iseralt - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > iseralt		Structured version Visualization version GIF version

Theorem iseralt 14263

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 14236.) (Contributed by Mario Carneiro, 7-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
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 12665	. . 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 14071	. . . . . . 7 ⊢ Rel ⇝
7	6	brrelexi 5082	. . . . . 6 ⊢ (𝐺 ⇝ 0 → 𝐺 ∈ V)
8	4, 7	syl 17	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ V)
9		eqidd 2611	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) = (𝐺‘𝑛))
10		iseralt.3	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝑍⟶ℝ)
11	10	ffvelrnda 6267	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) ∈ ℝ)
12	11	recnd 9947	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐺‘𝑛) ∈ ℂ)
13	1, 5, 8, 9, 12	clim0c 14086	. . . 4 ⊢ (𝜑 → (𝐺 ⇝ 0 ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)(abs‘(𝐺‘𝑛)) < 𝑥))
14	4, 13	mpbid 221	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)(abs‘(𝐺‘𝑛)) < 𝑥)
15		simpr 476	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍)
16	15, 1	syl6eleq 2698	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ_≥‘𝑀))
17		eluzelz 11573	. . . . . . . 8 ⊢ (𝑗 ∈ (ℤ_≥‘𝑀) → 𝑗 ∈ ℤ)
18		uzid 11578	. . . . . . . 8 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ_≥‘𝑗))
19	16, 17, 18	3syl 18	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ_≥‘𝑗))
20		peano2uz 11617	. . . . . . 7 ⊢ (𝑗 ∈ (ℤ_≥‘𝑗) → (𝑗 + 1) ∈ (ℤ_≥‘𝑗))
21		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑛 = (𝑗 + 1) → (𝐺‘𝑛) = (𝐺‘(𝑗 + 1)))
22	21	fveq2d 6107	. . . . . . . . 9 ⊢ (𝑛 = (𝑗 + 1) → (abs‘(𝐺‘𝑛)) = (abs‘(𝐺‘(𝑗 + 1))))
23	22	breq1d 4593	. . . . . . . 8 ⊢ (𝑛 = (𝑗 + 1) → ((abs‘(𝐺‘𝑛)) < 𝑥 ↔ (abs‘(𝐺‘(𝑗 + 1))) < 𝑥))
24	23	rspcv 3278	. . . . . . 7 ⊢ ((𝑗 + 1) ∈ (ℤ_≥‘𝑗) → (∀𝑛 ∈ (ℤ_≥‘𝑗)(abs‘(𝐺‘𝑛)) < 𝑥 → (abs‘(𝐺‘(𝑗 + 1))) < 𝑥))
25	19, 20, 24	3syl 18	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → (∀𝑛 ∈ (ℤ_≥‘𝑗)(abs‘(𝐺‘𝑛)) < 𝑥 → (abs‘(𝐺‘(𝑗 + 1))) < 𝑥))
26		eluzelz 11573	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ (ℤ_≥‘𝑗) → 𝑛 ∈ ℤ)
27	26	ad2antll 761	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝑛 ∈ ℤ)
28	27	zcnd 11359	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝑛 ∈ ℂ)
29	17, 1	eleq2s 2706	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ)
30	29	ad2antrl 760	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝑗 ∈ ℤ)
31	30	zcnd 11359	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝑗 ∈ ℂ)
32	28, 31	subcld 10271	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝑛 − 𝑗) ∈ ℂ)
33		2cnd 10970	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 2 ∈ ℂ)
34		2ne0 10990	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2 ≠ 0
35	34	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 2 ≠ 0)
36	32, 33, 35	divcan2d 10682	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (2 · ((𝑛 − 𝑗) / 2)) = (𝑛 − 𝑗))
37	36	oveq2d 6565	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝑗 + (2 · ((𝑛 − 𝑗) / 2))) = (𝑗 + (𝑛 − 𝑗)))
38	31, 28	pncan3d 10274	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝑗 + (𝑛 − 𝑗)) = 𝑛)
39	37, 38	eqtr2d 2645	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝑛 = (𝑗 + (2 · ((𝑛 − 𝑗) / 2))))
40	39	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ ((𝑛 − 𝑗) / 2) ∈ ℤ) → 𝑛 = (𝑗 + (2 · ((𝑛 − 𝑗) / 2))))
41	40	fveq2d 6107	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ ((𝑛 − 𝑗) / 2) ∈ ℤ) → (seq𝑀( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘(𝑗 + (2 · ((𝑛 − 𝑗) / 2)))))
42	41	oveq1d 6564	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ ((𝑛 − 𝑗) / 2) ∈ ℤ) → ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗)) = ((seq𝑀( + , 𝐹)‘(𝑗 + (2 · ((𝑛 − 𝑗) / 2)))) − (seq𝑀( + , 𝐹)‘𝑗)))
43	42	fveq2d 6107	. . . . . . . . . . . . . 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))))
44		simpll 786	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ ((𝑛 − 𝑗) / 2) ∈ ℤ) → 𝜑)
45		simpl 472	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑗 ∈ 𝑍)
46	45	ad2antlr 759	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ ((𝑛 − 𝑗) / 2) ∈ ℤ) → 𝑗 ∈ 𝑍)
47		simpr 476	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ ((𝑛 − 𝑗) / 2) ∈ ℤ) → ((𝑛 − 𝑗) / 2) ∈ ℤ)
48	27, 30	zsubcld 11363	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝑛 − 𝑗) ∈ ℤ)
49	48	zred 11358	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝑛 − 𝑗) ∈ ℝ)
50		eluzle 11576	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ (ℤ_≥‘𝑗) → 𝑗 ≤ 𝑛)
51	50	ad2antll 761	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝑗 ≤ 𝑛)
52	27	zred 11358	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝑛 ∈ ℝ)
53	30	zred 11358	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝑗 ∈ ℝ)
54	52, 53	subge0d 10496	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (0 ≤ (𝑛 − 𝑗) ↔ 𝑗 ≤ 𝑛))
55	51, 54	mpbird 246	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 0 ≤ (𝑛 − 𝑗))
56		2re 10967	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℝ
57	56	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 2 ∈ ℝ)
58		2pos 10989	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 < 2
59	58	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 0 < 2)
60		divge0 10771	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑛 − 𝑗) ∈ ℝ ∧ 0 ≤ (𝑛 − 𝑗)) ∧ (2 ∈ ℝ ∧ 0 < 2)) → 0 ≤ ((𝑛 − 𝑗) / 2))
61	49, 55, 57, 59, 60	syl22anc 1319	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 0 ≤ ((𝑛 − 𝑗) / 2))
62	61	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ ((𝑛 − 𝑗) / 2) ∈ ℤ) → 0 ≤ ((𝑛 − 𝑗) / 2))
63		elnn0z 11267	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 − 𝑗) / 2) ∈ ℕ₀ ↔ (((𝑛 − 𝑗) / 2) ∈ ℤ ∧ 0 ≤ ((𝑛 − 𝑗) / 2)))
64	47, 62, 63	sylanbrc 695	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ ((𝑛 − 𝑗) / 2) ∈ ℤ) → ((𝑛 − 𝑗) / 2) ∈ ℕ₀)
65		iseralt.4	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘(𝑘 + 1)) ≤ (𝐺‘𝑘))
66		iseralt.6	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = ((-1↑𝑘) · (𝐺‘𝑘)))
67	1, 5, 10, 65, 4, 66	iseraltlem3 14262	. . . . . . . . . . . . . . . 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))))
68	67	simpld 474	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ((𝑛 − 𝑗) / 2) ∈ ℕ₀) → (abs‘((seq𝑀( + , 𝐹)‘(𝑗 + (2 · ((𝑛 − 𝑗) / 2)))) − (seq𝑀( + , 𝐹)‘𝑗))) ≤ (𝐺‘(𝑗 + 1)))
69	44, 46, 64, 68	syl3anc 1318	. . . . . . . . . . . . . 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)))
70	43, 69	eqbrtrd 4605	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ ((𝑛 − 𝑗) / 2) ∈ ℤ) → (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗))) ≤ (𝐺‘(𝑗 + 1)))
71		2div2e1 11027	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (2 / 2) = 1
72	71	oveq2i 6560	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑛 − 𝑗) + 1) / 2) − (2 / 2)) = ((((𝑛 − 𝑗) + 1) / 2) − 1)
73		peano2cn 10087	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑛 − 𝑗) ∈ ℂ → ((𝑛 − 𝑗) + 1) ∈ ℂ)
74	32, 73	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((𝑛 − 𝑗) + 1) ∈ ℂ)
75	74, 33, 33, 35	divsubdird 10719	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((((𝑛 − 𝑗) + 1) − 2) / 2) = ((((𝑛 − 𝑗) + 1) / 2) − (2 / 2)))
76		df-2 10956	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 2 = (1 + 1)
77	76	oveq2i 6560	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑛 − 𝑗) + 1) − 2) = (((𝑛 − 𝑗) + 1) − (1 + 1))
78		ax-1cn 9873	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ 1 ∈ ℂ
79	78	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 1 ∈ ℂ)
80	32, 79, 79	pnpcan2d 10309	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (((𝑛 − 𝑗) + 1) − (1 + 1)) = ((𝑛 − 𝑗) − 1))
81	77, 80	syl5eq 2656	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (((𝑛 − 𝑗) + 1) − 2) = ((𝑛 − 𝑗) − 1))
82	81	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((((𝑛 − 𝑗) + 1) − 2) / 2) = (((𝑛 − 𝑗) − 1) / 2))
83	75, 82	eqtr3d 2646	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((((𝑛 − 𝑗) + 1) / 2) − (2 / 2)) = (((𝑛 − 𝑗) − 1) / 2))
84	72, 83	syl5eqr 2658	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((((𝑛 − 𝑗) + 1) / 2) − 1) = (((𝑛 − 𝑗) − 1) / 2))
85	84	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (2 · ((((𝑛 − 𝑗) + 1) / 2) − 1)) = (2 · (((𝑛 − 𝑗) − 1) / 2)))
86		subcl 10159	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑛 − 𝑗) ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 𝑗) − 1) ∈ ℂ)
87	32, 78, 86	sylancl 693	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((𝑛 − 𝑗) − 1) ∈ ℂ)
88	87, 33, 35	divcan2d 10682	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (2 · (((𝑛 − 𝑗) − 1) / 2)) = ((𝑛 − 𝑗) − 1))
89	28, 31, 79	sub32d 10303	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((𝑛 − 𝑗) − 1) = ((𝑛 − 1) − 𝑗))
90	85, 88, 89	3eqtrd 2648	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (2 · ((((𝑛 − 𝑗) + 1) / 2) − 1)) = ((𝑛 − 1) − 𝑗))
91	90	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝑗 + (2 · ((((𝑛 − 𝑗) + 1) / 2) − 1))) = (𝑗 + ((𝑛 − 1) − 𝑗)))
92		subcl 10159	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑛 − 1) ∈ ℂ)
93	28, 78, 92	sylancl 693	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝑛 − 1) ∈ ℂ)
94	31, 93	pncan3d 10274	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝑗 + ((𝑛 − 1) − 𝑗)) = (𝑛 − 1))
95	91, 94	eqtrd 2644	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝑗 + (2 · ((((𝑛 − 𝑗) + 1) / 2) − 1))) = (𝑛 − 1))
96	95	oveq1d 6564	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((𝑗 + (2 · ((((𝑛 − 𝑗) + 1) / 2) − 1))) + 1) = ((𝑛 − 1) + 1))
97		npcan 10169	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
98	28, 78, 97	sylancl 693	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((𝑛 − 1) + 1) = 𝑛)
99	96, 98	eqtr2d 2645	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝑛 = ((𝑗 + (2 · ((((𝑛 − 𝑗) + 1) / 2) − 1))) + 1))
100	99	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ (((𝑛 − 𝑗) + 1) / 2) ∈ ℤ) → 𝑛 = ((𝑗 + (2 · ((((𝑛 − 𝑗) + 1) / 2) − 1))) + 1))
101	100	fveq2d 6107	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ (((𝑛 − 𝑗) + 1) / 2) ∈ ℤ) → (seq𝑀( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘((𝑗 + (2 · ((((𝑛 − 𝑗) + 1) / 2) − 1))) + 1)))
102	101	oveq1d 6564	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ (((𝑛 − 𝑗) + 1) / 2) ∈ ℤ) → ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗)) = ((seq𝑀( + , 𝐹)‘((𝑗 + (2 · ((((𝑛 − 𝑗) + 1) / 2) − 1))) + 1)) − (seq𝑀( + , 𝐹)‘𝑗)))
103	102	fveq2d 6107	. . . . . . . . . . . . . 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))))
104		simpll 786	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ (((𝑛 − 𝑗) + 1) / 2) ∈ ℤ) → 𝜑)
105	45	ad2antlr 759	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ (((𝑛 − 𝑗) + 1) / 2) ∈ ℤ) → 𝑗 ∈ 𝑍)
106		simpr 476	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ (((𝑛 − 𝑗) + 1) / 2) ∈ ℤ) → (((𝑛 − 𝑗) + 1) / 2) ∈ ℤ)
107		uznn0sub 11595	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ (ℤ_≥‘𝑗) → (𝑛 − 𝑗) ∈ ℕ₀)
108	107	ad2antll 761	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝑛 − 𝑗) ∈ ℕ₀)
109		nn0p1nn 11209	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛 − 𝑗) ∈ ℕ₀ → ((𝑛 − 𝑗) + 1) ∈ ℕ)
110	108, 109	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((𝑛 − 𝑗) + 1) ∈ ℕ)
111	110	nnrpd 11746	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((𝑛 − 𝑗) + 1) ∈ ℝ⁺)
112	111	rphalfcld 11760	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (((𝑛 − 𝑗) + 1) / 2) ∈ ℝ⁺)
113	112	rpgt0d 11751	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 0 < (((𝑛 − 𝑗) + 1) / 2))
114	113	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ (((𝑛 − 𝑗) + 1) / 2) ∈ ℤ) → 0 < (((𝑛 − 𝑗) + 1) / 2))
115		elnnz 11264	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑛 − 𝑗) + 1) / 2) ∈ ℕ ↔ ((((𝑛 − 𝑗) + 1) / 2) ∈ ℤ ∧ 0 < (((𝑛 − 𝑗) + 1) / 2)))
116	106, 114, 115	sylanbrc 695	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ (((𝑛 − 𝑗) + 1) / 2) ∈ ℤ) → (((𝑛 − 𝑗) + 1) / 2) ∈ ℕ)
117		nnm1nn0 11211	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑛 − 𝑗) + 1) / 2) ∈ ℕ → ((((𝑛 − 𝑗) + 1) / 2) − 1) ∈ ℕ₀)
118	116, 117	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ (((𝑛 − 𝑗) + 1) / 2) ∈ ℤ) → ((((𝑛 − 𝑗) + 1) / 2) − 1) ∈ ℕ₀)
119	1, 5, 10, 65, 4, 66	iseraltlem3 14262	. . . . . . . . . . . . . . . 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))))
120	119	simprd 478	. . . . . . . . . . . . . . 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)))
121	104, 105, 118, 120	syl3anc 1318	. . . . . . . . . . . . . 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)))
122	103, 121	eqbrtrd 4605	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) ∧ (((𝑛 − 𝑗) + 1) / 2) ∈ ℤ) → (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗))) ≤ (𝐺‘(𝑗 + 1)))
123		zeo 11339	. . . . . . . . . . . . . 14 ⊢ ((𝑛 − 𝑗) ∈ ℤ → (((𝑛 − 𝑗) / 2) ∈ ℤ ∨ (((𝑛 − 𝑗) + 1) / 2) ∈ ℤ))
124	48, 123	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (((𝑛 − 𝑗) / 2) ∈ ℤ ∨ (((𝑛 − 𝑗) + 1) / 2) ∈ ℤ))
125	70, 122, 124	mpjaodan 823	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗))) ≤ (𝐺‘(𝑗 + 1)))
126	1	peano2uzs 11618	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ 𝑍 → (𝑗 + 1) ∈ 𝑍)
127	126	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → (𝑗 + 1) ∈ 𝑍)
128		ffvelrn 6265	. . . . . . . . . . . . . 14 ⊢ ((𝐺:𝑍⟶ℝ ∧ (𝑗 + 1) ∈ 𝑍) → (𝐺‘(𝑗 + 1)) ∈ ℝ)
129	10, 127, 128	syl2an 493	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (𝐺‘(𝑗 + 1)) ∈ ℝ)
130	1, 5, 10, 65, 4	iseraltlem1 14260	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑗 + 1) ∈ 𝑍) → 0 ≤ (𝐺‘(𝑗 + 1)))
131	127, 130	sylan2 490	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 0 ≤ (𝐺‘(𝑗 + 1)))
132	129, 131	absidd 14009	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (abs‘(𝐺‘(𝑗 + 1))) = (𝐺‘(𝑗 + 1)))
133	125, 132	breqtrrd 4611	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗))) ≤ (abs‘(𝐺‘(𝑗 + 1))))
134	133	adantlr 747	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗))) ≤ (abs‘(𝐺‘(𝑗 + 1))))
135		neg1rr 11002	. . . . . . . . . . . . . . . . . . . . 21 ⊢ -1 ∈ ℝ
136	135	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -1 ∈ ℝ)
137		neg1ne0 11003	. . . . . . . . . . . . . . . . . . . . 21 ⊢ -1 ≠ 0
138	137	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -1 ≠ 0)
139		eluzelz 11573	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → 𝑘 ∈ ℤ)
140	139, 1	eleq2s 2706	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ)
141	140	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ ℤ)
142	136, 138, 141	reexpclzd 12896	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (-1↑𝑘) ∈ ℝ)
143	10	ffvelrnda 6267	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ)
144	142, 143	remulcld 9949	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((-1↑𝑘) · (𝐺‘𝑘)) ∈ ℝ)
145	66, 144	eqeltrd 2688	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
146	1, 5, 145	serfre 12692	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℝ)
147	1	uztrn2 11581	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑛 ∈ 𝑍)
148		ffvelrn 6265	. . . . . . . . . . . . . . . 16 ⊢ ((seq𝑀( + , 𝐹):𝑍⟶ℝ ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℝ)
149	146, 147, 148	syl2an 493	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℝ)
150		ffvelrn 6265	. . . . . . . . . . . . . . . 16 ⊢ ((seq𝑀( + , 𝐹):𝑍⟶ℝ ∧ 𝑗 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑗) ∈ ℝ)
151	146, 45, 150	syl2an 493	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (seq𝑀( + , 𝐹)‘𝑗) ∈ ℝ)
152	149, 151	resubcld 10337	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗)) ∈ ℝ)
153	152	recnd 9947	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗)) ∈ ℂ)
154	153	abscld 14023	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗))) ∈ ℝ)
155	154	adantlr 747	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗))) ∈ ℝ)
156	132, 129	eqeltrd 2688	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (abs‘(𝐺‘(𝑗 + 1))) ∈ ℝ)
157	156	adantlr 747	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (abs‘(𝐺‘(𝑗 + 1))) ∈ ℝ)
158		rpre 11715	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
159	158	ad2antlr 759	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → 𝑥 ∈ ℝ)
160		lelttr 10007	. . . . . . . . . . 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))
161	155, 157, 159, 160	syl3anc 1318	. . . . . . . . . 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))
162	134, 161	mpand 707	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → ((abs‘(𝐺‘(𝑗 + 1))) < 𝑥 → (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗))) < 𝑥))
163	146	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → seq𝑀( + , 𝐹):𝑍⟶ℝ)
164	163, 147, 148	syl2an 493	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗))) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℝ)
165	162, 164	jctild 564	. . . . . . . 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)))
166	165	anassrs 678	. . . . . . 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)))
167	166	ralrimdva 2952	. . . . . 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)))
168	25, 167	syld 46	. . . . 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)))
169	168	reximdva 3000	. . . 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)))
170	169	ralimdva 2945	. . 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)))
171	14, 170	mpd 15	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℝ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑗))) < 𝑥))
172	1, 3, 171	caurcvg2 14256	1 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 Vcvv 3173 class class class wbr 4583 dom cdm 5038 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 ≤ cle 9954 − cmin 10145 -cneg 10146 / cdiv 10563 ℕcn 10897 2c2 10947 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 ℝ⁺crp 11708 seqcseq 12663 ↑cexp 12722 abscabs 13822 ⇝ cli 14063

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-ico 12052 df-fz 12198 df-fl 12455 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068

This theorem is referenced by: leibpi 24469

W3C validator