq𝑀( + , 𝐹):𝑍⟶ℝ)
25	8, 5	eleqtrdi 2843	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → 𝑁 ∈ (ℤ_≥‘𝑀))
26		2nn0 12485	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ₀
27		simp3 1138	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → 𝐾 ∈ ℕ₀)
28		nn0mulcl 12504	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℕ₀ ∧ 𝐾 ∈ ℕ₀) → (2 · 𝐾) ∈ ℕ₀)
29	26, 27, 28	sylancr 587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (2 · 𝐾) ∈ ℕ₀)
30		uzaddcl 12884	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ (2 · 𝐾) ∈ ℕ₀) → (𝑁 + (2 · 𝐾)) ∈ (ℤ_≥‘𝑀))
31	25, 29, 30	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (𝑁 + (2 · 𝐾)) ∈ (ℤ_≥‘𝑀))
32	31, 5	eleqtrrdi 2844	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (𝑁 + (2 · 𝐾)) ∈ 𝑍)
33	24, 32	ffvelcdmd 7084	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) ∈ ℝ)
34	33	recnd 11238	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) ∈ ℂ)
35	24, 8	ffvelcdmd 7084	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (seq𝑀( + , 𝐹)‘𝑁) ∈ ℝ)
36	35	recnd 11238	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (seq𝑀( + , 𝐹)‘𝑁) ∈ ℂ)
37	11, 34, 36	subdid 11666	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑𝑁) · ((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) − (seq𝑀( + , 𝐹)‘𝑁))) = (((-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾)))) − ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁))))
38	37	fveq2d 6892	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (abs‘((-1↑𝑁) · ((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) − (seq𝑀( + , 𝐹)‘𝑁)))) = (abs‘(((-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾)))) − ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)))))
39	33, 35	resubcld 11638	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) − (seq𝑀( + , 𝐹)‘𝑁)) ∈ ℝ)
40	39	recnd 11238	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) − (seq𝑀( + , 𝐹)‘𝑁)) ∈ ℂ)
41	11, 40	absmuld 15397	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (abs‘((-1↑𝑁) · ((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) − (seq𝑀( + , 𝐹)‘𝑁)))) = ((abs‘(-1↑𝑁)) · (abs‘((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) − (seq𝑀( + , 𝐹)‘𝑁)))))
42	38, 41	eqtr3d 2774	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (abs‘(((-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾)))) − ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)))) = ((abs‘(-1↑𝑁)) · (abs‘((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) − (seq𝑀( + , 𝐹)‘𝑁)))))
43	2	recnd 11238	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → -1 ∈ ℂ)
44		absexpz 15248	. . . . . . 7 ⊢ ((-1 ∈ ℂ ∧ -1 ≠ 0 ∧ 𝑁 ∈ ℤ) → (abs‘(-1↑𝑁)) = ((abs‘-1)↑𝑁))
45	43, 4, 9, 44	syl3anc 1371	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (abs‘(-1↑𝑁)) = ((abs‘-1)↑𝑁))
46		ax-1cn 11164	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
47	46	absnegi 15343	. . . . . . . . 9 ⊢ (abs‘-1) = (abs‘1)
48		abs1 15240	. . . . . . . . 9 ⊢ (abs‘1) = 1
49	47, 48	eqtri 2760	. . . . . . . 8 ⊢ (abs‘-1) = 1
50	49	oveq1i 7415	. . . . . . 7 ⊢ ((abs‘-1)↑𝑁) = (1↑𝑁)
51		1exp 14053	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (1↑𝑁) = 1)
52	9, 51	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (1↑𝑁) = 1)
53	50, 52	eqtrid 2784	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((abs‘-1)↑𝑁) = 1)
54	45, 53	eqtrd 2772	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (abs‘(-1↑𝑁)) = 1)
55	54	oveq1d 7420	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((abs‘(-1↑𝑁)) · (abs‘((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) − (seq𝑀( + , 𝐹)‘𝑁)))) = (1 · (abs‘((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) − (seq𝑀( + , 𝐹)‘𝑁)))))
56	40	abscld 15379	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (abs‘((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) − (seq𝑀( + , 𝐹)‘𝑁))) ∈ ℝ)
57	56	recnd 11238	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (abs‘((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) − (seq𝑀( + , 𝐹)‘𝑁))) ∈ ℂ)
58	57	mullidd 11228	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (1 · (abs‘((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) − (seq𝑀( + , 𝐹)‘𝑁)))) = (abs‘((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) − (seq𝑀( + , 𝐹)‘𝑁))))
59	42, 55, 58	3eqtrd 2776	. . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (abs‘(((-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾)))) − ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)))) = (abs‘((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) − (seq𝑀( + , 𝐹)‘𝑁))))
60	10, 35	remulcld 11240	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)) ∈ ℝ)
61	19	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → 𝐺:𝑍⟶ℝ)
62	5	peano2uzs 12882	. . . . . . . 8 ⊢ (𝑁 ∈ 𝑍 → (𝑁 + 1) ∈ 𝑍)
63	62	3ad2ant2 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (𝑁 + 1) ∈ 𝑍)
64	61, 63	ffvelcdmd 7084	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (𝐺‘(𝑁 + 1)) ∈ ℝ)
65	60, 64	resubcld 11638	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)) − (𝐺‘(𝑁 + 1))) ∈ ℝ)
66	5	peano2uzs 12882	. . . . . . . 8 ⊢ ((𝑁 + (2 · 𝐾)) ∈ 𝑍 → ((𝑁 + (2 · 𝐾)) + 1) ∈ 𝑍)
67	32, 66	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((𝑁 + (2 · 𝐾)) + 1) ∈ 𝑍)
68	24, 67	ffvelcdmd 7084	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)) ∈ ℝ)
69	10, 68	remulcld 11240	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1))) ∈ ℝ)
70	10, 33	remulcld 11240	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾)))) ∈ ℝ)
71		seqp1 13977	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (seq𝑀( + , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 + 1))))
72	25, 71	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (seq𝑀( + , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 + 1))))
73		fveq2 6888	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑁 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑁 + 1)))
74		oveq2 7413	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑁 + 1) → (-1↑𝑘) = (-1↑(𝑁 + 1)))
75		fveq2 6888	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑁 + 1) → (𝐺‘𝑘) = (𝐺‘(𝑁 + 1)))
76	74, 75	oveq12d 7423	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑁 + 1) → ((-1↑𝑘) · (𝐺‘𝑘)) = ((-1↑(𝑁 + 1)) · (𝐺‘(𝑁 + 1))))
77	73, 76	eqeq12d 2748	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑁 + 1) → ((𝐹‘𝑘) = ((-1↑𝑘) · (𝐺‘𝑘)) ↔ (𝐹‘(𝑁 + 1)) = ((-1↑(𝑁 + 1)) · (𝐺‘(𝑁 + 1)))))
78	13	ralrimiva 3146	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) = ((-1↑𝑘) · (𝐺‘𝑘)))
79	78	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) = ((-1↑𝑘) · (𝐺‘𝑘)))
80	77, 79, 63	rspcdva 3613	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (𝐹‘(𝑁 + 1)) = ((-1↑(𝑁 + 1)) · (𝐺‘(𝑁 + 1))))
81	80	oveq2d 7421	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((seq𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 + 1))) = ((seq𝑀( + , 𝐹)‘𝑁) + ((-1↑(𝑁 + 1)) · (𝐺‘(𝑁 + 1)))))
82	43, 4, 9	expp1zd 14116	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (-1↑(𝑁 + 1)) = ((-1↑𝑁) · -1))
83		neg1cn 12322	. . . . . . . . . . . . . . 15 ⊢ -1 ∈ ℂ
84		mulcom 11192	. . . . . . . . . . . . . . 15 ⊢ (((-1↑𝑁) ∈ ℂ ∧ -1 ∈ ℂ) → ((-1↑𝑁) · -1) = (-1 · (-1↑𝑁)))
85	11, 83, 84	sylancl 586	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑𝑁) · -1) = (-1 · (-1↑𝑁)))
86	11	mulm1d 11662	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (-1 · (-1↑𝑁)) = -(-1↑𝑁))
87	82, 85, 86	3eqtrd 2776	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (-1↑(𝑁 + 1)) = -(-1↑𝑁))
88	87	oveq1d 7420	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑(𝑁 + 1)) · (𝐺‘(𝑁 + 1))) = (-(-1↑𝑁) · (𝐺‘(𝑁 + 1))))
89	64	recnd 11238	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (𝐺‘(𝑁 + 1)) ∈ ℂ)
90	11, 89	mulneg1d 11663	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (-(-1↑𝑁) · (𝐺‘(𝑁 + 1))) = -((-1↑𝑁) · (𝐺‘(𝑁 + 1))))
91	88, 90	eqtrd 2772	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑(𝑁 + 1)) · (𝐺‘(𝑁 + 1))) = -((-1↑𝑁) · (𝐺‘(𝑁 + 1))))
92	91	oveq2d 7421	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((seq𝑀( + , 𝐹)‘𝑁) + ((-1↑(𝑁 + 1)) · (𝐺‘(𝑁 + 1)))) = ((seq𝑀( + , 𝐹)‘𝑁) + -((-1↑𝑁) · (𝐺‘(𝑁 + 1)))))
93	72, 81, 92	3eqtrd 2776	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (seq𝑀( + , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐹)‘𝑁) + -((-1↑𝑁) · (𝐺‘(𝑁 + 1)))))
94	10, 64	remulcld 11240	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑𝑁) · (𝐺‘(𝑁 + 1))) ∈ ℝ)
95	94	recnd 11238	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑𝑁) · (𝐺‘(𝑁 + 1))) ∈ ℂ)
96	36, 95	negsubd 11573	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((seq𝑀( + , 𝐹)‘𝑁) + -((-1↑𝑁) · (𝐺‘(𝑁 + 1)))) = ((seq𝑀( + , 𝐹)‘𝑁) − ((-1↑𝑁) · (𝐺‘(𝑁 + 1)))))
97	93, 96	eqtrd 2772	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (seq𝑀( + , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐹)‘𝑁) − ((-1↑𝑁) · (𝐺‘(𝑁 + 1)))))
98	97	oveq2d 7421	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + 1))) = ((-1↑𝑁) · ((seq𝑀( + , 𝐹)‘𝑁) − ((-1↑𝑁) · (𝐺‘(𝑁 + 1))))))
99	11, 36, 95	subdid 11666	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑𝑁) · ((seq𝑀( + , 𝐹)‘𝑁) − ((-1↑𝑁) · (𝐺‘(𝑁 + 1))))) = (((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)) − ((-1↑𝑁) · ((-1↑𝑁) · (𝐺‘(𝑁 + 1))))))
100	9	zcnd 12663	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → 𝑁 ∈ ℂ)
101	100	2timesd 12451	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (2 · 𝑁) = (𝑁 + 𝑁))
102	101	oveq2d 7421	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (-1↑(2 · 𝑁)) = (-1↑(𝑁 + 𝑁)))
103		2z 12590	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℤ
104	103	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → 2 ∈ ℤ)
105		expmulz 14070	. . . . . . . . . . . . . 14 ⊢ (((-1 ∈ ℂ ∧ -1 ≠ 0) ∧ (2 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (-1↑(2 · 𝑁)) = ((-1↑2)↑𝑁))
106	43, 4, 104, 9, 105	syl22anc 837	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (-1↑(2 · 𝑁)) = ((-1↑2)↑𝑁))
107	102, 106	eqtr3d 2774	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (-1↑(𝑁 + 𝑁)) = ((-1↑2)↑𝑁))
108		neg1sqe1 14156	. . . . . . . . . . . . 13 ⊢ (-1↑2) = 1
109	108	oveq1i 7415	. . . . . . . . . . . 12 ⊢ ((-1↑2)↑𝑁) = (1↑𝑁)
110	107, 109	eqtrdi 2788	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (-1↑(𝑁 + 𝑁)) = (1↑𝑁))
111		expaddz 14068	. . . . . . . . . . . 12 ⊢ (((-1 ∈ ℂ ∧ -1 ≠ 0) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (-1↑(𝑁 + 𝑁)) = ((-1↑𝑁) · (-1↑𝑁)))
112	43, 4, 9, 9, 111	syl22anc 837	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (-1↑(𝑁 + 𝑁)) = ((-1↑𝑁) · (-1↑𝑁)))
113	110, 112, 52	3eqtr3d 2780	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑𝑁) · (-1↑𝑁)) = 1)
114	113	oveq1d 7420	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (((-1↑𝑁) · (-1↑𝑁)) · (𝐺‘(𝑁 + 1))) = (1 · (𝐺‘(𝑁 + 1))))
115	11, 11, 89	mulassd 11233	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (((-1↑𝑁) · (-1↑𝑁)) · (𝐺‘(𝑁 + 1))) = ((-1↑𝑁) · ((-1↑𝑁) · (𝐺‘(𝑁 + 1)))))
116	89	mullidd 11228	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (1 · (𝐺‘(𝑁 + 1))) = (𝐺‘(𝑁 + 1)))
117	114, 115, 116	3eqtr3d 2780	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑𝑁) · ((-1↑𝑁) · (𝐺‘(𝑁 + 1)))) = (𝐺‘(𝑁 + 1)))
118	117	oveq2d 7421	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)) − ((-1↑𝑁) · ((-1↑𝑁) · (𝐺‘(𝑁 + 1))))) = (((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)) − (𝐺‘(𝑁 + 1))))
119	98, 99, 118	3eqtrd 2776	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + 1))) = (((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)) − (𝐺‘(𝑁 + 1))))
120		iseralt.4	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘(𝑘 + 1)) ≤ (𝐺‘𝑘))
121		iseralt.5	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ⇝ 0)
122	5, 12, 19, 120, 121, 13	iseraltlem2 15625	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑁 + 1) ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑(𝑁 + 1)) · (seq𝑀( + , 𝐹)‘((𝑁 + 1) + (2 · 𝐾)))) ≤ ((-1↑(𝑁 + 1)) · (seq𝑀( + , 𝐹)‘(𝑁 + 1))))
123	62, 122	syl3an2 1164	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑(𝑁 + 1)) · (seq𝑀( + , 𝐹)‘((𝑁 + 1) + (2 · 𝐾)))) ≤ ((-1↑(𝑁 + 1)) · (seq𝑀( + , 𝐹)‘(𝑁 + 1))))
124		1cnd 11205	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → 1 ∈ ℂ)
125	29	nn0cnd 12530	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (2 · 𝐾) ∈ ℂ)
126	100, 124, 125	add32d 11437	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((𝑁 + 1) + (2 · 𝐾)) = ((𝑁 + (2 · 𝐾)) + 1))
127	126	fveq2d 6892	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (seq𝑀( + , 𝐹)‘((𝑁 + 1) + (2 · 𝐾))) = (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)))
128	87, 127	oveq12d 7423	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑(𝑁 + 1)) · (seq𝑀( + , 𝐹)‘((𝑁 + 1) + (2 · 𝐾)))) = (-(-1↑𝑁) · (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1))))
129	87	oveq1d 7420	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑(𝑁 + 1)) · (seq𝑀( + , 𝐹)‘(𝑁 + 1))) = (-(-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + 1))))
130	123, 128, 129	3brtr3d 5178	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (-(-1↑𝑁) · (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1))) ≤ (-(-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + 1))))
131	68	recnd 11238	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)) ∈ ℂ)
132	11, 131	mulneg1d 11663	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (-(-1↑𝑁) · (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1))) = -((-1↑𝑁) · (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1))))
133	24, 63	ffvelcdmd 7084	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (seq𝑀( + , 𝐹)‘(𝑁 + 1)) ∈ ℝ)
134	133	recnd 11238	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (seq𝑀( + , 𝐹)‘(𝑁 + 1)) ∈ ℂ)
135	11, 134	mulneg1d 11663	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (-(-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + 1))) = -((-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + 1))))
136	130, 132, 135	3brtr3d 5178	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → -((-1↑𝑁) · (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1))) ≤ -((-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + 1))))
137	10, 133	remulcld 11240	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + 1))) ∈ ℝ)
138	137, 69	lenegd 11789	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (((-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + 1))) ≤ ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1))) ↔ -((-1↑𝑁) · (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1))) ≤ -((-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + 1)))))
139	136, 138	mpbird 256	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + 1))) ≤ ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1))))
140	119, 139	eqbrtrrd 5171	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)) − (𝐺‘(𝑁 + 1))) ≤ ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1))))
141		seqp1 13977	. . . . . . . . . 10 ⊢ ((𝑁 + (2 · 𝐾)) ∈ (ℤ_≥‘𝑀) → (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)) = ((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) + (𝐹‘((𝑁 + (2 · 𝐾)) + 1))))
142	31, 141	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)) = ((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) + (𝐹‘((𝑁 + (2 · 𝐾)) + 1))))
143		fveq2 6888	. . . . . . . . . . . . 13 ⊢ (𝑘 = ((𝑁 + (2 · 𝐾)) + 1) → (𝐹‘𝑘) = (𝐹‘((𝑁 + (2 · 𝐾)) + 1)))
144		oveq2 7413	. . . . . . . . . . . . . 14 ⊢ (𝑘 = ((𝑁 + (2 · 𝐾)) + 1) → (-1↑𝑘) = (-1↑((𝑁 + (2 · 𝐾)) + 1)))
145		fveq2 6888	. . . . . . . . . . . . . 14 ⊢ (𝑘 = ((𝑁 + (2 · 𝐾)) + 1) → (𝐺‘𝑘) = (𝐺‘((𝑁 + (2 · 𝐾)) + 1)))
146	144, 145	oveq12d 7423	. . . . . . . . . . . . 13 ⊢ (𝑘 = ((𝑁 + (2 · 𝐾)) + 1) → ((-1↑𝑘) · (𝐺‘𝑘)) = ((-1↑((𝑁 + (2 · 𝐾)) + 1)) · (𝐺‘((𝑁 + (2 · 𝐾)) + 1))))
147	143, 146	eqeq12d 2748	. . . . . . . . . . . 12 ⊢ (𝑘 = ((𝑁 + (2 · 𝐾)) + 1) → ((𝐹‘𝑘) = ((-1↑𝑘) · (𝐺‘𝑘)) ↔ (𝐹‘((𝑁 + (2 · 𝐾)) + 1)) = ((-1↑((𝑁 + (2 · 𝐾)) + 1)) · (𝐺‘((𝑁 + (2 · 𝐾)) + 1)))))
148	147, 79, 67	rspcdva 3613	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (𝐹‘((𝑁 + (2 · 𝐾)) + 1)) = ((-1↑((𝑁 + (2 · 𝐾)) + 1)) · (𝐺‘((𝑁 + (2 · 𝐾)) + 1))))
149	7, 63	sselid 3979	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (𝑁 + 1) ∈ ℤ)
150	29	nn0zd 12580	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (2 · 𝐾) ∈ ℤ)
151		expaddz 14068	. . . . . . . . . . . . . . 15 ⊢ (((-1 ∈ ℂ ∧ -1 ≠ 0) ∧ ((𝑁 + 1) ∈ ℤ ∧ (2 · 𝐾) ∈ ℤ)) → (-1↑((𝑁 + 1) + (2 · 𝐾))) = ((-1↑(𝑁 + 1)) · (-1↑(2 · 𝐾))))
152	43, 4, 149, 150, 151	syl22anc 837	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (-1↑((𝑁 + 1) + (2 · 𝐾))) = ((-1↑(𝑁 + 1)) · (-1↑(2 · 𝐾))))
153	27	nn0zd 12580	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → 𝐾 ∈ ℤ)
154		expmulz 14070	. . . . . . . . . . . . . . . . 17 ⊢ (((-1 ∈ ℂ ∧ -1 ≠ 0) ∧ (2 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → (-1↑(2 · 𝐾)) = ((-1↑2)↑𝐾))
155	43, 4, 104, 153, 154	syl22anc 837	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (-1↑(2 · 𝐾)) = ((-1↑2)↑𝐾))
156	108	oveq1i 7415	. . . . . . . . . . . . . . . . 17 ⊢ ((-1↑2)↑𝐾) = (1↑𝐾)
157		1exp 14053	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 ∈ ℤ → (1↑𝐾) = 1)
158	153, 157	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (1↑𝐾) = 1)
159	156, 158	eqtrid 2784	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑2)↑𝐾) = 1)
160	155, 159	eqtrd 2772	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (-1↑(2 · 𝐾)) = 1)
161	87, 160	oveq12d 7423	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑(𝑁 + 1)) · (-1↑(2 · 𝐾))) = (-(-1↑𝑁) · 1))
162	152, 161	eqtrd 2772	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (-1↑((𝑁 + 1) + (2 · 𝐾))) = (-(-1↑𝑁) · 1))
163	126	oveq2d 7421	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (-1↑((𝑁 + 1) + (2 · 𝐾))) = (-1↑((𝑁 + (2 · 𝐾)) + 1)))
164	11	negcld 11554	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → -(-1↑𝑁) ∈ ℂ)
165	164	mulridd 11227	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (-(-1↑𝑁) · 1) = -(-1↑𝑁))
166	162, 163, 165	3eqtr3d 2780	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (-1↑((𝑁 + (2 · 𝐾)) + 1)) = -(-1↑𝑁))
167	166	oveq1d 7420	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑((𝑁 + (2 · 𝐾)) + 1)) · (𝐺‘((𝑁 + (2 · 𝐾)) + 1))) = (-(-1↑𝑁) · (𝐺‘((𝑁 + (2 · 𝐾)) + 1))))
168	61, 67	ffvelcdmd 7084	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (𝐺‘((𝑁 + (2 · 𝐾)) + 1)) ∈ ℝ)
169	168	recnd 11238	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (𝐺‘((𝑁 + (2 · 𝐾)) + 1)) ∈ ℂ)
170	11, 169	mulneg1d 11663	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (-(-1↑𝑁) · (𝐺‘((𝑁 + (2 · 𝐾)) + 1))) = -((-1↑𝑁) · (𝐺‘((𝑁 + (2 · 𝐾)) + 1))))
171	148, 167, 170	3eqtrd 2776	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (𝐹‘((𝑁 + (2 · 𝐾)) + 1)) = -((-1↑𝑁) · (𝐺‘((𝑁 + (2 · 𝐾)) + 1))))
172	171	oveq2d 7421	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) + (𝐹‘((𝑁 + (2 · 𝐾)) + 1))) = ((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) + -((-1↑𝑁) · (𝐺‘((𝑁 + (2 · 𝐾)) + 1)))))
173	10, 168	remulcld 11240	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑𝑁) · (𝐺‘((𝑁 + (2 · 𝐾)) + 1))) ∈ ℝ)
174	173	recnd 11238	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑𝑁) · (𝐺‘((𝑁 + (2 · 𝐾)) + 1))) ∈ ℂ)
175	34, 174	negsubd 11573	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) + -((-1↑𝑁) · (𝐺‘((𝑁 + (2 · 𝐾)) + 1)))) = ((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) − ((-1↑𝑁) · (𝐺‘((𝑁 + (2 · 𝐾)) + 1)))))
176	142, 172, 175	3eqtrd 2776	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)) = ((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) − ((-1↑𝑁) · (𝐺‘((𝑁 + (2 · 𝐾)) + 1)))))
177	176	oveq2d 7421	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1))) = ((-1↑𝑁) · ((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) − ((-1↑𝑁) · (𝐺‘((𝑁 + (2 · 𝐾)) + 1))))))
178	11, 34, 174	subdid 11666	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑𝑁) · ((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) − ((-1↑𝑁) · (𝐺‘((𝑁 + (2 · 𝐾)) + 1))))) = (((-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾)))) − ((-1↑𝑁) · ((-1↑𝑁) · (𝐺‘((𝑁 + (2 · 𝐾)) + 1))))))
179	113	oveq1d 7420	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (((-1↑𝑁) · (-1↑𝑁)) · (𝐺‘((𝑁 + (2 · 𝐾)) + 1))) = (1 · (𝐺‘((𝑁 + (2 · 𝐾)) + 1))))
180	11, 11, 169	mulassd 11233	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (((-1↑𝑁) · (-1↑𝑁)) · (𝐺‘((𝑁 + (2 · 𝐾)) + 1))) = ((-1↑𝑁) · ((-1↑𝑁) · (𝐺‘((𝑁 + (2 · 𝐾)) + 1)))))
181	169	mullidd 11228	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (1 · (𝐺‘((𝑁 + (2 · 𝐾)) + 1))) = (𝐺‘((𝑁 + (2 · 𝐾)) + 1)))
182	179, 180, 181	3eqtr3d 2780	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑𝑁) · ((-1↑𝑁) · (𝐺‘((𝑁 + (2 · 𝐾)) + 1)))) = (𝐺‘((𝑁 + (2 · 𝐾)) + 1)))
183	182	oveq2d 7421	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (((-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾)))) − ((-1↑𝑁) · ((-1↑𝑁) · (𝐺‘((𝑁 + (2 · 𝐾)) + 1))))) = (((-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾)))) − (𝐺‘((𝑁 + (2 · 𝐾)) + 1))))
184	177, 178, 183	3eqtrd 2776	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1))) = (((-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾)))) − (𝐺‘((𝑁 + (2 · 𝐾)) + 1))))
185		simp1 1136	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → 𝜑)
186	5, 12, 19, 120, 121	iseraltlem1 15624	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑁 + (2 · 𝐾)) + 1) ∈ 𝑍) → 0 ≤ (𝐺‘((𝑁 + (2 · 𝐾)) + 1)))
187	185, 67, 186	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → 0 ≤ (𝐺‘((𝑁 + (2 · 𝐾)) + 1)))
188	70, 168	subge02d 11802	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (0 ≤ (𝐺‘((𝑁 + (2 · 𝐾)) + 1)) ↔ (((-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾)))) − (𝐺‘((𝑁 + (2 · 𝐾)) + 1))) ≤ ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))))))
189	187, 188	mpbid 231	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (((-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾)))) − (𝐺‘((𝑁 + (2 · 𝐾)) + 1))) ≤ ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾)))))
190	184, 189	eqbrtrd 5169	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1))) ≤ ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾)))))
191	65, 69, 70, 140, 190	letrd 11367	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)) − (𝐺‘(𝑁 + 1))) ≤ ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾)))))
192	60, 64	readdcld 11239	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)) + (𝐺‘(𝑁 + 1))) ∈ ℝ)
193	5, 12, 19, 120, 121, 13	iseraltlem2 15625	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾)))) ≤ ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)))
194	5, 12, 19, 120, 121	iseraltlem1 15624	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑁 + 1) ∈ 𝑍) → 0 ≤ (𝐺‘(𝑁 + 1)))
195	185, 63, 194	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → 0 ≤ (𝐺‘(𝑁 + 1)))
196	60, 64	addge01d 11798	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (0 ≤ (𝐺‘(𝑁 + 1)) ↔ ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)) ≤ (((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)) + (𝐺‘(𝑁 + 1)))))
197	195, 196	mpbid 231	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)) ≤ (((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)) + (𝐺‘(𝑁 + 1))))
198	70, 60, 192, 193, 197	letrd 11367	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾)))) ≤ (((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)) + (𝐺‘(𝑁 + 1))))
199	70, 60, 64	absdifled 15377	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((abs‘(((-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾)))) − ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)))) ≤ (𝐺‘(𝑁 + 1)) ↔ ((((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)) − (𝐺‘(𝑁 + 1))) ≤ ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾)))) ∧ ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾)))) ≤ (((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)) + (𝐺‘(𝑁 + 1))))))
200	191, 198, 199	mpbir2and 711	. . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (abs‘(((-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾)))) − ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)))) ≤ (𝐺‘(𝑁 + 1)))
201	59, 200	eqbrtrrd 5171	. 2 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (abs‘((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) − (seq𝑀( + , 𝐹)‘𝑁))) ≤ (𝐺‘(𝑁 + 1)))
202	11, 131, 36	subdid 11666	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑𝑁) · ((seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)) − (seq𝑀( + , 𝐹)‘𝑁))) = (((-1↑𝑁) · (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1))) − ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁))))
203	202	fveq2d 6892	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (abs‘((-1↑𝑁) · ((seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)) − (seq𝑀( + , 𝐹)‘𝑁)))) = (abs‘(((-1↑𝑁) · (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1))) − ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)))))
204	68, 35	resubcld 11638	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)) − (seq𝑀( + , 𝐹)‘𝑁)) ∈ ℝ)
205	204	recnd 11238	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)) − (seq𝑀( + , 𝐹)‘𝑁)) ∈ ℂ)
206	11, 205	absmuld 15397	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (abs‘((-1↑𝑁) · ((seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)) − (seq𝑀( + , 𝐹)‘𝑁)))) = ((abs‘(-1↑𝑁)) · (abs‘((seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)) − (seq𝑀( + , 𝐹)‘𝑁)))))
207	203, 206	eqtr3d 2774	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (abs‘(((-1↑𝑁) · (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1))) − ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)))) = ((abs‘(-1↑𝑁)) · (abs‘((seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)) − (seq𝑀( + , 𝐹)‘𝑁)))))
208	54	oveq1d 7420	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((abs‘(-1↑𝑁)) · (abs‘((seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)) − (seq𝑀( + , 𝐹)‘𝑁)))) = (1 · (abs‘((seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)) − (seq𝑀( + , 𝐹)‘𝑁)))))
209	205	abscld 15379	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (abs‘((seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)) − (seq𝑀( + , 𝐹)‘𝑁))) ∈ ℝ)
210	209	recnd 11238	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (abs‘((seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)) − (seq𝑀( + , 𝐹)‘𝑁))) ∈ ℂ)
211	210	mullidd 11228	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (1 · (abs‘((seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)) − (seq𝑀( + , 𝐹)‘𝑁)))) = (abs‘((seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)) − (seq𝑀( + , 𝐹)‘𝑁))))
212	207, 208, 211	3eqtrd 2776	. . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (abs‘(((-1↑𝑁) · (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1))) − ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)))) = (abs‘((seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)) − (seq𝑀( + , 𝐹)‘𝑁))))
213	69, 70, 192, 190, 198	letrd 11367	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1))) ≤ (((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)) + (𝐺‘(𝑁 + 1))))
214	69, 60, 64	absdifled 15377	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((abs‘(((-1↑𝑁) · (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1))) − ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)))) ≤ (𝐺‘(𝑁 + 1)) ↔ ((((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)) − (𝐺‘(𝑁 + 1))) ≤ ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1))) ∧ ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1))) ≤ (((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)) + (𝐺‘(𝑁 + 1))))))
215	140, 213, 214	mpbir2and 711	. . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (abs‘(((-1↑𝑁) · (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1))) − ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)))) ≤ (𝐺‘(𝑁 + 1)))
216	212, 215	eqbrtrrd 5171	. 2 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (abs‘((seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)) − (seq𝑀( + , 𝐹)‘𝑁))) ≤ (𝐺‘(𝑁 + 1)))
217	201, 216	jca 512	1 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((abs‘((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) − (seq𝑀( + , 𝐹)‘𝑁))) ≤ (𝐺‘(𝑁 + 1)) ∧ (abs‘((seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)) − (seq𝑀( + , 𝐹)‘𝑁))) ≤ (𝐺‘(𝑁 + 1))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 class class class wbr 5147 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 ≤ cle 11245 − cmin 11440 -cneg 11441 2c2 12263 ℕ₀cn0 12468 ℤcz 12554 ℤ_≥cuz 12818 seqcseq 13962 ↑cexp 14023 abscabs 15177 ⇝ cli 15424

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-fz 13481 df-fl 13753 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-rlim 15429

This theorem is referenced by: iseralt 15627

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