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Theorem iseralt 15606

Description: The alternating series test. If 𝐺(𝑘) is a decreasing sequence that converges to 0, then Σ𝑘 ∈ 𝑍(-1↑𝑘) · 𝐺(𝑘) is a convergent series. (Note that the first term is positive if 𝑀 is even, and negative if 𝑀 is odd. If the parity of your series does not match up with this, you will need to post-compose the series with multiplication by -1 using isermulc2 15579.) (Contributed by Mario Carneiro, 7-Apr-2015.) (Proof shortened by AV, 9-Jul-2022.)

**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
iseralt	⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )

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

Proof of Theorem iseralt Dummy variables 𝑗 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iseralt.1	. 2 ⊢ 𝑍 = (ℤ_≥‘𝑀)
2		seqex 13924	. . 3 ⊢ seq𝑀( + , 𝐹) ∈ V
3	2	a1i 11	. 2 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ V)
4		iseralt.5	. . . 4 ⊢ (𝜑 → 𝐺 ⇝ 0)
5		iseralt.2	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
6		climrel 15413	. . . . . . 7 ⊢ Rel ⇝
7	6	brrelex1i 5678	. . . . . 6 ⊢ (𝐺 ⇝ 0 → 𝐺 ∈ V)
8	4, 7	syl 17	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ V)
9		eqidd 2735	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) = (𝐺‘𝑛))
10		iseralt.3	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝑍⟶ℝ)
11	10	ffvelcdmda 7027	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) ∈ ℝ)
12	11	recnd 11158	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) ∈ ℂ)
13	1, 5, 8, 9, 12	clim0c 15428	. . . 4 ⊢ (𝜑 → (𝐺 ⇝ 0 ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)(abs‘(𝐺‘𝑛)) < 𝑥))
14	4, 13	mpbid 232	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)(abs‘(𝐺‘𝑛)) < 𝑥)
15		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍)
16	15, 1	eleqtrdi 2844	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ_≥‘𝑀))
17		eluzelz 12759	. . . . . . . 8 ⊢ (𝑗 ∈ (ℤ_≥‘𝑀) → 𝑗 ∈ ℤ)
18		uzid 12764	. . . . . . . 8 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ_≥‘𝑗))
19	16, 17, 18	3syl 18	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ_≥‘𝑗))
20		peano2uz 12812	. . . . . . 7 ⊢ (𝑗 ∈ (ℤ_≥‘𝑗) → (𝑗 + 1) ∈ (ℤ_≥‘𝑗))
21		2fveq3 6837	. . . . . . . . 9 ⊢ (𝑛 = (𝑗 + 1) → (abs‘(𝐺‘𝑛)) = (abs‘(𝐺‘(𝑗 + 1))))
22	21	breq1d 5106	. . . . . . . 8 ⊢ (𝑛 = (𝑗 + 1) → ((abs‘(𝐺‘𝑛)) < 𝑥 ↔ (abs‘(𝐺‘(𝑗 + 1))) < 𝑥))
23	22	rspcv 3570	. . . . . . 7 ⊢ ((𝑗 + 1) ∈ (ℤ_≥‘𝑗) → (∀𝑛 ∈ (ℤ_≥‘𝑗)(abs‘(𝐺‘𝑛)) < 𝑥 → (abs‘(𝐺‘(𝑗 + 1))) < 𝑥))
24	19, 20, 23	3syl 18	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → (∀𝑛 ∈ (ℤ_≥‘𝑗)(abs‘(𝐺‘𝑛)) < 𝑥 → (abs‘(𝐺‘(𝑗 + 1))) < 𝑥))
25		eluzelz 12759	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ (ℤ_≥‘𝑗) → 𝑛 ∈ ℤ)
26	25	ad2antll 729	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝑛 ∈ ℤ)
27	26	zcnd 12595	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝑛 ∈ ℂ)
28	17, 1	eleq2s 2852	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ)
29	28	ad2antrl 728	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝑗 ∈ ℤ)
30	29	zcnd 12595	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝑗 ∈ ℂ)
31	27, 30	subcld 11490	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝑛 − 𝑗) ∈ ℂ)
32		2cnd 12221	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 2 ∈ ℂ)
33		2ne0 12247	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ≠ 0
34	33	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 2 ≠ 0)
35	31, 32, 34	divcan2d 11917	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (2 · ((𝑛 − 𝑗) / 2)) = (𝑛 − 𝑗))
36	35	oveq2d 7372	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝑗 + (2 · ((𝑛 − 𝑗) / 2))) = (𝑗 + (𝑛 − 𝑗)))
37	30, 27	pncan3d 11493	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝑗 + (𝑛 − 𝑗)) = 𝑛)
38	36, 37	eqtr2d 2770	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝑛 = (𝑗 + (2 · ((𝑛 − 𝑗) / 2))))
39	38	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ ((𝑛 − 𝑗) / 2) ∈ ℤ) → 𝑛 = (𝑗 + (2 · ((𝑛 − 𝑗) / 2))))
40	39	fveq2d 6836	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ ((𝑛 − 𝑗) / 2) ∈ ℤ) → (seq𝑀( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘(𝑗 + (2 · ((𝑛 − 𝑗) / 2)))))
41	40	fvoveq1d 7378	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ ((𝑛 − 𝑗) / 2) ∈ ℤ) → (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗))) = (abs‘((seq𝑀( + , 𝐹)‘(𝑗 + (2 · ((𝑛 − 𝑗) / 2)))) − (seq𝑀( + , 𝐹)‘𝑗))))
42		simpll 766	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ ((𝑛 − 𝑗) / 2) ∈ ℤ) → 𝜑)
43		simpl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑗 ∈ 𝑍)
44	43	ad2antlr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ ((𝑛 − 𝑗) / 2) ∈ ℤ) → 𝑗 ∈ 𝑍)
45		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ ((𝑛 − 𝑗) / 2) ∈ ℤ) → ((𝑛 − 𝑗) / 2) ∈ ℤ)
46	26, 29	zsubcld 12599	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝑛 − 𝑗) ∈ ℤ)
47	46	zred 12594	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝑛 − 𝑗) ∈ ℝ)
48		2rp 12908	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℝ⁺
49	48	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 2 ∈ ℝ⁺)
50		eluzle 12762	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ (ℤ_≥‘𝑗) → 𝑗 ≤ 𝑛)
51	50	ad2antll 729	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝑗 ≤ 𝑛)
52	26	zred 12594	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝑛 ∈ ℝ)
53	29	zred 12594	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝑗 ∈ ℝ)
54	52, 53	subge0d 11725	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (0 ≤ (𝑛 − 𝑗) ↔ 𝑗 ≤ 𝑛))
55	51, 54	mpbird 257	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 0 ≤ (𝑛 − 𝑗))
56	47, 49, 55	divge0d 12987	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 0 ≤ ((𝑛 − 𝑗) / 2))
57	56	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ ((𝑛 − 𝑗) / 2) ∈ ℤ) → 0 ≤ ((𝑛 − 𝑗) / 2))
58		elnn0z 12499	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 − 𝑗) / 2) ∈ ℕ₀ ↔ (((𝑛 − 𝑗) / 2) ∈ ℤ ∧ 0 ≤ ((𝑛 − 𝑗) / 2)))
59	45, 57, 58	sylanbrc 583	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ ((𝑛 − 𝑗) / 2) ∈ ℤ) → ((𝑛 − 𝑗) / 2) ∈ ℕ₀)
60		iseralt.4	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘(𝑘 + 1)) ≤ (𝐺‘𝑘))
61		iseralt.6	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = ((-1↑𝑘) · (𝐺‘𝑘)))
62	1, 5, 10, 60, 4, 61	iseraltlem3 15605	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ((𝑛 − 𝑗) / 2) ∈ ℕ₀) → ((abs‘((seq𝑀( + , 𝐹)‘(𝑗 + (2 · ((𝑛 − 𝑗) / 2)))) − (seq𝑀( + , 𝐹)‘𝑗))) ≤ (𝐺‘(𝑗 + 1)) ∧ (abs‘((seq𝑀( + , 𝐹)‘((𝑗 + (2 · ((𝑛 − 𝑗) / 2))) + 1)) − (seq𝑀( + , 𝐹)‘𝑗))) ≤ (𝐺‘(𝑗 + 1))))
63	62	simpld 494	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ((𝑛 − 𝑗) / 2) ∈ ℕ₀) → (abs‘((seq𝑀( + , 𝐹)‘(𝑗 + (2 · ((𝑛 − 𝑗) / 2)))) − (seq𝑀( + , 𝐹)‘𝑗))) ≤ (𝐺‘(𝑗 + 1)))
64	42, 44, 59, 63	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ ((𝑛 − 𝑗) / 2) ∈ ℤ) → (abs‘((seq𝑀( + , 𝐹)‘(𝑗 + (2 · ((𝑛 − 𝑗) / 2)))) − (seq𝑀( + , 𝐹)‘𝑗))) ≤ (𝐺‘(𝑗 + 1)))
65	41, 64	eqbrtrd 5118	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ ((𝑛 − 𝑗) / 2) ∈ ℤ) → (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗))) ≤ (𝐺‘(𝑗 + 1)))
66		2div2e1 12279	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (2 / 2) = 1
67	66	oveq2i 7367	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑛 − 𝑗) + 1) / 2) − (2 / 2)) = ((((𝑛 − 𝑗) + 1) / 2) − 1)
68		peano2cn 11303	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑛 − 𝑗) ∈ ℂ → ((𝑛 − 𝑗) + 1) ∈ ℂ)
69	31, 68	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((𝑛 − 𝑗) + 1) ∈ ℂ)
70	69, 32, 32, 34	divsubdird 11954	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((((𝑛 − 𝑗) + 1) − 2) / 2) = ((((𝑛 − 𝑗) + 1) / 2) − (2 / 2)))
71		df-2 12206	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 2 = (1 + 1)
72	71	oveq2i 7367	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑛 − 𝑗) + 1) − 2) = (((𝑛 − 𝑗) + 1) − (1 + 1))
73		ax-1cn 11082	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 1 ∈ ℂ
74	73	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 1 ∈ ℂ)
75	31, 74, 74	pnpcan2d 11528	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (((𝑛 − 𝑗) + 1) − (1 + 1)) = ((𝑛 − 𝑗) − 1))
76	72, 75	eqtrid 2781	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (((𝑛 − 𝑗) + 1) − 2) = ((𝑛 − 𝑗) − 1))
77	76	oveq1d 7371	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((((𝑛 − 𝑗) + 1) − 2) / 2) = (((𝑛 − 𝑗) − 1) / 2))
78	70, 77	eqtr3d 2771	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((((𝑛 − 𝑗) + 1) / 2) − (2 / 2)) = (((𝑛 − 𝑗) − 1) / 2))
79	67, 78	eqtr3id 2783	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((((𝑛 − 𝑗) + 1) / 2) − 1) = (((𝑛 − 𝑗) − 1) / 2))
80	79	oveq2d 7372	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (2 · ((((𝑛 − 𝑗) + 1) / 2) − 1)) = (2 · (((𝑛 − 𝑗) − 1) / 2)))
81		subcl 11377	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑛 − 𝑗) ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 𝑗) − 1) ∈ ℂ)
82	31, 73, 81	sylancl 586	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((𝑛 − 𝑗) − 1) ∈ ℂ)
83	82, 32, 34	divcan2d 11917	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (2 · (((𝑛 − 𝑗) − 1) / 2)) = ((𝑛 − 𝑗) − 1))
84	27, 30, 74	sub32d 11522	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((𝑛 − 𝑗) − 1) = ((𝑛 − 1) − 𝑗))
85	80, 83, 84	3eqtrd 2773	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (2 · ((((𝑛 − 𝑗) + 1) / 2) − 1)) = ((𝑛 − 1) − 𝑗))
86	85	oveq2d 7372	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝑗 + (2 · ((((𝑛 − 𝑗) + 1) / 2) − 1))) = (𝑗 + ((𝑛 − 1) − 𝑗)))
87		subcl 11377	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑛 − 1) ∈ ℂ)
88	27, 73, 87	sylancl 586	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝑛 − 1) ∈ ℂ)
89	30, 88	pncan3d 11493	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝑗 + ((𝑛 − 1) − 𝑗)) = (𝑛 − 1))
90	86, 89	eqtrd 2769	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝑗 + (2 · ((((𝑛 − 𝑗) + 1) / 2) − 1))) = (𝑛 − 1))
91	90	oveq1d 7371	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((𝑗 + (2 · ((((𝑛 − 𝑗) + 1) / 2) − 1))) + 1) = ((𝑛 − 1) + 1))
92		npcan 11387	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
93	27, 73, 92	sylancl 586	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((𝑛 − 1) + 1) = 𝑛)
94	91, 93	eqtr2d 2770	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝑛 = ((𝑗 + (2 · ((((𝑛 − 𝑗) + 1) / 2) − 1))) + 1))
95	94	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ (((𝑛 − 𝑗) + 1) / 2) ∈ ℤ) → 𝑛 = ((𝑗 + (2 · ((((𝑛 − 𝑗) + 1) / 2) − 1))) + 1))
96	95	fveq2d 6836	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ (((𝑛 − 𝑗) + 1) / 2) ∈ ℤ) → (seq𝑀( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘((𝑗 + (2 · ((((𝑛 − 𝑗) + 1) / 2) − 1))) + 1)))
97	96	fvoveq1d 7378	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ (((𝑛 − 𝑗) + 1) / 2) ∈ ℤ) → (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗))) = (abs‘((seq𝑀( + , 𝐹)‘((𝑗 + (2 · ((((𝑛 − 𝑗) + 1) / 2) − 1))) + 1)) − (seq𝑀( + , 𝐹)‘𝑗))))
98		simpll 766	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ (((𝑛 − 𝑗) + 1) / 2) ∈ ℤ) → 𝜑)
99	43	ad2antlr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ (((𝑛 − 𝑗) + 1) / 2) ∈ ℤ) → 𝑗 ∈ 𝑍)
100		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ (((𝑛 − 𝑗) + 1) / 2) ∈ ℤ) → (((𝑛 − 𝑗) + 1) / 2) ∈ ℤ)
101		uznn0sub 12784	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ (ℤ_≥‘𝑗) → (𝑛 − 𝑗) ∈ ℕ₀)
102	101	ad2antll 729	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝑛 − 𝑗) ∈ ℕ₀)
103		nn0p1nn 12438	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛 − 𝑗) ∈ ℕ₀ → ((𝑛 − 𝑗) + 1) ∈ ℕ)
104	102, 103	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((𝑛 − 𝑗) + 1) ∈ ℕ)
105	104	nnrpd 12945	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((𝑛 − 𝑗) + 1) ∈ ℝ⁺)
106	105	rphalfcld 12959	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (((𝑛 − 𝑗) + 1) / 2) ∈ ℝ⁺)
107	106	rpgt0d 12950	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 0 < (((𝑛 − 𝑗) + 1) / 2))
108	107	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ (((𝑛 − 𝑗) + 1) / 2) ∈ ℤ) → 0 < (((𝑛 − 𝑗) + 1) / 2))
109		elnnz 12496	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 − 𝑗) + 1) / 2) ∈ ℕ ↔ ((((𝑛 − 𝑗) + 1) / 2) ∈ ℤ ∧ 0 < (((𝑛 − 𝑗) + 1) / 2)))
110	100, 108, 109	sylanbrc 583	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ (((𝑛 − 𝑗) + 1) / 2) ∈ ℤ) → (((𝑛 − 𝑗) + 1) / 2) ∈ ℕ)
111		nnm1nn0 12440	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑛 − 𝑗) + 1) / 2) ∈ ℕ → ((((𝑛 − 𝑗) + 1) / 2) − 1) ∈ ℕ₀)
112	110, 111	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ (((𝑛 − 𝑗) + 1) / 2) ∈ ℤ) → ((((𝑛 − 𝑗) + 1) / 2) − 1) ∈ ℕ₀)
113	1, 5, 10, 60, 4, 61	iseraltlem3 15605	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ((((𝑛 − 𝑗) + 1) / 2) − 1) ∈ ℕ₀) → ((abs‘((seq𝑀( + , 𝐹)‘(𝑗 + (2 · ((((𝑛 − 𝑗) + 1) / 2) − 1)))) − (seq𝑀( + , 𝐹)‘𝑗))) ≤ (𝐺‘(𝑗 + 1)) ∧ (abs‘((seq𝑀( + , 𝐹)‘((𝑗 + (2 · ((((𝑛 − 𝑗) + 1) / 2) − 1))) + 1)) − (seq𝑀( + , 𝐹)‘𝑗))) ≤ (𝐺‘(𝑗 + 1))))
114	113	simprd 495	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ((((𝑛 − 𝑗) + 1) / 2) − 1) ∈ ℕ₀) → (abs‘((seq𝑀( + , 𝐹)‘((𝑗 + (2 · ((((𝑛 − 𝑗) + 1) / 2) − 1))) + 1)) − (seq𝑀( + , 𝐹)‘𝑗))) ≤ (𝐺‘(𝑗 + 1)))
115	98, 99, 112, 114	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ (((𝑛 − 𝑗) + 1) / 2) ∈ ℤ) → (abs‘((seq𝑀( + , 𝐹)‘((𝑗 + (2 · ((((𝑛 − 𝑗) + 1) / 2) − 1))) + 1)) − (seq𝑀( + , 𝐹)‘𝑗))) ≤ (𝐺‘(𝑗 + 1)))
116	97, 115	eqbrtrd 5118	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ (((𝑛 − 𝑗) + 1) / 2) ∈ ℤ) → (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗))) ≤ (𝐺‘(𝑗 + 1)))
117		zeo 12576	. . . . . . . . . . . . . 14 ⊢ ((𝑛 − 𝑗) ∈ ℤ → (((𝑛 − 𝑗) / 2) ∈ ℤ ∨ (((𝑛 − 𝑗) + 1) / 2) ∈ ℤ))
118	46, 117	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (((𝑛 − 𝑗) / 2) ∈ ℤ ∨ (((𝑛 − 𝑗) + 1) / 2) ∈ ℤ))
119	65, 116, 118	mpjaodan 960	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗))) ≤ (𝐺‘(𝑗 + 1)))
120	1	peano2uzs 12813	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ 𝑍 → (𝑗 + 1) ∈ 𝑍)
121	120	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝑗 + 1) ∈ 𝑍)
122		ffvelcdm 7024	. . . . . . . . . . . . . 14 ⊢ ((𝐺:𝑍⟶ℝ ∧ (𝑗 + 1) ∈ 𝑍) → (𝐺‘(𝑗 + 1)) ∈ ℝ)
123	10, 121, 122	syl2an 596	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝐺‘(𝑗 + 1)) ∈ ℝ)
124	1, 5, 10, 60, 4	iseraltlem1 15603	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑗 + 1) ∈ 𝑍) → 0 ≤ (𝐺‘(𝑗 + 1)))
125	121, 124	sylan2 593	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 0 ≤ (𝐺‘(𝑗 + 1)))
126	123, 125	absidd 15344	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (abs‘(𝐺‘(𝑗 + 1))) = (𝐺‘(𝑗 + 1)))
127	119, 126	breqtrrd 5124	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗))) ≤ (abs‘(𝐺‘(𝑗 + 1))))
128	127	adantlr 715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗))) ≤ (abs‘(𝐺‘(𝑗 + 1))))
129		neg1rr 12129	. . . . . . . . . . . . . . . . . . . . 21 ⊢ -1 ∈ ℝ
130	129	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -1 ∈ ℝ)
131		neg1ne0 12130	. . . . . . . . . . . . . . . . . . . . 21 ⊢ -1 ≠ 0
132	131	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -1 ≠ 0)
133		eluzelz 12759	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → 𝑘 ∈ ℤ)
134	133, 1	eleq2s 2852	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ)
135	134	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ ℤ)
136	130, 132, 135	reexpclzd 14170	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (-1↑𝑘) ∈ ℝ)
137	10	ffvelcdmda 7027	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ)
138	136, 137	remulcld 11160	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((-1↑𝑘) · (𝐺‘𝑘)) ∈ ℝ)
139	61, 138	eqeltrd 2834	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
140	1, 5, 139	serfre 13952	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℝ)
141	1	uztrn2 12768	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑛 ∈ 𝑍)
142		ffvelcdm 7024	. . . . . . . . . . . . . . . 16 ⊢ ((seq𝑀( + , 𝐹):𝑍⟶ℝ ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℝ)
143	140, 141, 142	syl2an 596	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℝ)
144		ffvelcdm 7024	. . . . . . . . . . . . . . . 16 ⊢ ((seq𝑀( + , 𝐹):𝑍⟶ℝ ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑗) ∈ ℝ)
145	140, 43, 144	syl2an 596	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (seq𝑀( + , 𝐹)‘𝑗) ∈ ℝ)
146	143, 145	resubcld 11563	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗)) ∈ ℝ)
147	146	recnd 11158	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗)) ∈ ℂ)
148	147	abscld 15360	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗))) ∈ ℝ)
149	148	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗))) ∈ ℝ)
150	126, 123	eqeltrd 2834	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (abs‘(𝐺‘(𝑗 + 1))) ∈ ℝ)
151	150	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (abs‘(𝐺‘(𝑗 + 1))) ∈ ℝ)
152		rpre 12912	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
153	152	ad2antlr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝑥 ∈ ℝ)
154		lelttr 11221	. . . . . . . . . . 11 ⊢ (((abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗))) ∈ ℝ ∧ (abs‘(𝐺‘(𝑗 + 1))) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (((abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗))) ≤ (abs‘(𝐺‘(𝑗 + 1))) ∧ (abs‘(𝐺‘(𝑗 + 1))) < 𝑥) → (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗))) < 𝑥))
155	149, 151, 153, 154	syl3anc 1373	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (((abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗))) ≤ (abs‘(𝐺‘(𝑗 + 1))) ∧ (abs‘(𝐺‘(𝑗 + 1))) < 𝑥) → (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗))) < 𝑥))
156	128, 155	mpand 695	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((abs‘(𝐺‘(𝑗 + 1))) < 𝑥 → (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗))) < 𝑥))
157	140	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → seq𝑀( + , 𝐹):𝑍⟶ℝ)
158	157, 141, 142	syl2an 596	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℝ)
159	156, 158	jctild 525	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((abs‘(𝐺‘(𝑗 + 1))) < 𝑥 → ((seq𝑀( + , 𝐹)‘𝑛) ∈ ℝ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗))) < 𝑥)))
160	159	anassrs 467	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → ((abs‘(𝐺‘(𝑗 + 1))) < 𝑥 → ((seq𝑀( + , 𝐹)‘𝑛) ∈ ℝ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗))) < 𝑥)))
161	160	ralrimdva 3134	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → ((abs‘(𝐺‘(𝑗 + 1))) < 𝑥 → ∀𝑛 ∈ (ℤ_≥‘𝑗)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℝ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗))) < 𝑥)))
162	24, 161	syld 47	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → (∀𝑛 ∈ (ℤ_≥‘𝑗)(abs‘(𝐺‘𝑛)) < 𝑥 → ∀𝑛 ∈ (ℤ_≥‘𝑗)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℝ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗))) < 𝑥)))
163	162	reximdva 3147	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)(abs‘(𝐺‘𝑛)) < 𝑥 → ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℝ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗))) < 𝑥)))
164	163	ralimdva 3146	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)(abs‘(𝐺‘𝑛)) < 𝑥 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℝ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗))) < 𝑥)))
165	14, 164	mpd 15	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℝ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗))) < 𝑥))
166	1, 3, 165	caurcvg2 15599	1 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ≠ wne 2930 ∀wral 3049 ∃wrex 3058 Vcvv 3438 class class class wbr 5096 dom cdm 5622 ⟶wf 6486 ‘cfv 6490 (class class class)co 7356 ℂcc 11022 ℝcr 11023 0cc0 11024 1c1 11025 + caddc 11027 · cmul 11029 < clt 11164 ≤ cle 11165 − cmin 11362 -cneg 11363 / cdiv 11792 ℕcn 12143 2c2 12198 ℕ₀cn0 12399 ℤcz 12486 ℤ_≥cuz 12749 ℝ⁺crp 12903 seqcseq 13922 ↑cexp 13982 abscabs 15155 ⇝ cli 15405

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-inf2 9548 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8633 df-pm 8764 df-en 8882 df-dom 8883 df-sdom 8884 df-sup 9343 df-inf 9344 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-n0 12400 df-z 12487 df-uz 12750 df-rp 12904 df-ico 13265 df-fz 13422 df-fl 13710 df-seq 13923 df-exp 13983 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-limsup 15392 df-clim 15409 df-rlim 15410

This theorem is referenced by: leibpi 26906

W3C validator