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Theorem cvgratz 11587

Description: Ratio test for convergence of a complex infinite series. If the ratio 𝐴 of the absolute values of successive terms in an infinite sequence 𝐹 is less than 1 for all terms, then the infinite sum of the terms of 𝐹 converges to a complex number. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 11-Nov-2022.)

**Hypotheses**
Ref	Expression
cvgratz.1	⊢ 𝑍 = (ℤ_≥‘𝑀)
cvgratz.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
cvgratz.3	⊢ (𝜑 → 𝐴 ∈ ℝ)
cvgratz.4	⊢ (𝜑 → 𝐴 < 1)
cvgratz.gt0	⊢ (𝜑 → 0 < 𝐴)
cvgratz.6	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
cvgratz.7	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))

**Assertion**
Ref	Expression
cvgratz	⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )

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

Proof of Theorem cvgratz Dummy variables 𝑖 𝑥 𝑦 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvgratz.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
2	1	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 1 ≤ 𝑀) → 𝑀 ∈ ℤ)
3		fveq2 5541	. . . . . 6 ⊢ (𝑘 = 𝑥 → (𝐹‘𝑘) = (𝐹‘𝑥))
4	3	eleq1d 2258	. . . . 5 ⊢ (𝑘 = 𝑥 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑥) ∈ ℂ))
5		cvgratz.6	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
6	5	ralrimiva 2563	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ)
7	6	ad2antrr 488	. . . . 5 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ)
8		cvgratz.1	. . . . . . . 8 ⊢ 𝑍 = (ℤ_≥‘𝑀)
9	8	eleq2i 2256	. . . . . . 7 ⊢ (𝑥 ∈ 𝑍 ↔ 𝑥 ∈ (ℤ_≥‘𝑀))
10	9	biimpri 133	. . . . . 6 ⊢ (𝑥 ∈ (ℤ_≥‘𝑀) → 𝑥 ∈ 𝑍)
11	10	adantl 277	. . . . 5 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → 𝑥 ∈ 𝑍)
12	4, 7, 11	rspcdva 2865	. . . 4 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑥) ∈ ℂ)
13		eluzelz 9578	. . . . . . . 8 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → 𝑘 ∈ ℤ)
14	13	adantl 277	. . . . . . 7 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝑘 ∈ ℤ)
15		1red 8013	. . . . . . . 8 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 1 ∈ ℝ)
16	1	zred 9416	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℝ)
17	16	ad2antrr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝑀 ∈ ℝ)
18	14	zred 9416	. . . . . . . 8 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝑘 ∈ ℝ)
19		simplr 528	. . . . . . . 8 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 1 ≤ 𝑀)
20		eluzle 9581	. . . . . . . . 9 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → 𝑀 ≤ 𝑘)
21	20	adantl 277	. . . . . . . 8 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝑀 ≤ 𝑘)
22	15, 17, 18, 19, 21	letrd 8122	. . . . . . 7 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 1 ≤ 𝑘)
23		elnnz1 9317	. . . . . . 7 ⊢ (𝑘 ∈ ℕ ↔ (𝑘 ∈ ℤ ∧ 1 ≤ 𝑘))
24	14, 22, 23	sylanbrc 417	. . . . . 6 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → 𝑘 ∈ ℕ)
25		elnnuz 9606	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ_≥‘1))
26		fveq2 5541	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀))
27	26	eleq1d 2258	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑀) ∈ ℂ))
28		uzid 9583	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ_≥‘𝑀))
29	1, 28	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝑀))
30	29, 8	eleqtrrdi 2283	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
31	27, 6, 30	rspcdva 2865	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ℂ)
32	31	ad3antrrr 492	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ 𝑘 < 𝑀) → (𝐹‘𝑀) ∈ ℂ)
33		cvgratz.3	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ ℝ)
34		cvgratz.gt0	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 𝐴)
35	33, 34	elrpd 9735	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
36	35	ad3antrrr 492	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ 𝑘 < 𝑀) → 𝐴 ∈ ℝ⁺)
37	2	adantr 276	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘1)) → 𝑀 ∈ ℤ)
38	37	adantr 276	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ 𝑘 < 𝑀) → 𝑀 ∈ ℤ)
39	25	biimpri 133	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (ℤ_≥‘1) → 𝑘 ∈ ℕ)
40	39	adantl 277	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘1)) → 𝑘 ∈ ℕ)
41	40	nnzd 9415	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘1)) → 𝑘 ∈ ℤ)
42	41	adantr 276	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ 𝑘 < 𝑀) → 𝑘 ∈ ℤ)
43	38, 42	zsubcld 9421	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ 𝑘 < 𝑀) → (𝑀 − 𝑘) ∈ ℤ)
44	36, 43	rpexpcld 10724	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ 𝑘 < 𝑀) → (𝐴↑(𝑀 − 𝑘)) ∈ ℝ⁺)
45	44	rpcnd 9740	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ 𝑘 < 𝑀) → (𝐴↑(𝑀 − 𝑘)) ∈ ℂ)
46	44	rpap0d 9744	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ 𝑘 < 𝑀) → (𝐴↑(𝑀 − 𝑘)) # 0)
47	32, 45, 46	divclapd 8788	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ 𝑘 < 𝑀) → ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))) ∈ ℂ)
48		simplll 533	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑘 < 𝑀) → 𝜑)
49	37	adantr 276	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑘 < 𝑀) → 𝑀 ∈ ℤ)
50	41	adantr 276	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑘 < 𝑀) → 𝑘 ∈ ℤ)
51	16	ad3antrrr 492	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑘 < 𝑀) → 𝑀 ∈ ℝ)
52	50	zred 9416	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑘 < 𝑀) → 𝑘 ∈ ℝ)
53		simpr 110	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑘 < 𝑀) → ¬ 𝑘 < 𝑀)
54	51, 52, 53	nltled 8119	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑘 < 𝑀) → 𝑀 ≤ 𝑘)
55		eluz2 9575	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘))
56	49, 50, 54, 55	syl3anbrc 1183	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑘 < 𝑀) → 𝑘 ∈ (ℤ_≥‘𝑀))
57	56, 8	eleqtrrdi 2283	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑘 < 𝑀) → 𝑘 ∈ 𝑍)
58	48, 57, 5	syl2anc 411	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑘 < 𝑀) → (𝐹‘𝑘) ∈ ℂ)
59		zdclt 9371	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℤ ∧ 𝑀 ∈ ℤ) → DECID 𝑘 < 𝑀)
60	41, 37, 59	syl2anc 411	. . . . . . . . 9 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘1)) → DECID 𝑘 < 𝑀)
61	47, 58, 60	ifcldadc 3582	. . . . . . . 8 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘1)) → if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)) ∈ ℂ)
62	25, 61	sylan2b 287	. . . . . . 7 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)) ∈ ℂ)
63	24, 62	syldan 282	. . . . . 6 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)) ∈ ℂ)
64		breq1 4028	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → (𝑖 < 𝑀 ↔ 𝑘 < 𝑀))
65		oveq2 5912	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (𝑀 − 𝑖) = (𝑀 − 𝑘))
66	65	oveq2d 5920	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (𝐴↑(𝑀 − 𝑖)) = (𝐴↑(𝑀 − 𝑘)))
67	66	oveq2d 5920	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑖))) = ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))))
68		fveq2 5541	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → (𝐹‘𝑖) = (𝐹‘𝑘))
69	64, 67, 68	ifbieq12d 3579	. . . . . . 7 ⊢ (𝑖 = 𝑘 → if(𝑖 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑖))), (𝐹‘𝑖)) = if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)))
70		eqid 2189	. . . . . . 7 ⊢ (𝑖 ∈ ℕ ↦ if(𝑖 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑖))), (𝐹‘𝑖))) = (𝑖 ∈ ℕ ↦ if(𝑖 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑖))), (𝐹‘𝑖)))
71	69, 70	fvmptg 5620	. . . . . 6 ⊢ ((𝑘 ∈ ℕ ∧ if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)) ∈ ℂ) → ((𝑖 ∈ ℕ ↦ if(𝑖 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑖))), (𝐹‘𝑖)))‘𝑘) = if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)))
72	24, 63, 71	syl2anc 411	. . . . 5 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ((𝑖 ∈ ℕ ↦ if(𝑖 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑖))), (𝐹‘𝑖)))‘𝑘) = if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)))
73	17, 18, 21	lensymd 8120	. . . . . 6 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → ¬ 𝑘 < 𝑀)
74	73	iffalsed 3563	. . . . 5 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)) = (𝐹‘𝑘))
75	72, 74	eqtr2d 2223	. . . 4 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) = ((𝑖 ∈ ℕ ↦ if(𝑖 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑖))), (𝐹‘𝑖)))‘𝑘))
76		addcl 7976	. . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ)
77	76	adantl 277	. . . 4 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 + 𝑦) ∈ ℂ)
78	2, 12, 75, 77	seq3feq 10517	. . 3 ⊢ ((𝜑 ∧ 1 ≤ 𝑀) → seq𝑀( + , 𝐹) = seq𝑀( + , (𝑖 ∈ ℕ ↦ if(𝑖 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑖))), (𝐹‘𝑖)))))
79	33	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 1 ≤ 𝑀) → 𝐴 ∈ ℝ)
80		cvgratz.4	. . . . . 6 ⊢ (𝜑 → 𝐴 < 1)
81	80	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 1 ≤ 𝑀) → 𝐴 < 1)
82	34	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 1 ≤ 𝑀) → 0 < 𝐴)
83	71	eleq1d 2258	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)) ∈ ℂ) → (((𝑖 ∈ ℕ ↦ if(𝑖 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑖))), (𝐹‘𝑖)))‘𝑘) ∈ ℂ ↔ if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)) ∈ ℂ))
84	40, 61, 83	syl2anc 411	. . . . . . 7 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘1)) → (((𝑖 ∈ ℕ ↦ if(𝑖 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑖))), (𝐹‘𝑖)))‘𝑘) ∈ ℂ ↔ if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)) ∈ ℂ))
85	61, 84	mpbird 167	. . . . . 6 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ (ℤ_≥‘1)) → ((𝑖 ∈ ℕ ↦ if(𝑖 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑖))), (𝐹‘𝑖)))‘𝑘) ∈ ℂ)
86	25, 85	sylan2b 287	. . . . 5 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → ((𝑖 ∈ ℕ ↦ if(𝑖 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑖))), (𝐹‘𝑖)))‘𝑘) ∈ ℂ)
87	31	ad3antrrr 492	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) < 𝑀) → (𝐹‘𝑀) ∈ ℂ)
88	35	ad3antrrr 492	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) < 𝑀) → 𝐴 ∈ ℝ⁺)
89	2	ad2antrr 488	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) < 𝑀) → 𝑀 ∈ ℤ)
90	25, 41	sylan2b 287	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℤ)
91	90	adantr 276	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) < 𝑀) → 𝑘 ∈ ℤ)
92	91	peano2zd 9419	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) < 𝑀) → (𝑘 + 1) ∈ ℤ)
93	89, 92	zsubcld 9421	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) < 𝑀) → (𝑀 − (𝑘 + 1)) ∈ ℤ)
94	88, 93	rpexpcld 10724	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) < 𝑀) → (𝐴↑(𝑀 − (𝑘 + 1))) ∈ ℝ⁺)
95	94	rpcnd 9740	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) < 𝑀) → (𝐴↑(𝑀 − (𝑘 + 1))) ∈ ℂ)
96	94	rpap0d 9744	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) < 𝑀) → (𝐴↑(𝑀 − (𝑘 + 1))) # 0)
97	87, 95, 96	divclapd 8788	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) < 𝑀) → ((𝐹‘𝑀) / (𝐴↑(𝑀 − (𝑘 + 1)))) ∈ ℂ)
98		fveq2 5541	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝑘 + 1) → (𝐹‘𝑎) = (𝐹‘(𝑘 + 1)))
99	98	eleq1d 2258	. . . . . . . . . . 11 ⊢ (𝑎 = (𝑘 + 1) → ((𝐹‘𝑎) ∈ ℂ ↔ (𝐹‘(𝑘 + 1)) ∈ ℂ))
100		fveq2 5541	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑎 → (𝐹‘𝑘) = (𝐹‘𝑎))
101	100	eleq1d 2258	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑎 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑎) ∈ ℂ))
102	101	cbvralv 2722	. . . . . . . . . . . . 13 ⊢ (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ ↔ ∀𝑎 ∈ 𝑍 (𝐹‘𝑎) ∈ ℂ)
103	6, 102	sylib 122	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑎 ∈ 𝑍 (𝐹‘𝑎) ∈ ℂ)
104	103	ad3antrrr 492	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ ¬ (𝑘 + 1) < 𝑀) → ∀𝑎 ∈ 𝑍 (𝐹‘𝑎) ∈ ℂ)
105	2	ad2antrr 488	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ ¬ (𝑘 + 1) < 𝑀) → 𝑀 ∈ ℤ)
106		peano2nn 8972	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
107	106	adantl 277	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
108	107	nnzd 9415	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℤ)
109	108	adantr 276	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ ¬ (𝑘 + 1) < 𝑀) → (𝑘 + 1) ∈ ℤ)
110	16	ad3antrrr 492	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ ¬ (𝑘 + 1) < 𝑀) → 𝑀 ∈ ℝ)
111	107	nnred 8973	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℝ)
112	111	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ ¬ (𝑘 + 1) < 𝑀) → (𝑘 + 1) ∈ ℝ)
113		simpr 110	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ ¬ (𝑘 + 1) < 𝑀) → ¬ (𝑘 + 1) < 𝑀)
114	110, 112, 113	nltled 8119	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ ¬ (𝑘 + 1) < 𝑀) → 𝑀 ≤ (𝑘 + 1))
115		eluz2 9575	. . . . . . . . . . . . 13 ⊢ ((𝑘 + 1) ∈ (ℤ_≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ (𝑘 + 1) ∈ ℤ ∧ 𝑀 ≤ (𝑘 + 1)))
116	105, 109, 114, 115	syl3anbrc 1183	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ ¬ (𝑘 + 1) < 𝑀) → (𝑘 + 1) ∈ (ℤ_≥‘𝑀))
117	116, 8	eleqtrrdi 2283	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ ¬ (𝑘 + 1) < 𝑀) → (𝑘 + 1) ∈ 𝑍)
118	99, 104, 117	rspcdva 2865	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ ¬ (𝑘 + 1) < 𝑀) → (𝐹‘(𝑘 + 1)) ∈ ℂ)
119	2	adantr 276	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → 𝑀 ∈ ℤ)
120		zdclt 9371	. . . . . . . . . . 11 ⊢ (((𝑘 + 1) ∈ ℤ ∧ 𝑀 ∈ ℤ) → DECID (𝑘 + 1) < 𝑀)
121	108, 119, 120	syl2anc 411	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → DECID (𝑘 + 1) < 𝑀)
122	97, 118, 121	ifcldadc 3582	. . . . . . . . 9 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → if((𝑘 + 1) < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − (𝑘 + 1)))), (𝐹‘(𝑘 + 1))) ∈ ℂ)
123	122	abscld 11237	. . . . . . . 8 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → (abs‘if((𝑘 + 1) < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − (𝑘 + 1)))), (𝐹‘(𝑘 + 1)))) ∈ ℝ)
124	16	recnd 8027	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑀 ∈ ℂ)
125	124	ad2antrr 488	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → 𝑀 ∈ ℂ)
126		simpr 110	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
127	126	nncnd 8974	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℂ)
128		1cnd 8014	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → 1 ∈ ℂ)
129	125, 127, 128	subsub4d 8340	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → ((𝑀 − 𝑘) − 1) = (𝑀 − (𝑘 + 1)))
130	129	oveq2d 5920	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → (𝐴↑((𝑀 − 𝑘) − 1)) = (𝐴↑(𝑀 − (𝑘 + 1))))
131	33	recnd 8027	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ ℂ)
132	131	ad2antrr 488	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℂ)
133	33, 34	gt0ap0d 8627	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 # 0)
134	133	ad2antrr 488	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → 𝐴 # 0)
135	119, 90	zsubcld 9421	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → (𝑀 − 𝑘) ∈ ℤ)
136	132, 134, 135	expm1apd 10710	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → (𝐴↑((𝑀 − 𝑘) − 1)) = ((𝐴↑(𝑀 − 𝑘)) / 𝐴))
137	130, 136	eqtr3d 2224	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → (𝐴↑(𝑀 − (𝑘 + 1))) = ((𝐴↑(𝑀 − 𝑘)) / 𝐴))
138	137	oveq2d 5920	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑀) / (𝐴↑(𝑀 − (𝑘 + 1)))) = ((𝐹‘𝑀) / ((𝐴↑(𝑀 − 𝑘)) / 𝐴)))
139	138	adantr 276	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) < 𝑀) → ((𝐹‘𝑀) / (𝐴↑(𝑀 − (𝑘 + 1)))) = ((𝐹‘𝑀) / ((𝐴↑(𝑀 − 𝑘)) / 𝐴)))
140		simpr 110	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) < 𝑀) → (𝑘 + 1) < 𝑀)
141	140	iftrued 3560	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) < 𝑀) → if((𝑘 + 1) < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − (𝑘 + 1)))), (𝐹‘(𝑘 + 1))) = ((𝐹‘𝑀) / (𝐴↑(𝑀 − (𝑘 + 1)))))
142	126	nnred 8973	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ)
143	142	adantr 276	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) < 𝑀) → 𝑘 ∈ ℝ)
144		peano2re 8134	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
145	143, 144	syl 14	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) < 𝑀) → (𝑘 + 1) ∈ ℝ)
146	16	ad3antrrr 492	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) < 𝑀) → 𝑀 ∈ ℝ)
147	143	ltp1d 8928	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) < 𝑀) → 𝑘 < (𝑘 + 1))
148	143, 145, 146, 147, 140	lttrd 8124	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) < 𝑀) → 𝑘 < 𝑀)
149	148	iftrued 3560	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) < 𝑀) → if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)) = ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))))
150	149	oveq2d 5920	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) < 𝑀) → (𝐴 · if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘))) = (𝐴 · ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘)))))
151	31	ad2antrr 488	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑀) ∈ ℂ)
152	132, 134, 135	expclzapd 10705	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → (𝐴↑(𝑀 − 𝑘)) ∈ ℂ)
153	132, 134, 135	expap0d 10706	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → (𝐴↑(𝑀 − 𝑘)) # 0)
154	151, 152, 132, 153, 134	divdivap2d 8821	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑀) / ((𝐴↑(𝑀 − 𝑘)) / 𝐴)) = (((𝐹‘𝑀) · 𝐴) / (𝐴↑(𝑀 − 𝑘))))
155	151, 132	mulcomd 8020	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑀) · 𝐴) = (𝐴 · (𝐹‘𝑀)))
156	155	oveq1d 5919	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → (((𝐹‘𝑀) · 𝐴) / (𝐴↑(𝑀 − 𝑘))) = ((𝐴 · (𝐹‘𝑀)) / (𝐴↑(𝑀 − 𝑘))))
157	132, 151, 152, 153	divassapd 8824	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → ((𝐴 · (𝐹‘𝑀)) / (𝐴↑(𝑀 − 𝑘))) = (𝐴 · ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘)))))
158	154, 156, 157	3eqtrd 2226	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑀) / ((𝐴↑(𝑀 − 𝑘)) / 𝐴)) = (𝐴 · ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘)))))
159	158	adantr 276	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) < 𝑀) → ((𝐹‘𝑀) / ((𝐴↑(𝑀 − 𝑘)) / 𝐴)) = (𝐴 · ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘)))))
160	150, 159	eqtr4d 2225	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) < 𝑀) → (𝐴 · if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘))) = ((𝐹‘𝑀) / ((𝐴↑(𝑀 − 𝑘)) / 𝐴)))
161	139, 141, 160	3eqtr4d 2232	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) < 𝑀) → if((𝑘 + 1) < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − (𝑘 + 1)))), (𝐹‘(𝑘 + 1))) = (𝐴 · if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘))))
162	161	fveq2d 5545	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) < 𝑀) → (abs‘if((𝑘 + 1) < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − (𝑘 + 1)))), (𝐹‘(𝑘 + 1)))) = (abs‘(𝐴 · if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)))))
163	132, 62	absmuld 11250	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → (abs‘(𝐴 · if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)))) = ((abs‘𝐴) · (abs‘if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)))))
164	163	adantr 276	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) < 𝑀) → (abs‘(𝐴 · if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)))) = ((abs‘𝐴) · (abs‘if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)))))
165	35	rpge0d 9742	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ 𝐴)
166	33, 165	absidd 11223	. . . . . . . . . . 11 ⊢ (𝜑 → (abs‘𝐴) = 𝐴)
167	166	oveq1d 5919	. . . . . . . . . 10 ⊢ (𝜑 → ((abs‘𝐴) · (abs‘if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)))) = (𝐴 · (abs‘if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)))))
168	167	ad3antrrr 492	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) < 𝑀) → ((abs‘𝐴) · (abs‘if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)))) = (𝐴 · (abs‘if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)))))
169	162, 164, 168	3eqtrd 2226	. . . . . . . 8 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) < 𝑀) → (abs‘if((𝑘 + 1) < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − (𝑘 + 1)))), (𝐹‘(𝑘 + 1)))) = (𝐴 · (abs‘if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)))))
170		eqle 8090	. . . . . . . 8 ⊢ (((abs‘if((𝑘 + 1) < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − (𝑘 + 1)))), (𝐹‘(𝑘 + 1)))) ∈ ℝ ∧ (abs‘if((𝑘 + 1) < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − (𝑘 + 1)))), (𝐹‘(𝑘 + 1)))) = (𝐴 · (abs‘if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘))))) → (abs‘if((𝑘 + 1) < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − (𝑘 + 1)))), (𝐹‘(𝑘 + 1)))) ≤ (𝐴 · (abs‘if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)))))
171	123, 169, 170	syl2an2r 595	. . . . . . 7 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) < 𝑀) → (abs‘if((𝑘 + 1) < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − (𝑘 + 1)))), (𝐹‘(𝑘 + 1)))) ≤ (𝐴 · (abs‘if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)))))
172	16	ad2antrr 488	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → 𝑀 ∈ ℝ)
173	111, 172	lttri3d 8113	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → ((𝑘 + 1) = 𝑀 ↔ (¬ (𝑘 + 1) < 𝑀 ∧ ¬ 𝑀 < (𝑘 + 1))))
174	173	simprbda 383	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) = 𝑀) → ¬ (𝑘 + 1) < 𝑀)
175	174	iffalsed 3563	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) = 𝑀) → if((𝑘 + 1) < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − (𝑘 + 1)))), (𝐹‘(𝑘 + 1))) = (𝐹‘(𝑘 + 1)))
176		simpr 110	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) = 𝑀) → (𝑘 + 1) = 𝑀)
177	176	fveq2d 5545	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) = 𝑀) → (𝐹‘(𝑘 + 1)) = (𝐹‘𝑀))
178	175, 177	eqtrd 2222	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) = 𝑀) → if((𝑘 + 1) < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − (𝑘 + 1)))), (𝐹‘(𝑘 + 1))) = (𝐹‘𝑀))
179	178	fveq2d 5545	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) = 𝑀) → (abs‘if((𝑘 + 1) < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − (𝑘 + 1)))), (𝐹‘(𝑘 + 1)))) = (abs‘(𝐹‘𝑀)))
180	142	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) = 𝑀) → 𝑘 ∈ ℝ)
181	180	ltp1d 8928	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) = 𝑀) → 𝑘 < (𝑘 + 1))
182		breq2 4029	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 + 1) = 𝑀 → (𝑘 < (𝑘 + 1) ↔ 𝑘 < 𝑀))
183	182	adantl 277	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) = 𝑀) → (𝑘 < (𝑘 + 1) ↔ 𝑘 < 𝑀))
184	181, 183	mpbid 147	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) = 𝑀) → 𝑘 < 𝑀)
185	184	iftrued 3560	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) = 𝑀) → if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)) = ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))))
186	176	oveq1d 5919	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) = 𝑀) → ((𝑘 + 1) − 𝑘) = (𝑀 − 𝑘))
187	127	adantr 276	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) = 𝑀) → 𝑘 ∈ ℂ)
188		1cnd 8014	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) = 𝑀) → 1 ∈ ℂ)
189	187, 188	pncan2d 8311	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) = 𝑀) → ((𝑘 + 1) − 𝑘) = 1)
190	186, 189	eqtr3d 2224	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) = 𝑀) → (𝑀 − 𝑘) = 1)
191	190	oveq2d 5920	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) = 𝑀) → (𝐴↑(𝑀 − 𝑘)) = (𝐴↑1))
192	132	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) = 𝑀) → 𝐴 ∈ ℂ)
193	192	exp1d 10695	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) = 𝑀) → (𝐴↑1) = 𝐴)
194	191, 193	eqtrd 2222	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) = 𝑀) → (𝐴↑(𝑀 − 𝑘)) = 𝐴)
195	194	oveq2d 5920	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) = 𝑀) → ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))) = ((𝐹‘𝑀) / 𝐴))
196	185, 195	eqtrd 2222	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) = 𝑀) → if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)) = ((𝐹‘𝑀) / 𝐴))
197	196	oveq2d 5920	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) = 𝑀) → (𝐴 · if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘))) = (𝐴 · ((𝐹‘𝑀) / 𝐴)))
198	31, 131, 133	divcanap2d 8790	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 · ((𝐹‘𝑀) / 𝐴)) = (𝐹‘𝑀))
199	198	ad3antrrr 492	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) = 𝑀) → (𝐴 · ((𝐹‘𝑀) / 𝐴)) = (𝐹‘𝑀))
200	197, 199	eqtrd 2222	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) = 𝑀) → (𝐴 · if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘))) = (𝐹‘𝑀))
201	200	fveq2d 5545	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) = 𝑀) → (abs‘(𝐴 · if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)))) = (abs‘(𝐹‘𝑀)))
202	167	ad2antrr 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → ((abs‘𝐴) · (abs‘if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)))) = (𝐴 · (abs‘if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)))))
203	163, 202	eqtrd 2222	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → (abs‘(𝐴 · if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)))) = (𝐴 · (abs‘if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)))))
204	203	adantr 276	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) = 𝑀) → (abs‘(𝐴 · if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)))) = (𝐴 · (abs‘if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)))))
205	179, 201, 204	3eqtr2d 2228	. . . . . . . 8 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) = 𝑀) → (abs‘if((𝑘 + 1) < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − (𝑘 + 1)))), (𝐹‘(𝑘 + 1)))) = (𝐴 · (abs‘if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)))))
206	123, 205, 170	syl2an2r 595	. . . . . . 7 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ (𝑘 + 1) = 𝑀) → (abs‘if((𝑘 + 1) < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − (𝑘 + 1)))), (𝐹‘(𝑘 + 1)))) ≤ (𝐴 · (abs‘if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)))))
207		simplll 533	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ 𝑀 < (𝑘 + 1)) → 𝜑)
208	119	adantr 276	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ 𝑀 < (𝑘 + 1)) → 𝑀 ∈ ℤ)
209	90	adantr 276	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ 𝑀 < (𝑘 + 1)) → 𝑘 ∈ ℤ)
210		simpr 110	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ 𝑀 < (𝑘 + 1)) → 𝑀 < (𝑘 + 1))
211		zleltp1 9349	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑀 ≤ 𝑘 ↔ 𝑀 < (𝑘 + 1)))
212	119, 209, 211	syl2an2r 595	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ 𝑀 < (𝑘 + 1)) → (𝑀 ≤ 𝑘 ↔ 𝑀 < (𝑘 + 1)))
213	210, 212	mpbird 167	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ 𝑀 < (𝑘 + 1)) → 𝑀 ≤ 𝑘)
214	208, 209, 213, 55	syl3anbrc 1183	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ 𝑀 < (𝑘 + 1)) → 𝑘 ∈ (ℤ_≥‘𝑀))
215	214, 8	eleqtrrdi 2283	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ 𝑀 < (𝑘 + 1)) → 𝑘 ∈ 𝑍)
216		cvgratz.7	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))
217	207, 215, 216	syl2anc 411	. . . . . . . 8 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ 𝑀 < (𝑘 + 1)) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))
218	172	adantr 276	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ 𝑀 < (𝑘 + 1)) → 𝑀 ∈ ℝ)
219	111	adantr 276	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ 𝑀 < (𝑘 + 1)) → (𝑘 + 1) ∈ ℝ)
220	218, 219, 210	ltnsymd 8118	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ 𝑀 < (𝑘 + 1)) → ¬ (𝑘 + 1) < 𝑀)
221	220	iffalsed 3563	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ 𝑀 < (𝑘 + 1)) → if((𝑘 + 1) < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − (𝑘 + 1)))), (𝐹‘(𝑘 + 1))) = (𝐹‘(𝑘 + 1)))
222	221	fveq2d 5545	. . . . . . . 8 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ 𝑀 < (𝑘 + 1)) → (abs‘if((𝑘 + 1) < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − (𝑘 + 1)))), (𝐹‘(𝑘 + 1)))) = (abs‘(𝐹‘(𝑘 + 1))))
223	142	adantr 276	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ 𝑀 < (𝑘 + 1)) → 𝑘 ∈ ℝ)
224	218, 223, 213	lensymd 8120	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ 𝑀 < (𝑘 + 1)) → ¬ 𝑘 < 𝑀)
225	224	iffalsed 3563	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ 𝑀 < (𝑘 + 1)) → if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)) = (𝐹‘𝑘))
226	225	fveq2d 5545	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ 𝑀 < (𝑘 + 1)) → (abs‘if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘))) = (abs‘(𝐹‘𝑘)))
227	226	oveq2d 5920	. . . . . . . 8 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ 𝑀 < (𝑘 + 1)) → (𝐴 · (abs‘if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)))) = (𝐴 · (abs‘(𝐹‘𝑘))))
228	217, 222, 227	3brtr4d 4057	. . . . . . 7 ⊢ ((((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) ∧ 𝑀 < (𝑘 + 1)) → (abs‘if((𝑘 + 1) < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − (𝑘 + 1)))), (𝐹‘(𝑘 + 1)))) ≤ (𝐴 · (abs‘if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)))))
229		ztri3or 9337	. . . . . . . 8 ⊢ (((𝑘 + 1) ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑘 + 1) < 𝑀 ∨ (𝑘 + 1) = 𝑀 ∨ 𝑀 < (𝑘 + 1)))
230	108, 119, 229	syl2anc 411	. . . . . . 7 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → ((𝑘 + 1) < 𝑀 ∨ (𝑘 + 1) = 𝑀 ∨ 𝑀 < (𝑘 + 1)))
231	171, 206, 228, 230	mpjao3dan 1318	. . . . . 6 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → (abs‘if((𝑘 + 1) < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − (𝑘 + 1)))), (𝐹‘(𝑘 + 1)))) ≤ (𝐴 · (abs‘if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)))))
232		breq1 4028	. . . . . . . . . 10 ⊢ (𝑖 = (𝑘 + 1) → (𝑖 < 𝑀 ↔ (𝑘 + 1) < 𝑀))
233		oveq2 5912	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝑘 + 1) → (𝑀 − 𝑖) = (𝑀 − (𝑘 + 1)))
234	233	oveq2d 5920	. . . . . . . . . . 11 ⊢ (𝑖 = (𝑘 + 1) → (𝐴↑(𝑀 − 𝑖)) = (𝐴↑(𝑀 − (𝑘 + 1))))
235	234	oveq2d 5920	. . . . . . . . . 10 ⊢ (𝑖 = (𝑘 + 1) → ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑖))) = ((𝐹‘𝑀) / (𝐴↑(𝑀 − (𝑘 + 1)))))
236		fveq2 5541	. . . . . . . . . 10 ⊢ (𝑖 = (𝑘 + 1) → (𝐹‘𝑖) = (𝐹‘(𝑘 + 1)))
237	232, 235, 236	ifbieq12d 3579	. . . . . . . . 9 ⊢ (𝑖 = (𝑘 + 1) → if(𝑖 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑖))), (𝐹‘𝑖)) = if((𝑘 + 1) < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − (𝑘 + 1)))), (𝐹‘(𝑘 + 1))))
238	237, 70	fvmptg 5620	. . . . . . . 8 ⊢ (((𝑘 + 1) ∈ ℕ ∧ if((𝑘 + 1) < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − (𝑘 + 1)))), (𝐹‘(𝑘 + 1))) ∈ ℂ) → ((𝑖 ∈ ℕ ↦ if(𝑖 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑖))), (𝐹‘𝑖)))‘(𝑘 + 1)) = if((𝑘 + 1) < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − (𝑘 + 1)))), (𝐹‘(𝑘 + 1))))
239	107, 122, 238	syl2anc 411	. . . . . . 7 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → ((𝑖 ∈ ℕ ↦ if(𝑖 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑖))), (𝐹‘𝑖)))‘(𝑘 + 1)) = if((𝑘 + 1) < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − (𝑘 + 1)))), (𝐹‘(𝑘 + 1))))
240	239	fveq2d 5545	. . . . . 6 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → (abs‘((𝑖 ∈ ℕ ↦ if(𝑖 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑖))), (𝐹‘𝑖)))‘(𝑘 + 1))) = (abs‘if((𝑘 + 1) < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − (𝑘 + 1)))), (𝐹‘(𝑘 + 1)))))
241	126, 62, 71	syl2anc 411	. . . . . . . 8 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → ((𝑖 ∈ ℕ ↦ if(𝑖 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑖))), (𝐹‘𝑖)))‘𝑘) = if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)))
242	241	fveq2d 5545	. . . . . . 7 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → (abs‘((𝑖 ∈ ℕ ↦ if(𝑖 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑖))), (𝐹‘𝑖)))‘𝑘)) = (abs‘if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘))))
243	242	oveq2d 5920	. . . . . 6 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → (𝐴 · (abs‘((𝑖 ∈ ℕ ↦ if(𝑖 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑖))), (𝐹‘𝑖)))‘𝑘))) = (𝐴 · (abs‘if(𝑘 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑘))), (𝐹‘𝑘)))))
244	231, 240, 243	3brtr4d 4057	. . . . 5 ⊢ (((𝜑 ∧ 1 ≤ 𝑀) ∧ 𝑘 ∈ ℕ) → (abs‘((𝑖 ∈ ℕ ↦ if(𝑖 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑖))), (𝐹‘𝑖)))‘(𝑘 + 1))) ≤ (𝐴 · (abs‘((𝑖 ∈ ℕ ↦ if(𝑖 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑖))), (𝐹‘𝑖)))‘𝑘))))
245	79, 81, 82, 86, 244	cvgratnn 11586	. . . 4 ⊢ ((𝜑 ∧ 1 ≤ 𝑀) → seq1( + , (𝑖 ∈ ℕ ↦ if(𝑖 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑖))), (𝐹‘𝑖)))) ∈ dom ⇝ )
246		eqid 2189	. . . . 5 ⊢ (ℤ_≥‘1) = (ℤ_≥‘1)
247		1zzd 9321	. . . . . 6 ⊢ ((𝜑 ∧ 1 ≤ 𝑀) → 1 ∈ ℤ)
248		simpr 110	. . . . . 6 ⊢ ((𝜑 ∧ 1 ≤ 𝑀) → 1 ≤ 𝑀)
249		eluz2 9575	. . . . . 6 ⊢ (𝑀 ∈ (ℤ_≥‘1) ↔ (1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 1 ≤ 𝑀))
250	247, 2, 248, 249	syl3anbrc 1183	. . . . 5 ⊢ ((𝜑 ∧ 1 ≤ 𝑀) → 𝑀 ∈ (ℤ_≥‘1))
251	246, 250, 85	iserex 11394	. . . 4 ⊢ ((𝜑 ∧ 1 ≤ 𝑀) → (seq1( + , (𝑖 ∈ ℕ ↦ if(𝑖 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑖))), (𝐹‘𝑖)))) ∈ dom ⇝ ↔ seq𝑀( + , (𝑖 ∈ ℕ ↦ if(𝑖 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑖))), (𝐹‘𝑖)))) ∈ dom ⇝ ))
252	245, 251	mpbid 147	. . 3 ⊢ ((𝜑 ∧ 1 ≤ 𝑀) → seq𝑀( + , (𝑖 ∈ ℕ ↦ if(𝑖 < 𝑀, ((𝐹‘𝑀) / (𝐴↑(𝑀 − 𝑖))), (𝐹‘𝑖)))) ∈ dom ⇝ )
253	78, 252	eqeltrd 2266	. 2 ⊢ ((𝜑 ∧ 1 ≤ 𝑀) → seq𝑀( + , 𝐹) ∈ dom ⇝ )
254	33	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ≤ 1) → 𝐴 ∈ ℝ)
255	80	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ≤ 1) → 𝐴 < 1)
256	34	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ≤ 1) → 0 < 𝐴)
257	1	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ≤ 1) → 𝑀 ∈ ℤ)
258	257	adantr 276	. . . . . 6 ⊢ (((𝜑 ∧ 𝑀 ≤ 1) ∧ 𝑘 ∈ ℕ) → 𝑀 ∈ ℤ)
259		nnz 9313	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
260	259	adantl 277	. . . . . 6 ⊢ (((𝜑 ∧ 𝑀 ≤ 1) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℤ)
261	258	zred 9416	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑀 ≤ 1) ∧ 𝑘 ∈ ℕ) → 𝑀 ∈ ℝ)
262		1red 8013	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑀 ≤ 1) ∧ 𝑘 ∈ ℕ) → 1 ∈ ℝ)
263	260	zred 9416	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑀 ≤ 1) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ)
264		simplr 528	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑀 ≤ 1) ∧ 𝑘 ∈ ℕ) → 𝑀 ≤ 1)
265		nnge1 8983	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → 1 ≤ 𝑘)
266	265	adantl 277	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑀 ≤ 1) ∧ 𝑘 ∈ ℕ) → 1 ≤ 𝑘)
267	261, 262, 263, 264, 266	letrd 8122	. . . . . 6 ⊢ (((𝜑 ∧ 𝑀 ≤ 1) ∧ 𝑘 ∈ ℕ) → 𝑀 ≤ 𝑘)
268	258, 260, 267, 55	syl3anbrc 1183	. . . . 5 ⊢ (((𝜑 ∧ 𝑀 ≤ 1) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ_≥‘𝑀))
269	8	eleq2i 2256	. . . . . . 7 ⊢ (𝑘 ∈ 𝑍 ↔ 𝑘 ∈ (ℤ_≥‘𝑀))
270	269, 5	sylan2br 288	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)
271	270	adantlr 477	. . . . 5 ⊢ (((𝜑 ∧ 𝑀 ≤ 1) ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ)
272	268, 271	syldan 282	. . . 4 ⊢ (((𝜑 ∧ 𝑀 ≤ 1) ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)
273	269, 216	sylan2br 288	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))
274	273	adantlr 477	. . . . 5 ⊢ (((𝜑 ∧ 𝑀 ≤ 1) ∧ 𝑘 ∈ (ℤ_≥‘𝑀)) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))
275	268, 274	syldan 282	. . . 4 ⊢ (((𝜑 ∧ 𝑀 ≤ 1) ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))))
276	254, 255, 256, 272, 275	cvgratnn 11586	. . 3 ⊢ ((𝜑 ∧ 𝑀 ≤ 1) → seq1( + , 𝐹) ∈ dom ⇝ )
277		eqid 2189	. . . 4 ⊢ (ℤ_≥‘𝑀) = (ℤ_≥‘𝑀)
278		1zzd 9321	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ≤ 1) → 1 ∈ ℤ)
279		simpr 110	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ≤ 1) → 𝑀 ≤ 1)
280		eluz2 9575	. . . . 5 ⊢ (1 ∈ (ℤ_≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑀 ≤ 1))
281	257, 278, 279, 280	syl3anbrc 1183	. . . 4 ⊢ ((𝜑 ∧ 𝑀 ≤ 1) → 1 ∈ (ℤ_≥‘𝑀))
282	277, 281, 271	iserex 11394	. . 3 ⊢ ((𝜑 ∧ 𝑀 ≤ 1) → (seq𝑀( + , 𝐹) ∈ dom ⇝ ↔ seq1( + , 𝐹) ∈ dom ⇝ ))
283	276, 282	mpbird 167	. 2 ⊢ ((𝜑 ∧ 𝑀 ≤ 1) → seq𝑀( + , 𝐹) ∈ dom ⇝ )
284		1z 9320	. . 3 ⊢ 1 ∈ ℤ
285		zletric 9338	. . 3 ⊢ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (1 ≤ 𝑀 ∨ 𝑀 ≤ 1))
286	284, 1, 285	sylancr 414	. 2 ⊢ (𝜑 → (1 ≤ 𝑀 ∨ 𝑀 ≤ 1))
287	253, 283, 286	mpjaodan 799	1 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 709 DECID wdc 835 ∨ w3o 979 = wceq 1364 ∈ wcel 2160 ∀wral 2468 ifcif 3553 class class class wbr 4025 ↦ cmpt 4086 dom cdm 4651 ‘cfv 5242 (class class class)co 5904 ℂcc 7849 ℝcr 7850 0cc0 7851 1c1 7852 + caddc 7854 · cmul 7856 < clt 8033 ≤ cle 8034 − cmin 8169 # cap 8579 / cdiv 8670 ℕcn 8960 ℤcz 9294 ℤ_≥cuz 9569 ℝ⁺crp 9695 seqcseq 10490 ↑cexp 10565 abscabs 11053 ⇝ cli 11333

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4140 ax-sep 4143 ax-nul 4151 ax-pow 4199 ax-pr 4234 ax-un 4458 ax-setind 4561 ax-iinf 4612 ax-cnex 7942 ax-resscn 7943 ax-1cn 7944 ax-1re 7945 ax-icn 7946 ax-addcl 7947 ax-addrcl 7948 ax-mulcl 7949 ax-mulrcl 7950 ax-addcom 7951 ax-mulcom 7952 ax-addass 7953 ax-mulass 7954 ax-distr 7955 ax-i2m1 7956 ax-0lt1 7957 ax-1rid 7958 ax-0id 7959 ax-rnegex 7960 ax-precex 7961 ax-cnre 7962 ax-pre-ltirr 7963 ax-pre-ltwlin 7964 ax-pre-lttrn 7965 ax-pre-apti 7966 ax-pre-ltadd 7967 ax-pre-mulgt0 7968 ax-pre-mulext 7969 ax-arch 7970 ax-caucvg 7971

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2758 df-sbc 2982 df-csb 3077 df-dif 3150 df-un 3152 df-in 3154 df-ss 3161 df-nul 3442 df-if 3554 df-pw 3599 df-sn 3620 df-pr 3621 df-op 3623 df-uni 3832 df-int 3867 df-iun 3910 df-br 4026 df-opab 4087 df-mpt 4088 df-tr 4124 df-id 4318 df-po 4321 df-iso 4322 df-iord 4391 df-on 4393 df-ilim 4394 df-suc 4396 df-iom 4615 df-xp 4657 df-rel 4658 df-cnv 4659 df-co 4660 df-dm 4661 df-rn 4662 df-res 4663 df-ima 4664 df-iota 5203 df-fun 5244 df-fn 5245 df-f 5246 df-f1 5247 df-fo 5248 df-f1o 5249 df-fv 5250 df-isom 5251 df-riota 5859 df-ov 5907 df-oprab 5908 df-mpo 5909 df-1st 6173 df-2nd 6174 df-recs 6338 df-irdg 6403 df-frec 6424 df-1o 6449 df-oadd 6453 df-er 6567 df-en 6775 df-dom 6776 df-fin 6777 df-pnf 8035 df-mnf 8036 df-xr 8037 df-ltxr 8038 df-le 8039 df-sub 8171 df-neg 8172 df-reap 8573 df-ap 8580 df-div 8671 df-inn 8961 df-2 9019 df-3 9020 df-4 9021 df-n0 9218 df-z 9295 df-uz 9570 df-q 9662 df-rp 9696 df-ico 9936 df-fz 10051 df-fzo 10185 df-seqfrec 10491 df-exp 10566 df-ihash 10803 df-cj 10898 df-re 10899 df-im 10900 df-rsqrt 11054 df-abs 11055 df-clim 11334 df-sumdc 11409

This theorem is referenced by: cvgratgt0 11588

W3C validator