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Theorem iseraltlem3 15880

Description: Lemma for iseralt 15881. From iseraltlem2 15879, 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 12333	. . . . . . . . . 10 ⊢ -1 ∈ ℝ
2	1	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → -1 ∈ ℝ)
3		neg1ne0 12334	. . . . . . . . . 10 ⊢ -1 ≠ 0
4	3	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → -1 ≠ 0)
5		iseralt.1	. . . . . . . . . . 11 ⊢ 𝑍 = (ℤ_≥‘𝑀)
6		uzssz 13013	. . . . . . . . . . 11 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
7	5, 6	eqsstri 3959	. . . . . . . . . 10 ⊢ 𝑍 ⊆ ℤ
8		simp2 1155	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → 𝑁 ∈ 𝑍)
9	7, 8	sselid 3911	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → 𝑁 ∈ ℤ)
10	2, 4, 9	reexpclzd 14420	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ( -1↑𝑁) ∈ ℝ)
11	10	recnd 11364	. . . . . . 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 490	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍)
17	7, 16	sselid 3911	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ ℤ)
18	14, 15, 17	reexpclzd 14420	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ( -1↑𝑘) ∈ ℝ)
19		iseralt.3	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺:𝑍⟶ℝ)
20	19	ffvelcdmda 7108	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ)
21	18, 20	remulcld 11366	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (( -1↑𝑘) · (𝐺‘𝑘)) ∈ ℝ)
22	13, 21	eqeltrd 2864	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
23	5, 12, 22	serfre 14201	. . . . . . . . . 10 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℝ)
24	23	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → seq𝑀( + , 𝐹):𝑍⟶ℝ)
25	8, 5	eleqtrdi 2874	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → 𝑁 ∈ (ℤ_≥‘𝑀))
26		2nn0 12650	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ₀
27		simp3 1156	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → 𝐾 ∈ ℕ₀)
28		nn0mulcl 12669	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℕ₀ ∧ 𝐾 ∈ ℕ₀) → (2 · 𝐾) ∈ ℕ₀)
29	26, 27, 28	sylancr 599	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (2 · 𝐾) ∈ ℕ₀)
30		uzaddcl 13058	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ (2 · 𝐾) ∈ ℕ₀) → (𝑁 + (2 · 𝐾)) ∈ (ℤ_≥‘𝑀))
31	25, 29, 30	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (𝑁 + (2 · 𝐾)) ∈ (ℤ_≥‘𝑀))
32	31, 5	eleqtrrdi 2875	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (𝑁 + (2 · 𝐾)) ∈ 𝑍)
33	24, 32	ffvelcdmd 7109	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) ∈ ℝ)
34	33	recnd 11364	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) ∈ ℂ)
35	24, 8	ffvelcdmd 7109	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (seq𝑀( + , 𝐹)‘𝑁) ∈ ℝ)
36	35	recnd 11364	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (seq𝑀( + , 𝐹)‘𝑁) ∈ ℂ)
37	11, 34, 36	subdid 11799	. . . . . 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 6912	. . . . 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 11771	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) − (seq𝑀( + , 𝐹)‘𝑁)) ∈ ℝ)
40	39	recnd 11364	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) − (seq𝑀( + , 𝐹)‘𝑁)) ∈ ℂ)
41	11, 40	absmuld 15653	. . . . 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 2800	. . . 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 11364	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → -1 ∈ ℂ)
44		absexpz 15501	. . . . . . 7 ⊢ (( -1 ∈ ℂ ∧ -1 ≠ 0 ∧ 𝑁 ∈ ℤ) → (abs‘( -1↑𝑁)) = ((abs‘ -1)↑𝑁))
45	43, 4, 9, 44	syl3anc 1399	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (abs‘( -1↑𝑁)) = ((abs‘ -1)↑𝑁))
46		ax-1cn 11285	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
47	46	absnegi 15597	. . . . . . . . 9 ⊢ (abs‘ -1) = (abs‘1)
48		abs1 15493	. . . . . . . . 9 ⊢ (abs‘1) = 1
49	47, 48	eqtri 2786	. . . . . . . 8 ⊢ (abs‘ -1) = 1
50	49	oveq1i 7453	. . . . . . 7 ⊢ ((abs‘ -1)↑𝑁) = (1↑𝑁)
51		1exp 14261	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (1↑𝑁) = 1)
52	9, 51	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (1↑𝑁) = 1)
53	50, 52	eqtrid 2810	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((abs‘ -1)↑𝑁) = 1)
54	45, 53	eqtrd 2798	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (abs‘( -1↑𝑁)) = 1)
55	54	oveq1d 7458	. . . 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 15635	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (abs‘((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) − (seq𝑀( + , 𝐹)‘𝑁))) ∈ ℝ)
57	56	recnd 11364	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (abs‘((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) − (seq𝑀( + , 𝐹)‘𝑁))) ∈ ℂ)
58	57	mullidd 11354	. . . 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 2802	. . 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 11366	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (( -1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)) ∈ ℝ)
61	19	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → 𝐺:𝑍⟶ℝ)
62	5	peano2uzs 13056	. . . . . . . 8 ⊢ (𝑁 ∈ 𝑍 → (𝑁 + 1) ∈ 𝑍)
63	62	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (𝑁 + 1) ∈ 𝑍)
64	61, 63	ffvelcdmd 7109	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (𝐺‘(𝑁 + 1)) ∈ ℝ)
65	60, 64	resubcld 11771	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((( -1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)) − (𝐺‘(𝑁 + 1))) ∈ ℝ)
66	5	peano2uzs 13056	. . . . . . . 8 ⊢ ((𝑁 + (2 · 𝐾)) ∈ 𝑍 → ((𝑁 + (2 · 𝐾)) + 1) ∈ 𝑍)
67	32, 66	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((𝑁 + (2 · 𝐾)) + 1) ∈ 𝑍)
68	24, 67	ffvelcdmd 7109	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)) ∈ ℝ)
69	10, 68	remulcld 11366	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (( -1↑𝑁) · (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1))) ∈ ℝ)
70	10, 33	remulcld 11366	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (( -1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾)))) ∈ ℝ)
71		seqp1 14186	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (seq𝑀( + , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 + 1))))
72	25, 71	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (seq𝑀( + , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 + 1))))
73		fveq2 6908	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑁 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑁 + 1)))
74		oveq2 7451	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑁 + 1) → ( -1↑𝑘) = ( -1↑(𝑁 + 1)))
75		fveq2 6908	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑁 + 1) → (𝐺‘𝑘) = (𝐺‘(𝑁 + 1)))
76	74, 75	oveq12d 7461	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑁 + 1) → (( -1↑𝑘) · (𝐺‘𝑘)) = (( -1↑(𝑁 + 1)) · (𝐺‘(𝑁 + 1))))
77	73, 76	eqeq12d 2779	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑁 + 1) → ((𝐹‘𝑘) = (( -1↑𝑘) · (𝐺‘𝑘)) ↔ (𝐹‘(𝑁 + 1)) = (( -1↑(𝑁 + 1)) · (𝐺‘(𝑁 + 1)))))
78	13	ralrimiva 3158	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) = (( -1↑𝑘) · (𝐺‘𝑘)))
79	78	3ad2ant1 1151	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) = (( -1↑𝑘) · (𝐺‘𝑘)))
80	77, 79, 63	rspcdva 3581	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (𝐹‘(𝑁 + 1)) = (( -1↑(𝑁 + 1)) · (𝐺‘(𝑁 + 1))))
81	80	oveq2d 7459	. . . . . . . . . 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 14325	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ( -1↑(𝑁 + 1)) = (( -1↑𝑁) · -1))
83		neg1cn 12332	. . . . . . . . . . . . . . 15 ⊢ -1 ∈ ℂ
84		mulcom 11313	. . . . . . . . . . . . . . 15 ⊢ ((( -1↑𝑁) ∈ ℂ ∧ -1 ∈ ℂ) → (( -1↑𝑁) · -1) = ( -1 · ( -1↑𝑁)))
85	11, 83, 84	sylancl 598	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (( -1↑𝑁) · -1) = ( -1 · ( -1↑𝑁)))
86	11	mulm1d 11795	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ( -1 · ( -1↑𝑁)) = -( -1↑𝑁))
87	82, 85, 86	3eqtrd 2802	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ( -1↑(𝑁 + 1)) = -( -1↑𝑁))
88	87	oveq1d 7458	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (( -1↑(𝑁 + 1)) · (𝐺‘(𝑁 + 1))) = ( -( -1↑𝑁) · (𝐺‘(𝑁 + 1))))
89	64	recnd 11364	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (𝐺‘(𝑁 + 1)) ∈ ℂ)
90	11, 89	mulneg1d 11796	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ( -( -1↑𝑁) · (𝐺‘(𝑁 + 1))) = -(( -1↑𝑁) · (𝐺‘(𝑁 + 1))))
91	88, 90	eqtrd 2798	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (( -1↑(𝑁 + 1)) · (𝐺‘(𝑁 + 1))) = -(( -1↑𝑁) · (𝐺‘(𝑁 + 1))))
92	91	oveq2d 7459	. . . . . . . . . 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 2802	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (seq𝑀( + , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐹)‘𝑁) + -(( -1↑𝑁) · (𝐺‘(𝑁 + 1)))))
94	10, 64	remulcld 11366	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (( -1↑𝑁) · (𝐺‘(𝑁 + 1))) ∈ ℝ)
95	94	recnd 11364	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (( -1↑𝑁) · (𝐺‘(𝑁 + 1))) ∈ ℂ)
96	36, 95	negsubd 11702	. . . . . . . . 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 2798	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (seq𝑀( + , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐹)‘𝑁) − (( -1↑𝑁) · (𝐺‘(𝑁 + 1)))))
98	97	oveq2d 7459	. . . . . . 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 11799	. . . . . . 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 12831	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → 𝑁 ∈ ℂ)
101	100	2timesd 12616	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (2 · 𝑁) = (𝑁 + 𝑁))
102	101	oveq2d 7459	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ( -1↑(2 · 𝑁)) = ( -1↑(𝑁 + 𝑁)))
103		2z 12755	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℤ
104	103	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → 2 ∈ ℤ)
105		expmulz 14278	. . . . . . . . . . . . . 14 ⊢ ((( -1 ∈ ℂ ∧ -1 ≠ 0) ∧ (2 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ( -1↑(2 · 𝑁)) = (( -1↑2)↑𝑁))
106	43, 4, 104, 9, 105	syl22anc 852	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ( -1↑(2 · 𝑁)) = (( -1↑2)↑𝑁))
107	102, 106	eqtr3d 2800	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ( -1↑(𝑁 + 𝑁)) = (( -1↑2)↑𝑁))
108		neg1sqe1 14366	. . . . . . . . . . . . 13 ⊢ ( -1↑2) = 1
109	108	oveq1i 7453	. . . . . . . . . . . 12 ⊢ (( -1↑2)↑𝑁) = (1↑𝑁)
110	107, 109	eqtrdi 2814	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ( -1↑(𝑁 + 𝑁)) = (1↑𝑁))
111		expaddz 14276	. . . . . . . . . . . 12 ⊢ ((( -1 ∈ ℂ ∧ -1 ≠ 0) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ( -1↑(𝑁 + 𝑁)) = (( -1↑𝑁) · ( -1↑𝑁)))
112	43, 4, 9, 9, 111	syl22anc 852	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ( -1↑(𝑁 + 𝑁)) = (( -1↑𝑁) · ( -1↑𝑁)))
113	110, 112, 52	3eqtr3d 2806	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (( -1↑𝑁) · ( -1↑𝑁)) = 1)
114	113	oveq1d 7458	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((( -1↑𝑁) · ( -1↑𝑁)) · (𝐺‘(𝑁 + 1))) = (1 · (𝐺‘(𝑁 + 1))))
115	11, 11, 89	mulassd 11359	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((( -1↑𝑁) · ( -1↑𝑁)) · (𝐺‘(𝑁 + 1))) = (( -1↑𝑁) · (( -1↑𝑁) · (𝐺‘(𝑁 + 1)))))
116	89	mullidd 11354	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (1 · (𝐺‘(𝑁 + 1))) = (𝐺‘(𝑁 + 1)))
117	114, 115, 116	3eqtr3d 2806	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (( -1↑𝑁) · (( -1↑𝑁) · (𝐺‘(𝑁 + 1)))) = (𝐺‘(𝑁 + 1)))
118	117	oveq2d 7459	. . . . . . 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 2802	. . . . . 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 15879	. . . . . . . . . 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 1182	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (( -1↑(𝑁 + 1)) · (seq𝑀( + , 𝐹)‘((𝑁 + 1) + (2 · 𝐾)))) ≤ (( -1↑(𝑁 + 1)) · (seq𝑀( + , 𝐹)‘(𝑁 + 1))))
124		1cnd 11329	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → 1 ∈ ℂ)
125	29	nn0cnd 12696	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (2 · 𝐾) ∈ ℂ)
126	100, 124, 125	add32d 11565	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((𝑁 + 1) + (2 · 𝐾)) = ((𝑁 + (2 · 𝐾)) + 1))
127	126	fveq2d 6912	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (seq𝑀( + , 𝐹)‘((𝑁 + 1) + (2 · 𝐾))) = (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)))
128	87, 127	oveq12d 7461	. . . . . . . . 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 7458	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (( -1↑(𝑁 + 1)) · (seq𝑀( + , 𝐹)‘(𝑁 + 1))) = ( -( -1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + 1))))
130	123, 128, 129	3brtr3d 5139	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ( -( -1↑𝑁) · (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1))) ≤ ( -( -1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + 1))))
131	68	recnd 11364	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)) ∈ ℂ)
132	11, 131	mulneg1d 11796	. . . . . . . 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 7109	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (seq𝑀( + , 𝐹)‘(𝑁 + 1)) ∈ ℝ)
134	133	recnd 11364	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (seq𝑀( + , 𝐹)‘(𝑁 + 1)) ∈ ℂ)
135	11, 134	mulneg1d 11796	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ( -( -1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + 1))) = -(( -1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + 1))))
136	130, 132, 135	3brtr3d 5139	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → -(( -1↑𝑁) · (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1))) ≤ -(( -1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + 1))))
137	10, 133	remulcld 11366	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (( -1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + 1))) ∈ ℝ)
138	137, 69	lenegd 11922	. . . . . . 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 260	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (( -1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + 1))) ≤ (( -1↑𝑁) · (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1))))
140	119, 139	eqbrtrrd 5132	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((( -1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)) − (𝐺‘(𝑁 + 1))) ≤ (( -1↑𝑁) · (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1))))
141		seqp1 14186	. . . . . . . . . 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 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)) = ((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) + (𝐹‘((𝑁 + (2 · 𝐾)) + 1))))
143		fveq2 6908	. . . . . . . . . . . . 13 ⊢ (𝑘 = ((𝑁 + (2 · 𝐾)) + 1) → (𝐹‘𝑘) = (𝐹‘((𝑁 + (2 · 𝐾)) + 1)))
144		oveq2 7451	. . . . . . . . . . . . . 14 ⊢ (𝑘 = ((𝑁 + (2 · 𝐾)) + 1) → ( -1↑𝑘) = ( -1↑((𝑁 + (2 · 𝐾)) + 1)))
145		fveq2 6908	. . . . . . . . . . . . . 14 ⊢ (𝑘 = ((𝑁 + (2 · 𝐾)) + 1) → (𝐺‘𝑘) = (𝐺‘((𝑁 + (2 · 𝐾)) + 1)))
146	144, 145	oveq12d 7461	. . . . . . . . . . . . 13 ⊢ (𝑘 = ((𝑁 + (2 · 𝐾)) + 1) → (( -1↑𝑘) · (𝐺‘𝑘)) = (( -1↑((𝑁 + (2 · 𝐾)) + 1)) · (𝐺‘((𝑁 + (2 · 𝐾)) + 1))))
147	143, 146	eqeq12d 2779	. . . . . . . . . . . 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 3581	. . . . . . . . . . 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 3911	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (𝑁 + 1) ∈ ℤ)
150	29	nn0zd 12745	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (2 · 𝐾) ∈ ℤ)
151		expaddz 14276	. . . . . . . . . . . . . . 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 852	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ( -1↑((𝑁 + 1) + (2 · 𝐾))) = (( -1↑(𝑁 + 1)) · ( -1↑(2 · 𝐾))))
153	27	nn0zd 12745	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → 𝐾 ∈ ℤ)
154		expmulz 14278	. . . . . . . . . . . . . . . . 17 ⊢ ((( -1 ∈ ℂ ∧ -1 ≠ 0) ∧ (2 ∈ ℤ ∧ 𝐾 ∈ ℤ)) → ( -1↑(2 · 𝐾)) = (( -1↑2)↑𝐾))
155	43, 4, 104, 153, 154	syl22anc 852	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ( -1↑(2 · 𝐾)) = (( -1↑2)↑𝐾))
156	108	oveq1i 7453	. . . . . . . . . . . . . . . . 17 ⊢ (( -1↑2)↑𝐾) = (1↑𝐾)
157		1exp 14261	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 ∈ ℤ → (1↑𝐾) = 1)
158	153, 157	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (1↑𝐾) = 1)
159	156, 158	eqtrid 2810	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (( -1↑2)↑𝐾) = 1)
160	155, 159	eqtrd 2798	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ( -1↑(2 · 𝐾)) = 1)
161	87, 160	oveq12d 7461	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (( -1↑(𝑁 + 1)) · ( -1↑(2 · 𝐾))) = ( -( -1↑𝑁) · 1))
162	152, 161	eqtrd 2798	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ( -1↑((𝑁 + 1) + (2 · 𝐾))) = ( -( -1↑𝑁) · 1))
163	126	oveq2d 7459	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ( -1↑((𝑁 + 1) + (2 · 𝐾))) = ( -1↑((𝑁 + (2 · 𝐾)) + 1)))
164	11	negcld 11683	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → -( -1↑𝑁) ∈ ℂ)
165	164	mulridd 11353	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ( -( -1↑𝑁) · 1) = -( -1↑𝑁))
166	162, 163, 165	3eqtr3d 2806	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ( -1↑((𝑁 + (2 · 𝐾)) + 1)) = -( -1↑𝑁))
167	166	oveq1d 7458	. . . . . . . . . . 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 7109	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (𝐺‘((𝑁 + (2 · 𝐾)) + 1)) ∈ ℝ)
169	168	recnd 11364	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (𝐺‘((𝑁 + (2 · 𝐾)) + 1)) ∈ ℂ)
170	11, 169	mulneg1d 11796	. . . . . . . . . . 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 2802	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (𝐹‘((𝑁 + (2 · 𝐾)) + 1)) = -(( -1↑𝑁) · (𝐺‘((𝑁 + (2 · 𝐾)) + 1))))
172	171	oveq2d 7459	. . . . . . . . 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 11366	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (( -1↑𝑁) · (𝐺‘((𝑁 + (2 · 𝐾)) + 1))) ∈ ℝ)
174	173	recnd 11364	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (( -1↑𝑁) · (𝐺‘((𝑁 + (2 · 𝐾)) + 1))) ∈ ℂ)
175	34, 174	negsubd 11702	. . . . . . . . 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 2802	. . . . . . . 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 7459	. . . . . . 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 11799	. . . . . . 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 7458	. . . . . . . . 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 11359	. . . . . . . . 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 11354	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (1 · (𝐺‘((𝑁 + (2 · 𝐾)) + 1))) = (𝐺‘((𝑁 + (2 · 𝐾)) + 1)))
182	179, 180, 181	3eqtr3d 2806	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (( -1↑𝑁) · (( -1↑𝑁) · (𝐺‘((𝑁 + (2 · 𝐾)) + 1)))) = (𝐺‘((𝑁 + (2 · 𝐾)) + 1)))
183	182	oveq2d 7459	. . . . . . 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 2802	. . . . . 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 1154	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → 𝜑)
186	5, 12, 19, 120, 121	iseraltlem1 15878	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑁 + (2 · 𝐾)) + 1) ∈ 𝑍) → 0 ≤ (𝐺‘((𝑁 + (2 · 𝐾)) + 1)))
187	185, 67, 186	syl2anc 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → 0 ≤ (𝐺‘((𝑁 + (2 · 𝐾)) + 1)))
188	70, 168	subge02d 11935	. . . . . . 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 235	. . . . . 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 5130	. . . . 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 11494	. . . 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 11365	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((( -1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)) + (𝐺‘(𝑁 + 1))) ∈ ℝ)
193	5, 12, 19, 120, 121, 13	iseraltlem2 15879	. . . . 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 15878	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑁 + 1) ∈ 𝑍) → 0 ≤ (𝐺‘(𝑁 + 1)))
195	185, 63, 194	syl2anc 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → 0 ≤ (𝐺‘(𝑁 + 1)))
196	60, 64	addge01d 11931	. . . . . 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 235	. . . . 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 11494	. . . 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 15633	. . . 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 726	. . 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 5132	. 2 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (abs‘((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) − (seq𝑀( + , 𝐹)‘𝑁))) ≤ (𝐺‘(𝑁 + 1)))
202	11, 131, 36	subdid 11799	. . . . . 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 6912	. . . . 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 11771	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)) − (seq𝑀( + , 𝐹)‘𝑁)) ∈ ℝ)
205	204	recnd 11364	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → ((seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)) − (seq𝑀( + , 𝐹)‘𝑁)) ∈ ℂ)
206	11, 205	absmuld 15653	. . . . 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 2800	. . . 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 7458	. . . 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 15635	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (abs‘((seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)) − (seq𝑀( + , 𝐹)‘𝑁))) ∈ ℝ)
210	209	recnd 11364	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (abs‘((seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)) − (seq𝑀( + , 𝐹)‘𝑁))) ∈ ℂ)
211	210	mullidd 11354	. . . 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 2802	. . 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 11494	. . . 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 15633	. . . 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 726	. . 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 5132	. 2 ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ₀) → (abs‘((seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)) − (seq𝑀( + , 𝐹)‘𝑁))) ≤ (𝐺‘(𝑁 + 1)))
217	201, 216	jca 521	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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∀wral 3080 class class class wbr 5106 ⟶wf 6554 ‘cfv 6558 (class class class)co 7443 ℂcc 11225 ℝcr 11226 0cc0 11227 1c1 11228 + caddc 11230 · cmul 11232 ≤ cle 11371 − cmin 11568 -cneg 11569 2c2 12424 ℕ₀cn0 12633 ℤcz 12720 ℤ_≥cuz 12992 seqcseq 14171 ↑cexp 14231 abscabs 15430 ⇝ cli 15680

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304 ax-pre-sup 11305

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-om 7902 df-1st 8025 df-2nd 8026 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-er 8745 df-pm 8878 df-en 9002 df-dom 9003 df-sdom 9004 df-sup 9461 df-inf 9462 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-div 12001 df-nn 12363 df-2 12432 df-3 12433 df-n0 12634 df-z 12721 df-uz 12993 df-rp 13148 df-fz 13667 df-fl 13959 df-seq 14172 df-exp 14232 df-cj 15295 df-re 15296 df-im 15297 df-sqrt 15431 df-abs 15432 df-clim 15684 df-rlim 15685

This theorem is used by: iseralt 15881

W3C validator