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Theorem iseralt 15592

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 15565.) (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 13910	. . 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 15399	. . . . . . 7 ⊢ Rel ⇝
7	6	brrelex1i 5670	. . . . . 6 ⊢ (𝐺 ⇝ 0 → 𝐺 ∈ V)
8	4, 7	syl 17	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ V)
9		eqidd 2732	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) = (𝐺‘𝑛))
10		iseralt.3	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝑍⟶ℝ)
11	10	ffvelcdmda 7017	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) ∈ ℝ)
12	11	recnd 11140	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) ∈ ℂ)
13	1, 5, 8, 9, 12	clim0c 15414	. . . 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 2841	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ_≥‘𝑀))
17		eluzelz 12742	. . . . . . . 8 ⊢ (𝑗 ∈ (ℤ_≥‘𝑀) → 𝑗 ∈ ℤ)
18		uzid 12747	. . . . . . . 8 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ_≥‘𝑗))
19	16, 17, 18	3syl 18	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ_≥‘𝑗))
20		peano2uz 12799	. . . . . . 7 ⊢ (𝑗 ∈ (ℤ_≥‘𝑗) → (𝑗 + 1) ∈ (ℤ_≥‘𝑗))
21		2fveq3 6827	. . . . . . . . 9 ⊢ (𝑛 = (𝑗 + 1) → (abs‘(𝐺‘𝑛)) = (abs‘(𝐺‘(𝑗 + 1))))
22	21	breq1d 5099	. . . . . . . 8 ⊢ (𝑛 = (𝑗 + 1) → ((abs‘(𝐺‘𝑛)) < 𝑥 ↔ (abs‘(𝐺‘(𝑗 + 1))) < 𝑥))
23	22	rspcv 3568	. . . . . . 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 12742	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ (ℤ_≥‘𝑗) → 𝑛 ∈ ℤ)
26	25	ad2antll 729	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝑛 ∈ ℤ)
27	26	zcnd 12578	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝑛 ∈ ℂ)
28	17, 1	eleq2s 2849	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ)
29	28	ad2antrl 728	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝑗 ∈ ℤ)
30	29	zcnd 12578	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝑗 ∈ ℂ)
31	27, 30	subcld 11472	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝑛 − 𝑗) ∈ ℂ)
32		2cnd 12203	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 2 ∈ ℂ)
33		2ne0 12229	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ≠ 0
34	33	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 2 ≠ 0)
35	31, 32, 34	divcan2d 11899	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (2 · ((𝑛 − 𝑗) / 2)) = (𝑛 − 𝑗))
36	35	oveq2d 7362	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝑗 + (2 · ((𝑛 − 𝑗) / 2))) = (𝑗 + (𝑛 − 𝑗)))
37	30, 27	pncan3d 11475	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝑗 + (𝑛 − 𝑗)) = 𝑛)
38	36, 37	eqtr2d 2767	. . . . . . . . . . . . . . . . 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 6826	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ ((𝑛 − 𝑗) / 2) ∈ ℤ) → (seq𝑀( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘(𝑗 + (2 · ((𝑛 − 𝑗) / 2)))))
41	40	fvoveq1d 7368	. . . . . . . . . . . . . 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 12582	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝑛 − 𝑗) ∈ ℤ)
47	46	zred 12577	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝑛 − 𝑗) ∈ ℝ)
48		2rp 12895	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℝ⁺
49	48	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 2 ∈ ℝ⁺)
50		eluzle 12745	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ (ℤ_≥‘𝑗) → 𝑗 ≤ 𝑛)
51	50	ad2antll 729	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝑗 ≤ 𝑛)
52	26	zred 12577	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝑛 ∈ ℝ)
53	29	zred 12577	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝑗 ∈ ℝ)
54	52, 53	subge0d 11707	. . . . . . . . . . . . . . . . . . 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 12974	. . . . . . . . . . . . . . . . 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 12481	. . . . . . . . . . . . . . . 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 15591	. . . . . . . . . . . . . . . 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 5111	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ ((𝑛 − 𝑗) / 2) ∈ ℤ) → (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗))) ≤ (𝐺‘(𝑗 + 1)))
66		2div2e1 12261	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (2 / 2) = 1
67	66	oveq2i 7357	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑛 − 𝑗) + 1) / 2) − (2 / 2)) = ((((𝑛 − 𝑗) + 1) / 2) − 1)
68		peano2cn 11285	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 11936	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((((𝑛 − 𝑗) + 1) − 2) / 2) = ((((𝑛 − 𝑗) + 1) / 2) − (2 / 2)))
71		df-2 12188	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 2 = (1 + 1)
72	71	oveq2i 7357	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑛 − 𝑗) + 1) − 2) = (((𝑛 − 𝑗) + 1) − (1 + 1))
73		ax-1cn 11064	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 1 ∈ ℂ
74	73	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 1 ∈ ℂ)
75	31, 74, 74	pnpcan2d 11510	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (((𝑛 − 𝑗) + 1) − (1 + 1)) = ((𝑛 − 𝑗) − 1))
76	72, 75	eqtrid 2778	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (((𝑛 − 𝑗) + 1) − 2) = ((𝑛 − 𝑗) − 1))
77	76	oveq1d 7361	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((((𝑛 − 𝑗) + 1) − 2) / 2) = (((𝑛 − 𝑗) − 1) / 2))
78	70, 77	eqtr3d 2768	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((((𝑛 − 𝑗) + 1) / 2) − (2 / 2)) = (((𝑛 − 𝑗) − 1) / 2))
79	67, 78	eqtr3id 2780	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((((𝑛 − 𝑗) + 1) / 2) − 1) = (((𝑛 − 𝑗) − 1) / 2))
80	79	oveq2d 7362	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (2 · ((((𝑛 − 𝑗) + 1) / 2) − 1)) = (2 · (((𝑛 − 𝑗) − 1) / 2)))
81		subcl 11359	. . . . . . . . . . . . . . . . . . . . . . . 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 11899	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (2 · (((𝑛 − 𝑗) − 1) / 2)) = ((𝑛 − 𝑗) − 1))
84	27, 30, 74	sub32d 11504	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((𝑛 − 𝑗) − 1) = ((𝑛 − 1) − 𝑗))
85	80, 83, 84	3eqtrd 2770	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (2 · ((((𝑛 − 𝑗) + 1) / 2) − 1)) = ((𝑛 − 1) − 𝑗))
86	85	oveq2d 7362	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝑗 + (2 · ((((𝑛 − 𝑗) + 1) / 2) − 1))) = (𝑗 + ((𝑛 − 1) − 𝑗)))
87		subcl 11359	. . . . . . . . . . . . . . . . . . . . . 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 11475	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝑗 + ((𝑛 − 1) − 𝑗)) = (𝑛 − 1))
90	86, 89	eqtrd 2766	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝑗 + (2 · ((((𝑛 − 𝑗) + 1) / 2) − 1))) = (𝑛 − 1))
91	90	oveq1d 7361	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((𝑗 + (2 · ((((𝑛 − 𝑗) + 1) / 2) − 1))) + 1) = ((𝑛 − 1) + 1))
92		npcan 11369	. . . . . . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . . . . 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 6826	. . . . . . . . . . . . . . 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 7368	. . . . . . . . . . . . . 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 12771	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ (ℤ_≥‘𝑗) → (𝑛 − 𝑗) ∈ ℕ₀)
102	101	ad2antll 729	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝑛 − 𝑗) ∈ ℕ₀)
103		nn0p1nn 12420	. . . . . . . . . . . . . . . . . . . . . 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 12932	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((𝑛 − 𝑗) + 1) ∈ ℝ⁺)
106	105	rphalfcld 12946	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (((𝑛 − 𝑗) + 1) / 2) ∈ ℝ⁺)
107	106	rpgt0d 12937	. . . . . . . . . . . . . . . . . 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 12478	. . . . . . . . . . . . . . . . 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 12422	. . . . . . . . . . . . . . . 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 15591	. . . . . . . . . . . . . . . 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 5111	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ (((𝑛 − 𝑗) + 1) / 2) ∈ ℤ) → (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗))) ≤ (𝐺‘(𝑗 + 1)))
117		zeo 12559	. . . . . . . . . . . . . 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 12800	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ 𝑍 → (𝑗 + 1) ∈ 𝑍)
121	120	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝑗 + 1) ∈ 𝑍)
122		ffvelcdm 7014	. . . . . . . . . . . . . 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 15589	. . . . . . . . . . . . . 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 15330	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (abs‘(𝐺‘(𝑗 + 1))) = (𝐺‘(𝑗 + 1)))
127	119, 126	breqtrrd 5117	. . . . . . . . . . 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 12111	. . . . . . . . . . . . . . . . . . . . 21 ⊢ -1 ∈ ℝ
130	129	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -1 ∈ ℝ)
131		neg1ne0 12112	. . . . . . . . . . . . . . . . . . . . 21 ⊢ -1 ≠ 0
132	131	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -1 ≠ 0)
133		eluzelz 12742	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → 𝑘 ∈ ℤ)
134	133, 1	eleq2s 2849	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ)
135	134	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ ℤ)
136	130, 132, 135	reexpclzd 14156	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (-1↑𝑘) ∈ ℝ)
137	10	ffvelcdmda 7017	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ)
138	136, 137	remulcld 11142	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((-1↑𝑘) · (𝐺‘𝑘)) ∈ ℝ)
139	61, 138	eqeltrd 2831	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
140	1, 5, 139	serfre 13938	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℝ)
141	1	uztrn2 12751	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑛 ∈ 𝑍)
142		ffvelcdm 7014	. . . . . . . . . . . . . . . 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 7014	. . . . . . . . . . . . . . . 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 11545	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗)) ∈ ℝ)
147	146	recnd 11140	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗)) ∈ ℂ)
148	147	abscld 15346	. . . . . . . . . . . 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 2831	. . . . . . . . . . . 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 12899	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
153	152	ad2antlr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝑥 ∈ ℝ)
154		lelttr 11203	. . . . . . . . . . 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 3132	. . . . . 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 3145	. . . 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 3144	. . 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 15585	1 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 ∃wrex 3056 Vcvv 3436 class class class wbr 5089 dom cdm 5614 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 ℂcc 11004 ℝcr 11005 0cc0 11006 1c1 11007 + caddc 11009 · cmul 11011 < clt 11146 ≤ cle 11147 − cmin 11344 -cneg 11345 / cdiv 11774 ℕcn 12125 2c2 12180 ℕ₀cn0 12381 ℤcz 12468 ℤ_≥cuz 12732 ℝ⁺crp 12890 seqcseq 13908 ↑cexp 13968 abscabs 15141 ⇝ cli 15391

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 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-inf2 9531 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084

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 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-er 8622 df-pm 8753 df-en 8870 df-dom 8871 df-sdom 8872 df-sup 9326 df-inf 9327 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-n0 12382 df-z 12469 df-uz 12733 df-rp 12891 df-ico 13251 df-fz 13408 df-fl 13696 df-seq 13909 df-exp 13969 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396

This theorem is referenced by: leibpi 26879

W3C validator