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Theorem ostth2lem2 25323

Description: Lemma for ostth2 25326. (Contributed by Mario Carneiro, 10-Sep-2014.)

**Hypotheses**
Ref	Expression
qrng.q	⊢ 𝑄 = (ℂ_fld ↾_s ℚ)
qabsabv.a	⊢ 𝐴 = (AbsVal‘𝑄)
padic.j	⊢ 𝐽 = (𝑞 ∈ ℙ ↦ (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑞↑-(𝑞 pCnt 𝑥)))))
ostth.k	⊢ 𝐾 = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, 1))
ostth.1	⊢ (𝜑 → 𝐹 ∈ 𝐴)
ostth2.2	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘2))
ostth2.3	⊢ (𝜑 → 1 < (𝐹‘𝑁))
ostth2.4	⊢ 𝑅 = ((log‘(𝐹‘𝑁)) / (log‘𝑁))
ostth2.5	⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘2))
ostth2.6	⊢ 𝑆 = ((log‘(𝐹‘𝑀)) / (log‘𝑀))
ostth2.7	⊢ 𝑇 = if((𝐹‘𝑀) ≤ 1, 1, (𝐹‘𝑀))

**Assertion**
Ref	Expression
ostth2lem2	⊢ ((𝜑 ∧ 𝑋 ∈ ℕ₀ ∧ 𝑌 ∈ (0...((𝑀↑𝑋) − 1))) → (𝐹‘𝑌) ≤ ((𝑀 · 𝑋) · (𝑇↑𝑋)))

Distinct variable groups: 𝑥,𝑀 𝑥,𝑞,𝜑 𝑥,𝑇 𝑥,𝑋 𝐴,𝑞,𝑥 𝑥,𝑁 𝑥,𝑄 𝐹,𝑞 𝑅,𝑞 𝑥,𝐹 Allowed substitution hints: 𝑄(𝑞) 𝑅(𝑥) 𝑆(𝑥,𝑞) 𝑇(𝑞) 𝐽(𝑥,𝑞) 𝐾(𝑥,𝑞) 𝑀(𝑞) 𝑁(𝑞) 𝑋(𝑞) 𝑌(𝑥,𝑞)

Proof of Theorem ostth2lem2 Dummy variables 𝑘 𝑛 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6658	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑀↑𝑥) = (𝑀↑0))
2	1	oveq1d 6665	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝑀↑𝑥) − 1) = ((𝑀↑0) − 1))
3	2	oveq2d 6666	. . . . . . 7 ⊢ (𝑥 = 0 → (0...((𝑀↑𝑥) − 1)) = (0...((𝑀↑0) − 1)))
4		oveq2 6658	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑀 · 𝑥) = (𝑀 · 0))
5		oveq2 6658	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑇↑𝑥) = (𝑇↑0))
6	4, 5	oveq12d 6668	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝑀 · 𝑥) · (𝑇↑𝑥)) = ((𝑀 · 0) · (𝑇↑0)))
7	6	breq2d 4665	. . . . . . 7 ⊢ (𝑥 = 0 → ((𝐹‘𝑘) ≤ ((𝑀 · 𝑥) · (𝑇↑𝑥)) ↔ (𝐹‘𝑘) ≤ ((𝑀 · 0) · (𝑇↑0))))
8	3, 7	raleqbidv 3152	. . . . . 6 ⊢ (𝑥 = 0 → (∀𝑘 ∈ (0...((𝑀↑𝑥) − 1))(𝐹‘𝑘) ≤ ((𝑀 · 𝑥) · (𝑇↑𝑥)) ↔ ∀𝑘 ∈ (0...((𝑀↑0) − 1))(𝐹‘𝑘) ≤ ((𝑀 · 0) · (𝑇↑0))))
9	8	imbi2d 330	. . . . 5 ⊢ (𝑥 = 0 → ((𝜑 → ∀𝑘 ∈ (0...((𝑀↑𝑥) − 1))(𝐹‘𝑘) ≤ ((𝑀 · 𝑥) · (𝑇↑𝑥))) ↔ (𝜑 → ∀𝑘 ∈ (0...((𝑀↑0) − 1))(𝐹‘𝑘) ≤ ((𝑀 · 0) · (𝑇↑0)))))
10		oveq2 6658	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (𝑀↑𝑥) = (𝑀↑𝑛))
11	10	oveq1d 6665	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → ((𝑀↑𝑥) − 1) = ((𝑀↑𝑛) − 1))
12	11	oveq2d 6666	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (0...((𝑀↑𝑥) − 1)) = (0...((𝑀↑𝑛) − 1)))
13		oveq2 6658	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (𝑀 · 𝑥) = (𝑀 · 𝑛))
14		oveq2 6658	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (𝑇↑𝑥) = (𝑇↑𝑛))
15	13, 14	oveq12d 6668	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → ((𝑀 · 𝑥) · (𝑇↑𝑥)) = ((𝑀 · 𝑛) · (𝑇↑𝑛)))
16	15	breq2d 4665	. . . . . . 7 ⊢ (𝑥 = 𝑛 → ((𝐹‘𝑘) ≤ ((𝑀 · 𝑥) · (𝑇↑𝑥)) ↔ (𝐹‘𝑘) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛))))
17	12, 16	raleqbidv 3152	. . . . . 6 ⊢ (𝑥 = 𝑛 → (∀𝑘 ∈ (0...((𝑀↑𝑥) − 1))(𝐹‘𝑘) ≤ ((𝑀 · 𝑥) · (𝑇↑𝑥)) ↔ ∀𝑘 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑘) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛))))
18	17	imbi2d 330	. . . . 5 ⊢ (𝑥 = 𝑛 → ((𝜑 → ∀𝑘 ∈ (0...((𝑀↑𝑥) − 1))(𝐹‘𝑘) ≤ ((𝑀 · 𝑥) · (𝑇↑𝑥))) ↔ (𝜑 → ∀𝑘 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑘) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))))
19		oveq2 6658	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → (𝑀↑𝑥) = (𝑀↑(𝑛 + 1)))
20	19	oveq1d 6665	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → ((𝑀↑𝑥) − 1) = ((𝑀↑(𝑛 + 1)) − 1))
21	20	oveq2d 6666	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (0...((𝑀↑𝑥) − 1)) = (0...((𝑀↑(𝑛 + 1)) − 1)))
22		oveq2 6658	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → (𝑀 · 𝑥) = (𝑀 · (𝑛 + 1)))
23		oveq2 6658	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → (𝑇↑𝑥) = (𝑇↑(𝑛 + 1)))
24	22, 23	oveq12d 6668	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → ((𝑀 · 𝑥) · (𝑇↑𝑥)) = ((𝑀 · (𝑛 + 1)) · (𝑇↑(𝑛 + 1))))
25	24	breq2d 4665	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → ((𝐹‘𝑘) ≤ ((𝑀 · 𝑥) · (𝑇↑𝑥)) ↔ (𝐹‘𝑘) ≤ ((𝑀 · (𝑛 + 1)) · (𝑇↑(𝑛 + 1)))))
26	21, 25	raleqbidv 3152	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → (∀𝑘 ∈ (0...((𝑀↑𝑥) − 1))(𝐹‘𝑘) ≤ ((𝑀 · 𝑥) · (𝑇↑𝑥)) ↔ ∀𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1))(𝐹‘𝑘) ≤ ((𝑀 · (𝑛 + 1)) · (𝑇↑(𝑛 + 1)))))
27	26	imbi2d 330	. . . . 5 ⊢ (𝑥 = (𝑛 + 1) → ((𝜑 → ∀𝑘 ∈ (0...((𝑀↑𝑥) − 1))(𝐹‘𝑘) ≤ ((𝑀 · 𝑥) · (𝑇↑𝑥))) ↔ (𝜑 → ∀𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1))(𝐹‘𝑘) ≤ ((𝑀 · (𝑛 + 1)) · (𝑇↑(𝑛 + 1))))))
28		oveq2 6658	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑀↑𝑥) = (𝑀↑𝑋))
29	28	oveq1d 6665	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝑀↑𝑥) − 1) = ((𝑀↑𝑋) − 1))
30	29	oveq2d 6666	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (0...((𝑀↑𝑥) − 1)) = (0...((𝑀↑𝑋) − 1)))
31		oveq2 6658	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑀 · 𝑥) = (𝑀 · 𝑋))
32		oveq2 6658	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑇↑𝑥) = (𝑇↑𝑋))
33	31, 32	oveq12d 6668	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝑀 · 𝑥) · (𝑇↑𝑥)) = ((𝑀 · 𝑋) · (𝑇↑𝑋)))
34	33	breq2d 4665	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑘) ≤ ((𝑀 · 𝑥) · (𝑇↑𝑥)) ↔ (𝐹‘𝑘) ≤ ((𝑀 · 𝑋) · (𝑇↑𝑋))))
35	30, 34	raleqbidv 3152	. . . . . 6 ⊢ (𝑥 = 𝑋 → (∀𝑘 ∈ (0...((𝑀↑𝑥) − 1))(𝐹‘𝑘) ≤ ((𝑀 · 𝑥) · (𝑇↑𝑥)) ↔ ∀𝑘 ∈ (0...((𝑀↑𝑋) − 1))(𝐹‘𝑘) ≤ ((𝑀 · 𝑋) · (𝑇↑𝑋))))
36	35	imbi2d 330	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝜑 → ∀𝑘 ∈ (0...((𝑀↑𝑥) − 1))(𝐹‘𝑘) ≤ ((𝑀 · 𝑥) · (𝑇↑𝑥))) ↔ (𝜑 → ∀𝑘 ∈ (0...((𝑀↑𝑋) − 1))(𝐹‘𝑘) ≤ ((𝑀 · 𝑋) · (𝑇↑𝑋)))))
37		ostth2.5	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘2))
38		eluz2nn 11726	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ (ℤ_≥‘2) → 𝑀 ∈ ℕ)
39	37, 38	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℕ)
40	39	nncnd 11036	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℂ)
41	40	exp0d 13002	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀↑0) = 1)
42	41	oveq1d 6665	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑀↑0) − 1) = (1 − 1))
43		1m1e0 11089	. . . . . . . . . 10 ⊢ (1 − 1) = 0
44	42, 43	syl6eq 2672	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀↑0) − 1) = 0)
45	44	oveq2d 6666	. . . . . . . 8 ⊢ (𝜑 → (0...((𝑀↑0) − 1)) = (0...0))
46	45	eleq2d 2687	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ (0...((𝑀↑0) − 1)) ↔ 𝑘 ∈ (0...0)))
47		0le0 11110	. . . . . . . . . 10 ⊢ 0 ≤ 0
48	47	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ 0)
49		ostth.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ 𝐴)
50		qabsabv.a	. . . . . . . . . . 11 ⊢ 𝐴 = (AbsVal‘𝑄)
51		qrng.q	. . . . . . . . . . . 12 ⊢ 𝑄 = (ℂ_fld ↾_s ℚ)
52	51	qrng0 25310	. . . . . . . . . . 11 ⊢ 0 = (0_g‘𝑄)
53	50, 52	abv0 18831	. . . . . . . . . 10 ⊢ (𝐹 ∈ 𝐴 → (𝐹‘0) = 0)
54	49, 53	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘0) = 0)
55	40	mul01d 10235	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 · 0) = 0)
56	55	oveq1d 6665	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑀 · 0) · (𝑇↑0)) = (0 · (𝑇↑0)))
57		ostth2.7	. . . . . . . . . . . . . 14 ⊢ 𝑇 = if((𝐹‘𝑀) ≤ 1, 1, (𝐹‘𝑀))
58		1re 10039	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℝ
59		nnq 11801	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℚ)
60	39, 59	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ ℚ)
61	51	qrngbas 25308	. . . . . . . . . . . . . . . . 17 ⊢ ℚ = (Base‘𝑄)
62	50, 61	abvcl 18824	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑀 ∈ ℚ) → (𝐹‘𝑀) ∈ ℝ)
63	49, 60, 62	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ℝ)
64		ifcl 4130	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℝ ∧ (𝐹‘𝑀) ∈ ℝ) → if((𝐹‘𝑀) ≤ 1, 1, (𝐹‘𝑀)) ∈ ℝ)
65	58, 63, 64	sylancr 695	. . . . . . . . . . . . . 14 ⊢ (𝜑 → if((𝐹‘𝑀) ≤ 1, 1, (𝐹‘𝑀)) ∈ ℝ)
66	57, 65	syl5eqel 2705	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑇 ∈ ℝ)
67	66	recnd 10068	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ ℂ)
68		0nn0 11307	. . . . . . . . . . . 12 ⊢ 0 ∈ ℕ₀
69		expcl 12878	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ ℂ ∧ 0 ∈ ℕ₀) → (𝑇↑0) ∈ ℂ)
70	67, 68, 69	sylancl 694	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑇↑0) ∈ ℂ)
71	70	mul02d 10234	. . . . . . . . . 10 ⊢ (𝜑 → (0 · (𝑇↑0)) = 0)
72	56, 71	eqtrd 2656	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀 · 0) · (𝑇↑0)) = 0)
73	48, 54, 72	3brtr4d 4685	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘0) ≤ ((𝑀 · 0) · (𝑇↑0)))
74		elfz1eq 12352	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...0) → 𝑘 = 0)
75	74	fveq2d 6195	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...0) → (𝐹‘𝑘) = (𝐹‘0))
76	75	breq1d 4663	. . . . . . . 8 ⊢ (𝑘 ∈ (0...0) → ((𝐹‘𝑘) ≤ ((𝑀 · 0) · (𝑇↑0)) ↔ (𝐹‘0) ≤ ((𝑀 · 0) · (𝑇↑0))))
77	73, 76	syl5ibrcom 237	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ (0...0) → (𝐹‘𝑘) ≤ ((𝑀 · 0) · (𝑇↑0))))
78	46, 77	sylbid 230	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ (0...((𝑀↑0) − 1)) → (𝐹‘𝑘) ≤ ((𝑀 · 0) · (𝑇↑0))))
79	78	ralrimiv 2965	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (0...((𝑀↑0) − 1))(𝐹‘𝑘) ≤ ((𝑀 · 0) · (𝑇↑0)))
80		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
81	80	breq1d 4663	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)) ↔ (𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛))))
82	81	cbvralv 3171	. . . . . . . 8 ⊢ (∀𝑘 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑘) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)) ↔ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))
83	49	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 𝐹 ∈ 𝐴)
84		elfzelz 12342	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) → 𝑘 ∈ ℤ)
85	84	ad2antrl 764	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 𝑘 ∈ ℤ)
86		zq 11794	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℤ → 𝑘 ∈ ℚ)
87	85, 86	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 𝑘 ∈ ℚ)
88	50, 61	abvcl 18824	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑘 ∈ ℚ) → (𝐹‘𝑘) ∈ ℝ)
89	83, 87, 88	syl2anc 693	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝐹‘𝑘) ∈ ℝ)
90	39	ad2antrr 762	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 𝑀 ∈ ℕ)
91		simplr 792	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 𝑛 ∈ ℕ₀)
92	90, 91	nnexpcld 13030	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑀↑𝑛) ∈ ℕ)
93	85, 92	zmodcld 12691	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑘 mod (𝑀↑𝑛)) ∈ ℕ₀)
94	93	nn0zd 11480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑘 mod (𝑀↑𝑛)) ∈ ℤ)
95		zq 11794	. . . . . . . . . . . . . 14 ⊢ ((𝑘 mod (𝑀↑𝑛)) ∈ ℤ → (𝑘 mod (𝑀↑𝑛)) ∈ ℚ)
96	94, 95	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑘 mod (𝑀↑𝑛)) ∈ ℚ)
97	50, 61	abvcl 18824	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑘 mod (𝑀↑𝑛)) ∈ ℚ) → (𝐹‘(𝑘 mod (𝑀↑𝑛))) ∈ ℝ)
98	83, 96, 97	syl2anc 693	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝐹‘(𝑘 mod (𝑀↑𝑛))) ∈ ℝ)
99	90, 59	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 𝑀 ∈ ℚ)
100	83, 99, 62	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝐹‘𝑀) ∈ ℝ)
101	100, 91	reexpcld 13025	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → ((𝐹‘𝑀)↑𝑛) ∈ ℝ)
102	85	zred 11482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 𝑘 ∈ ℝ)
103	102, 92	nndivred 11069	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑘 / (𝑀↑𝑛)) ∈ ℝ)
104	103	flcld 12599	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (⌊‘(𝑘 / (𝑀↑𝑛))) ∈ ℤ)
105		zq 11794	. . . . . . . . . . . . . . 15 ⊢ ((⌊‘(𝑘 / (𝑀↑𝑛))) ∈ ℤ → (⌊‘(𝑘 / (𝑀↑𝑛))) ∈ ℚ)
106	104, 105	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (⌊‘(𝑘 / (𝑀↑𝑛))) ∈ ℚ)
107	50, 61	abvcl 18824	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ 𝐴 ∧ (⌊‘(𝑘 / (𝑀↑𝑛))) ∈ ℚ) → (𝐹‘(⌊‘(𝑘 / (𝑀↑𝑛)))) ∈ ℝ)
108	83, 106, 107	syl2anc 693	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝐹‘(⌊‘(𝑘 / (𝑀↑𝑛)))) ∈ ℝ)
109	101, 108	remulcld 10070	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (((𝐹‘𝑀)↑𝑛) · (𝐹‘(⌊‘(𝑘 / (𝑀↑𝑛))))) ∈ ℝ)
110	98, 109	readdcld 10069	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → ((𝐹‘(𝑘 mod (𝑀↑𝑛))) + (((𝐹‘𝑀)↑𝑛) · (𝐹‘(⌊‘(𝑘 / (𝑀↑𝑛)))))) ∈ ℝ)
111	90	nnred 11035	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 𝑀 ∈ ℝ)
112		nn0p1nn 11332	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ)
113	112	ad2antlr 763	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑛 + 1) ∈ ℕ)
114	113	nnred 11035	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑛 + 1) ∈ ℝ)
115	111, 114	remulcld 10070	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑀 · (𝑛 + 1)) ∈ ℝ)
116	66	ad2antrr 762	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 𝑇 ∈ ℝ)
117		peano2nn0 11333	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ₀)
118	117	ad2antlr 763	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑛 + 1) ∈ ℕ₀)
119	116, 118	reexpcld 13025	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑇↑(𝑛 + 1)) ∈ ℝ)
120	115, 119	remulcld 10070	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → ((𝑀 · (𝑛 + 1)) · (𝑇↑(𝑛 + 1))) ∈ ℝ)
121		nnq 11801	. . . . . . . . . . . . . . 15 ⊢ ((𝑀↑𝑛) ∈ ℕ → (𝑀↑𝑛) ∈ ℚ)
122	92, 121	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑀↑𝑛) ∈ ℚ)
123		qmulcl 11806	. . . . . . . . . . . . . 14 ⊢ (((𝑀↑𝑛) ∈ ℚ ∧ (⌊‘(𝑘 / (𝑀↑𝑛))) ∈ ℚ) → ((𝑀↑𝑛) · (⌊‘(𝑘 / (𝑀↑𝑛)))) ∈ ℚ)
124	122, 106, 123	syl2anc 693	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → ((𝑀↑𝑛) · (⌊‘(𝑘 / (𝑀↑𝑛)))) ∈ ℚ)
125		qex 11800	. . . . . . . . . . . . . . 15 ⊢ ℚ ∈ V
126		cnfldadd 19751	. . . . . . . . . . . . . . . 16 ⊢ + = (+_g‘ℂ_fld)
127	51, 126	ressplusg 15993	. . . . . . . . . . . . . . 15 ⊢ (ℚ ∈ V → + = (+_g‘𝑄))
128	125, 127	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ + = (+_g‘𝑄)
129	50, 61, 128	abvtri 18830	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑘 mod (𝑀↑𝑛)) ∈ ℚ ∧ ((𝑀↑𝑛) · (⌊‘(𝑘 / (𝑀↑𝑛)))) ∈ ℚ) → (𝐹‘((𝑘 mod (𝑀↑𝑛)) + ((𝑀↑𝑛) · (⌊‘(𝑘 / (𝑀↑𝑛)))))) ≤ ((𝐹‘(𝑘 mod (𝑀↑𝑛))) + (𝐹‘((𝑀↑𝑛) · (⌊‘(𝑘 / (𝑀↑𝑛)))))))
130	83, 96, 124, 129	syl3anc 1326	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝐹‘((𝑘 mod (𝑀↑𝑛)) + ((𝑀↑𝑛) · (⌊‘(𝑘 / (𝑀↑𝑛)))))) ≤ ((𝐹‘(𝑘 mod (𝑀↑𝑛))) + (𝐹‘((𝑀↑𝑛) · (⌊‘(𝑘 / (𝑀↑𝑛)))))))
131	92	nnrpd 11870	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑀↑𝑛) ∈ ℝ⁺)
132		modval 12670	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℝ ∧ (𝑀↑𝑛) ∈ ℝ⁺) → (𝑘 mod (𝑀↑𝑛)) = (𝑘 − ((𝑀↑𝑛) · (⌊‘(𝑘 / (𝑀↑𝑛))))))
133	102, 131, 132	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑘 mod (𝑀↑𝑛)) = (𝑘 − ((𝑀↑𝑛) · (⌊‘(𝑘 / (𝑀↑𝑛))))))
134	133	oveq1d 6665	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → ((𝑘 mod (𝑀↑𝑛)) + ((𝑀↑𝑛) · (⌊‘(𝑘 / (𝑀↑𝑛))))) = ((𝑘 − ((𝑀↑𝑛) · (⌊‘(𝑘 / (𝑀↑𝑛))))) + ((𝑀↑𝑛) · (⌊‘(𝑘 / (𝑀↑𝑛))))))
135	102	recnd 10068	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 𝑘 ∈ ℂ)
136		qcn 11802	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀↑𝑛) · (⌊‘(𝑘 / (𝑀↑𝑛)))) ∈ ℚ → ((𝑀↑𝑛) · (⌊‘(𝑘 / (𝑀↑𝑛)))) ∈ ℂ)
137	124, 136	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → ((𝑀↑𝑛) · (⌊‘(𝑘 / (𝑀↑𝑛)))) ∈ ℂ)
138	135, 137	npcand 10396	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → ((𝑘 − ((𝑀↑𝑛) · (⌊‘(𝑘 / (𝑀↑𝑛))))) + ((𝑀↑𝑛) · (⌊‘(𝑘 / (𝑀↑𝑛))))) = 𝑘)
139	134, 138	eqtrd 2656	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → ((𝑘 mod (𝑀↑𝑛)) + ((𝑀↑𝑛) · (⌊‘(𝑘 / (𝑀↑𝑛))))) = 𝑘)
140	139	fveq2d 6195	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝐹‘((𝑘 mod (𝑀↑𝑛)) + ((𝑀↑𝑛) · (⌊‘(𝑘 / (𝑀↑𝑛)))))) = (𝐹‘𝑘))
141		cnfldmul 19752	. . . . . . . . . . . . . . . . . 18 ⊢ · = (._r‘ℂ_fld)
142	51, 141	ressmulr 16006	. . . . . . . . . . . . . . . . 17 ⊢ (ℚ ∈ V → · = (._r‘𝑄))
143	125, 142	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ · = (._r‘𝑄)
144	50, 61, 143	abvmul 18829	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑀↑𝑛) ∈ ℚ ∧ (⌊‘(𝑘 / (𝑀↑𝑛))) ∈ ℚ) → (𝐹‘((𝑀↑𝑛) · (⌊‘(𝑘 / (𝑀↑𝑛))))) = ((𝐹‘(𝑀↑𝑛)) · (𝐹‘(⌊‘(𝑘 / (𝑀↑𝑛))))))
145	83, 122, 106, 144	syl3anc 1326	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝐹‘((𝑀↑𝑛) · (⌊‘(𝑘 / (𝑀↑𝑛))))) = ((𝐹‘(𝑀↑𝑛)) · (𝐹‘(⌊‘(𝑘 / (𝑀↑𝑛))))))
146	51, 50	qabvexp 25315	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑀 ∈ ℚ ∧ 𝑛 ∈ ℕ₀) → (𝐹‘(𝑀↑𝑛)) = ((𝐹‘𝑀)↑𝑛))
147	83, 99, 91, 146	syl3anc 1326	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝐹‘(𝑀↑𝑛)) = ((𝐹‘𝑀)↑𝑛))
148	147	oveq1d 6665	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → ((𝐹‘(𝑀↑𝑛)) · (𝐹‘(⌊‘(𝑘 / (𝑀↑𝑛))))) = (((𝐹‘𝑀)↑𝑛) · (𝐹‘(⌊‘(𝑘 / (𝑀↑𝑛))))))
149	145, 148	eqtrd 2656	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝐹‘((𝑀↑𝑛) · (⌊‘(𝑘 / (𝑀↑𝑛))))) = (((𝐹‘𝑀)↑𝑛) · (𝐹‘(⌊‘(𝑘 / (𝑀↑𝑛))))))
150	149	oveq2d 6666	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → ((𝐹‘(𝑘 mod (𝑀↑𝑛))) + (𝐹‘((𝑀↑𝑛) · (⌊‘(𝑘 / (𝑀↑𝑛)))))) = ((𝐹‘(𝑘 mod (𝑀↑𝑛))) + (((𝐹‘𝑀)↑𝑛) · (𝐹‘(⌊‘(𝑘 / (𝑀↑𝑛)))))))
151	130, 140, 150	3brtr3d 4684	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝐹‘𝑘) ≤ ((𝐹‘(𝑘 mod (𝑀↑𝑛))) + (((𝐹‘𝑀)↑𝑛) · (𝐹‘(⌊‘(𝑘 / (𝑀↑𝑛)))))))
152	116, 91	reexpcld 13025	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑇↑𝑛) ∈ ℝ)
153	115, 152	remulcld 10070	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → ((𝑀 · (𝑛 + 1)) · (𝑇↑𝑛)) ∈ ℝ)
154		nn0re 11301	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℝ)
155	154	ad2antlr 763	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 𝑛 ∈ ℝ)
156	111, 155	remulcld 10070	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑀 · 𝑛) ∈ ℝ)
157	156, 152	remulcld 10070	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → ((𝑀 · 𝑛) · (𝑇↑𝑛)) ∈ ℝ)
158	111, 152	remulcld 10070	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑀 · (𝑇↑𝑛)) ∈ ℝ)
159		zmodfz 12692	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℤ ∧ (𝑀↑𝑛) ∈ ℕ) → (𝑘 mod (𝑀↑𝑛)) ∈ (0...((𝑀↑𝑛) − 1)))
160	85, 92, 159	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑘 mod (𝑀↑𝑛)) ∈ (0...((𝑀↑𝑛) − 1)))
161		simprr 796	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))
162		fveq2 6191	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = (𝑘 mod (𝑀↑𝑛)) → (𝐹‘𝑗) = (𝐹‘(𝑘 mod (𝑀↑𝑛))))
163	162	breq1d 4663	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = (𝑘 mod (𝑀↑𝑛)) → ((𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)) ↔ (𝐹‘(𝑘 mod (𝑀↑𝑛))) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛))))
164	163	rspcv 3305	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 mod (𝑀↑𝑛)) ∈ (0...((𝑀↑𝑛) − 1)) → (∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)) → (𝐹‘(𝑘 mod (𝑀↑𝑛))) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛))))
165	160, 161, 164	sylc 65	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝐹‘(𝑘 mod (𝑀↑𝑛))) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))
166	111, 101	remulcld 10070	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑀 · ((𝐹‘𝑀)↑𝑛)) ∈ ℝ)
167	101	recnd 10068	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → ((𝐹‘𝑀)↑𝑛) ∈ ℂ)
168	108	recnd 10068	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝐹‘(⌊‘(𝑘 / (𝑀↑𝑛)))) ∈ ℂ)
169	167, 168	mulcomd 10061	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (((𝐹‘𝑀)↑𝑛) · (𝐹‘(⌊‘(𝑘 / (𝑀↑𝑛))))) = ((𝐹‘(⌊‘(𝑘 / (𝑀↑𝑛)))) · ((𝐹‘𝑀)↑𝑛)))
170	50, 61	abvge0 18825	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑀 ∈ ℚ) → 0 ≤ (𝐹‘𝑀))
171	83, 99, 170	syl2anc 693	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 0 ≤ (𝐹‘𝑀))
172	100, 91, 171	expge0d 13026	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 0 ≤ ((𝐹‘𝑀)↑𝑛))
173	104	zred 11482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (⌊‘(𝑘 / (𝑀↑𝑛))) ∈ ℝ)
174		elfzle1 12344	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) → 0 ≤ 𝑘)
175	174	ad2antrl 764	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 0 ≤ 𝑘)
176	92	nnred 11035	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑀↑𝑛) ∈ ℝ)
177	92	nngt0d 11064	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 0 < (𝑀↑𝑛))
178		divge0 10892	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑘 ∈ ℝ ∧ 0 ≤ 𝑘) ∧ ((𝑀↑𝑛) ∈ ℝ ∧ 0 < (𝑀↑𝑛))) → 0 ≤ (𝑘 / (𝑀↑𝑛)))
179	102, 175, 176, 177, 178	syl22anc 1327	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 0 ≤ (𝑘 / (𝑀↑𝑛)))
180		flge0nn0 12621	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑘 / (𝑀↑𝑛)) ∈ ℝ ∧ 0 ≤ (𝑘 / (𝑀↑𝑛))) → (⌊‘(𝑘 / (𝑀↑𝑛))) ∈ ℕ₀)
181	103, 179, 180	syl2anc 693	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (⌊‘(𝑘 / (𝑀↑𝑛))) ∈ ℕ₀)
182	51, 50	qabvle 25314	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹 ∈ 𝐴 ∧ (⌊‘(𝑘 / (𝑀↑𝑛))) ∈ ℕ₀) → (𝐹‘(⌊‘(𝑘 / (𝑀↑𝑛)))) ≤ (⌊‘(𝑘 / (𝑀↑𝑛))))
183	83, 181, 182	syl2anc 693	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝐹‘(⌊‘(𝑘 / (𝑀↑𝑛)))) ≤ (⌊‘(𝑘 / (𝑀↑𝑛))))
184		simprl 794	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)))
185		0z 11388	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 0 ∈ ℤ
186	90, 118	nnexpcld 13030	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑀↑(𝑛 + 1)) ∈ ℕ)
187	186	nnzd 11481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑀↑(𝑛 + 1)) ∈ ℤ)
188		elfzm11 12411	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((0 ∈ ℤ ∧ (𝑀↑(𝑛 + 1)) ∈ ℤ) → (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ↔ (𝑘 ∈ ℤ ∧ 0 ≤ 𝑘 ∧ 𝑘 < (𝑀↑(𝑛 + 1)))))
189	185, 187, 188	sylancr 695	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ↔ (𝑘 ∈ ℤ ∧ 0 ≤ 𝑘 ∧ 𝑘 < (𝑀↑(𝑛 + 1)))))
190	184, 189	mpbid 222	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑘 ∈ ℤ ∧ 0 ≤ 𝑘 ∧ 𝑘 < (𝑀↑(𝑛 + 1))))
191	190	simp3d 1075	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 𝑘 < (𝑀↑(𝑛 + 1)))
192	90	nncnd 11036	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 𝑀 ∈ ℂ)
193	192, 91	expp1d 13009	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑀↑(𝑛 + 1)) = ((𝑀↑𝑛) · 𝑀))
194	191, 193	breqtrd 4679	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 𝑘 < ((𝑀↑𝑛) · 𝑀))
195		ltdivmul 10898	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ ((𝑀↑𝑛) ∈ ℝ ∧ 0 < (𝑀↑𝑛))) → ((𝑘 / (𝑀↑𝑛)) < 𝑀 ↔ 𝑘 < ((𝑀↑𝑛) · 𝑀)))
196	102, 111, 176, 177, 195	syl112anc 1330	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → ((𝑘 / (𝑀↑𝑛)) < 𝑀 ↔ 𝑘 < ((𝑀↑𝑛) · 𝑀)))
197	194, 196	mpbird 247	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑘 / (𝑀↑𝑛)) < 𝑀)
198	90	nnzd 11481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 𝑀 ∈ ℤ)
199		fllt 12607	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑘 / (𝑀↑𝑛)) ∈ ℝ ∧ 𝑀 ∈ ℤ) → ((𝑘 / (𝑀↑𝑛)) < 𝑀 ↔ (⌊‘(𝑘 / (𝑀↑𝑛))) < 𝑀))
200	103, 198, 199	syl2anc 693	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → ((𝑘 / (𝑀↑𝑛)) < 𝑀 ↔ (⌊‘(𝑘 / (𝑀↑𝑛))) < 𝑀))
201	197, 200	mpbid 222	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (⌊‘(𝑘 / (𝑀↑𝑛))) < 𝑀)
202	173, 111, 201	ltled 10185	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (⌊‘(𝑘 / (𝑀↑𝑛))) ≤ 𝑀)
203	108, 173, 111, 183, 202	letrd 10194	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝐹‘(⌊‘(𝑘 / (𝑀↑𝑛)))) ≤ 𝑀)
204	108, 111, 101, 172, 203	lemul1ad 10963	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → ((𝐹‘(⌊‘(𝑘 / (𝑀↑𝑛)))) · ((𝐹‘𝑀)↑𝑛)) ≤ (𝑀 · ((𝐹‘𝑀)↑𝑛)))
205	169, 204	eqbrtrd 4675	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (((𝐹‘𝑀)↑𝑛) · (𝐹‘(⌊‘(𝑘 / (𝑀↑𝑛))))) ≤ (𝑀 · ((𝐹‘𝑀)↑𝑛)))
206	90	nnnn0d 11351	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 𝑀 ∈ ℕ₀)
207	206	nn0ge0d 11354	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 0 ≤ 𝑀)
208		max1 12016	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹‘𝑀) ∈ ℝ ∧ 1 ∈ ℝ) → (𝐹‘𝑀) ≤ if((𝐹‘𝑀) ≤ 1, 1, (𝐹‘𝑀)))
209	100, 58, 208	sylancl 694	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝐹‘𝑀) ≤ if((𝐹‘𝑀) ≤ 1, 1, (𝐹‘𝑀)))
210	209, 57	syl6breqr 4695	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝐹‘𝑀) ≤ 𝑇)
211		leexp1a 12919	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹‘𝑀) ∈ ℝ ∧ 𝑇 ∈ ℝ ∧ 𝑛 ∈ ℕ₀) ∧ (0 ≤ (𝐹‘𝑀) ∧ (𝐹‘𝑀) ≤ 𝑇)) → ((𝐹‘𝑀)↑𝑛) ≤ (𝑇↑𝑛))
212	100, 116, 91, 171, 210, 211	syl32anc 1334	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → ((𝐹‘𝑀)↑𝑛) ≤ (𝑇↑𝑛))
213	101, 152, 111, 207, 212	lemul2ad 10964	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑀 · ((𝐹‘𝑀)↑𝑛)) ≤ (𝑀 · (𝑇↑𝑛)))
214	109, 166, 158, 205, 213	letrd 10194	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (((𝐹‘𝑀)↑𝑛) · (𝐹‘(⌊‘(𝑘 / (𝑀↑𝑛))))) ≤ (𝑀 · (𝑇↑𝑛)))
215	98, 109, 157, 158, 165, 214	le2addd 10646	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → ((𝐹‘(𝑘 mod (𝑀↑𝑛))) + (((𝐹‘𝑀)↑𝑛) · (𝐹‘(⌊‘(𝑘 / (𝑀↑𝑛)))))) ≤ (((𝑀 · 𝑛) · (𝑇↑𝑛)) + (𝑀 · (𝑇↑𝑛))))
216		nn0cn 11302	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℂ)
217	216	ad2antlr 763	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 𝑛 ∈ ℂ)
218		1cnd 10056	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 1 ∈ ℂ)
219	192, 217, 218	adddid 10064	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑀 · (𝑛 + 1)) = ((𝑀 · 𝑛) + (𝑀 · 1)))
220	192	mulid1d 10057	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑀 · 1) = 𝑀)
221	220	oveq2d 6666	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → ((𝑀 · 𝑛) + (𝑀 · 1)) = ((𝑀 · 𝑛) + 𝑀))
222	219, 221	eqtrd 2656	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑀 · (𝑛 + 1)) = ((𝑀 · 𝑛) + 𝑀))
223	222	oveq1d 6665	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → ((𝑀 · (𝑛 + 1)) · (𝑇↑𝑛)) = (((𝑀 · 𝑛) + 𝑀) · (𝑇↑𝑛)))
224	192, 217	mulcld 10060	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑀 · 𝑛) ∈ ℂ)
225	152	recnd 10068	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑇↑𝑛) ∈ ℂ)
226	224, 192, 225	adddird 10065	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (((𝑀 · 𝑛) + 𝑀) · (𝑇↑𝑛)) = (((𝑀 · 𝑛) · (𝑇↑𝑛)) + (𝑀 · (𝑇↑𝑛))))
227	223, 226	eqtrd 2656	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → ((𝑀 · (𝑛 + 1)) · (𝑇↑𝑛)) = (((𝑀 · 𝑛) · (𝑇↑𝑛)) + (𝑀 · (𝑇↑𝑛))))
228	215, 227	breqtrrd 4681	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → ((𝐹‘(𝑘 mod (𝑀↑𝑛))) + (((𝐹‘𝑀)↑𝑛) · (𝐹‘(⌊‘(𝑘 / (𝑀↑𝑛)))))) ≤ ((𝑀 · (𝑛 + 1)) · (𝑇↑𝑛)))
229		max2 12018	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑀) ∈ ℝ ∧ 1 ∈ ℝ) → 1 ≤ if((𝐹‘𝑀) ≤ 1, 1, (𝐹‘𝑀)))
230	100, 58, 229	sylancl 694	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 1 ≤ if((𝐹‘𝑀) ≤ 1, 1, (𝐹‘𝑀)))
231	230, 57	syl6breqr 4695	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 1 ≤ 𝑇)
232		nn0z 11400	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℤ)
233	232	ad2antlr 763	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 𝑛 ∈ ℤ)
234		uzid 11702	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ_≥‘𝑛))
235	233, 234	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 𝑛 ∈ (ℤ_≥‘𝑛))
236		peano2uz 11741	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (ℤ_≥‘𝑛) → (𝑛 + 1) ∈ (ℤ_≥‘𝑛))
237	235, 236	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑛 + 1) ∈ (ℤ_≥‘𝑛))
238	116, 231, 237	leexp2ad 13041	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑇↑𝑛) ≤ (𝑇↑(𝑛 + 1)))
239	90, 113	nnmulcld 11068	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑀 · (𝑛 + 1)) ∈ ℕ)
240	239	nngt0d 11064	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 0 < (𝑀 · (𝑛 + 1)))
241		lemul2 10876	. . . . . . . . . . . . . 14 ⊢ (((𝑇↑𝑛) ∈ ℝ ∧ (𝑇↑(𝑛 + 1)) ∈ ℝ ∧ ((𝑀 · (𝑛 + 1)) ∈ ℝ ∧ 0 < (𝑀 · (𝑛 + 1)))) → ((𝑇↑𝑛) ≤ (𝑇↑(𝑛 + 1)) ↔ ((𝑀 · (𝑛 + 1)) · (𝑇↑𝑛)) ≤ ((𝑀 · (𝑛 + 1)) · (𝑇↑(𝑛 + 1)))))
242	152, 119, 115, 240, 241	syl112anc 1330	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → ((𝑇↑𝑛) ≤ (𝑇↑(𝑛 + 1)) ↔ ((𝑀 · (𝑛 + 1)) · (𝑇↑𝑛)) ≤ ((𝑀 · (𝑛 + 1)) · (𝑇↑(𝑛 + 1)))))
243	238, 242	mpbid 222	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → ((𝑀 · (𝑛 + 1)) · (𝑇↑𝑛)) ≤ ((𝑀 · (𝑛 + 1)) · (𝑇↑(𝑛 + 1))))
244	110, 153, 120, 228, 243	letrd 10194	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → ((𝐹‘(𝑘 mod (𝑀↑𝑛))) + (((𝐹‘𝑀)↑𝑛) · (𝐹‘(⌊‘(𝑘 / (𝑀↑𝑛)))))) ≤ ((𝑀 · (𝑛 + 1)) · (𝑇↑(𝑛 + 1))))
245	89, 110, 120, 151, 244	letrd 10194	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝐹‘𝑘) ≤ ((𝑀 · (𝑛 + 1)) · (𝑇↑(𝑛 + 1))))
246	245	expr 643	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1))) → (∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)) → (𝐹‘𝑘) ≤ ((𝑀 · (𝑛 + 1)) · (𝑇↑(𝑛 + 1)))))
247	246	ralrimdva 2969	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)) → ∀𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1))(𝐹‘𝑘) ≤ ((𝑀 · (𝑛 + 1)) · (𝑇↑(𝑛 + 1)))))
248	82, 247	syl5bi 232	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (∀𝑘 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑘) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)) → ∀𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1))(𝐹‘𝑘) ≤ ((𝑀 · (𝑛 + 1)) · (𝑇↑(𝑛 + 1)))))
249	248	expcom 451	. . . . . 6 ⊢ (𝑛 ∈ ℕ₀ → (𝜑 → (∀𝑘 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑘) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)) → ∀𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1))(𝐹‘𝑘) ≤ ((𝑀 · (𝑛 + 1)) · (𝑇↑(𝑛 + 1))))))
250	249	a2d 29	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ → ((𝜑 → ∀𝑘 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑘) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛))) → (𝜑 → ∀𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1))(𝐹‘𝑘) ≤ ((𝑀 · (𝑛 + 1)) · (𝑇↑(𝑛 + 1))))))
251	9, 18, 27, 36, 79, 250	nn0ind 11472	. . . 4 ⊢ (𝑋 ∈ ℕ₀ → (𝜑 → ∀𝑘 ∈ (0...((𝑀↑𝑋) − 1))(𝐹‘𝑘) ≤ ((𝑀 · 𝑋) · (𝑇↑𝑋))))
252	251	impcom 446	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ ℕ₀) → ∀𝑘 ∈ (0...((𝑀↑𝑋) − 1))(𝐹‘𝑘) ≤ ((𝑀 · 𝑋) · (𝑇↑𝑋)))
253		fveq2 6191	. . . . 5 ⊢ (𝑘 = 𝑌 → (𝐹‘𝑘) = (𝐹‘𝑌))
254	253	breq1d 4663	. . . 4 ⊢ (𝑘 = 𝑌 → ((𝐹‘𝑘) ≤ ((𝑀 · 𝑋) · (𝑇↑𝑋)) ↔ (𝐹‘𝑌) ≤ ((𝑀 · 𝑋) · (𝑇↑𝑋))))
255	254	rspccv 3306	. . 3 ⊢ (∀𝑘 ∈ (0...((𝑀↑𝑋) − 1))(𝐹‘𝑘) ≤ ((𝑀 · 𝑋) · (𝑇↑𝑋)) → (𝑌 ∈ (0...((𝑀↑𝑋) − 1)) → (𝐹‘𝑌) ≤ ((𝑀 · 𝑋) · (𝑇↑𝑋))))
256	252, 255	syl 17	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ ℕ₀) → (𝑌 ∈ (0...((𝑀↑𝑋) − 1)) → (𝐹‘𝑌) ≤ ((𝑀 · 𝑋) · (𝑇↑𝑋))))
257	256	3impia 1261	1 ⊢ ((𝜑 ∧ 𝑋 ∈ ℕ₀ ∧ 𝑌 ∈ (0...((𝑀↑𝑋) − 1))) → (𝐹‘𝑌) ≤ ((𝑀 · 𝑋) · (𝑇↑𝑋)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ifcif 4086 class class class wbr 4653 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 < clt 10074 ≤ cle 10075 − cmin 10266 -cneg 10267 / cdiv 10684 ℕcn 11020 2c2 11070 ℕ₀cn0 11292 ℤcz 11377 ℤ_≥cuz 11687 ℚcq 11788 ℝ⁺crp 11832 ...cfz 12326 ⌊cfl 12591 mod cmo 12668 ↑cexp 12860 ℙcprime 15385 pCnt cpc 15541 ↾_s cress 15858 +_gcplusg 15941 ._rcmulr 15942 AbsValcabv 18816 ℂ_fldccnfld 19746 logclog 24301

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-ico 12181 df-fz 12327 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-subg 17591 df-cmn 18195 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-subrg 18778 df-abv 18817 df-cnfld 19747

This theorem is referenced by: ostth2lem3 25324

W3C validator