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Theorem ostth2lem2 27612

Description: Lemma for ostth2 27615. (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 7427	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑀↑𝑥) = (𝑀↑0))
2	1	oveq1d 7434	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝑀↑𝑥) − 1) = ((𝑀↑0) − 1))
3	2	oveq2d 7435	. . . . . . 7 ⊢ (𝑥 = 0 → (0...((𝑀↑𝑥) − 1)) = (0...((𝑀↑0) − 1)))
4		oveq2 7427	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑀 · 𝑥) = (𝑀 · 0))
5		oveq2 7427	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑇↑𝑥) = (𝑇↑0))
6	4, 5	oveq12d 7437	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝑀 · 𝑥) · (𝑇↑𝑥)) = ((𝑀 · 0) · (𝑇↑0)))
7	6	breq2d 5161	. . . . . . 7 ⊢ (𝑥 = 0 → ((𝐹‘𝑘) ≤ ((𝑀 · 𝑥) · (𝑇↑𝑥)) ↔ (𝐹‘𝑘) ≤ ((𝑀 · 0) · (𝑇↑0))))
8	3, 7	raleqbidv 3329	. . . . . 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 339	. . . . 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 7427	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (𝑀↑𝑥) = (𝑀↑𝑛))
11	10	oveq1d 7434	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → ((𝑀↑𝑥) − 1) = ((𝑀↑𝑛) − 1))
12	11	oveq2d 7435	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (0...((𝑀↑𝑥) − 1)) = (0...((𝑀↑𝑛) − 1)))
13		oveq2 7427	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (𝑀 · 𝑥) = (𝑀 · 𝑛))
14		oveq2 7427	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → (𝑇↑𝑥) = (𝑇↑𝑛))
15	13, 14	oveq12d 7437	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → ((𝑀 · 𝑥) · (𝑇↑𝑥)) = ((𝑀 · 𝑛) · (𝑇↑𝑛)))
16	15	breq2d 5161	. . . . . . 7 ⊢ (𝑥 = 𝑛 → ((𝐹‘𝑘) ≤ ((𝑀 · 𝑥) · (𝑇↑𝑥)) ↔ (𝐹‘𝑘) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛))))
17	12, 16	raleqbidv 3329	. . . . . 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 339	. . . . 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 7427	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → (𝑀↑𝑥) = (𝑀↑(𝑛 + 1)))
20	19	oveq1d 7434	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → ((𝑀↑𝑥) − 1) = ((𝑀↑(𝑛 + 1)) − 1))
21	20	oveq2d 7435	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (0...((𝑀↑𝑥) − 1)) = (0...((𝑀↑(𝑛 + 1)) − 1)))
22		oveq2 7427	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → (𝑀 · 𝑥) = (𝑀 · (𝑛 + 1)))
23		oveq2 7427	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → (𝑇↑𝑥) = (𝑇↑(𝑛 + 1)))
24	22, 23	oveq12d 7437	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → ((𝑀 · 𝑥) · (𝑇↑𝑥)) = ((𝑀 · (𝑛 + 1)) · (𝑇↑(𝑛 + 1))))
25	24	breq2d 5161	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → ((𝐹‘𝑘) ≤ ((𝑀 · 𝑥) · (𝑇↑𝑥)) ↔ (𝐹‘𝑘) ≤ ((𝑀 · (𝑛 + 1)) · (𝑇↑(𝑛 + 1)))))
26	21, 25	raleqbidv 3329	. . . . . 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 339	. . . . 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 7427	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑀↑𝑥) = (𝑀↑𝑋))
29	28	oveq1d 7434	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝑀↑𝑥) − 1) = ((𝑀↑𝑋) − 1))
30	29	oveq2d 7435	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (0...((𝑀↑𝑥) − 1)) = (0...((𝑀↑𝑋) − 1)))
31		oveq2 7427	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑀 · 𝑥) = (𝑀 · 𝑋))
32		oveq2 7427	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑇↑𝑥) = (𝑇↑𝑋))
33	31, 32	oveq12d 7437	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝑀 · 𝑥) · (𝑇↑𝑥)) = ((𝑀 · 𝑋) · (𝑇↑𝑋)))
34	33	breq2d 5161	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑘) ≤ ((𝑀 · 𝑥) · (𝑇↑𝑥)) ↔ (𝐹‘𝑘) ≤ ((𝑀 · 𝑋) · (𝑇↑𝑋))))
35	30, 34	raleqbidv 3329	. . . . . 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 339	. . . . 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 12901	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ (ℤ_≥‘2) → 𝑀 ∈ ℕ)
39	37, 38	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℕ)
40	39	nncnd 12261	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℂ)
41	40	exp0d 14140	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀↑0) = 1)
42	41	oveq1d 7434	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑀↑0) − 1) = (1 − 1))
43		1m1e0 12317	. . . . . . . . . 10 ⊢ (1 − 1) = 0
44	42, 43	eqtrdi 2781	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀↑0) − 1) = 0)
45	44	oveq2d 7435	. . . . . . . 8 ⊢ (𝜑 → (0...((𝑀↑0) − 1)) = (0...0))
46	45	eleq2d 2811	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ (0...((𝑀↑0) − 1)) ↔ 𝑘 ∈ (0...0)))
47		0le0 12346	. . . . . . . . . 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 27599	. . . . . . . . . . 11 ⊢ 0 = (0_g‘𝑄)
53	50, 52	abv0 20723	. . . . . . . . . 10 ⊢ (𝐹 ∈ 𝐴 → (𝐹‘0) = 0)
54	49, 53	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘0) = 0)
55	40	mul01d 11445	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 · 0) = 0)
56	55	oveq1d 7434	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑀 · 0) · (𝑇↑0)) = (0 · (𝑇↑0)))
57		ostth2.7	. . . . . . . . . . . . . 14 ⊢ 𝑇 = if((𝐹‘𝑀) ≤ 1, 1, (𝐹‘𝑀))
58		1re 11246	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℝ
59		nnq 12979	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℚ)
60	39, 59	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ ℚ)
61	51	qrngbas 27597	. . . . . . . . . . . . . . . . 17 ⊢ ℚ = (Base‘𝑄)
62	50, 61	abvcl 20716	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑀 ∈ ℚ) → (𝐹‘𝑀) ∈ ℝ)
63	49, 60, 62	syl2anc 582	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ℝ)
64		ifcl 4575	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℝ ∧ (𝐹‘𝑀) ∈ ℝ) → if((𝐹‘𝑀) ≤ 1, 1, (𝐹‘𝑀)) ∈ ℝ)
65	58, 63, 64	sylancr 585	. . . . . . . . . . . . . 14 ⊢ (𝜑 → if((𝐹‘𝑀) ≤ 1, 1, (𝐹‘𝑀)) ∈ ℝ)
66	57, 65	eqeltrid 2829	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑇 ∈ ℝ)
67	66	recnd 11274	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ ℂ)
68		0nn0 12520	. . . . . . . . . . . 12 ⊢ 0 ∈ ℕ₀
69		expcl 14080	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ ℂ ∧ 0 ∈ ℕ₀) → (𝑇↑0) ∈ ℂ)
70	67, 68, 69	sylancl 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑇↑0) ∈ ℂ)
71	70	mul02d 11444	. . . . . . . . . 10 ⊢ (𝜑 → (0 · (𝑇↑0)) = 0)
72	56, 71	eqtrd 2765	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀 · 0) · (𝑇↑0)) = 0)
73	48, 54, 72	3brtr4d 5181	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘0) ≤ ((𝑀 · 0) · (𝑇↑0)))
74		elfz1eq 13547	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...0) → 𝑘 = 0)
75	74	fveq2d 6900	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...0) → (𝐹‘𝑘) = (𝐹‘0))
76	75	breq1d 5159	. . . . . . . 8 ⊢ (𝑘 ∈ (0...0) → ((𝐹‘𝑘) ≤ ((𝑀 · 0) · (𝑇↑0)) ↔ (𝐹‘0) ≤ ((𝑀 · 0) · (𝑇↑0))))
77	73, 76	syl5ibrcom 246	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ (0...0) → (𝐹‘𝑘) ≤ ((𝑀 · 0) · (𝑇↑0))))
78	46, 77	sylbid 239	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ (0...((𝑀↑0) − 1)) → (𝐹‘𝑘) ≤ ((𝑀 · 0) · (𝑇↑0))))
79	78	ralrimiv 3134	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (0...((𝑀↑0) − 1))(𝐹‘𝑘) ≤ ((𝑀 · 0) · (𝑇↑0)))
80		fveq2 6896	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
81	80	breq1d 5159	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)) ↔ (𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛))))
82	81	cbvralvw 3224	. . . . . . . 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 724	. . . . . . . . . . . 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 13536	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) → 𝑘 ∈ ℤ)
85	84	ad2antrl 726	. . . . . . . . . . . . 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 12971	. . . . . . . . . . . . 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 20716	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑘 ∈ ℚ) → (𝐹‘𝑘) ∈ ℝ)
89	83, 87, 88	syl2anc 582	. . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . . 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 14243	. . . . . . . . . . . . . . . 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 13893	. . . . . . . . . . . . . . 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 12617	. . . . . . . . . . . . . 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 12971	. . . . . . . . . . . . . 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 20716	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑘 mod (𝑀↑𝑛)) ∈ ℚ) → (𝐹‘(𝑘 mod (𝑀↑𝑛))) ∈ ℝ)
98	83, 96, 97	syl2anc 582	. . . . . . . . . . . 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 582	. . . . . . . . . . . . . 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 14163	. . . . . . . . . . . . 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 12699	. . . . . . . . . . . . . . . . 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 12299	. . . . . . . . . . . . . . . 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 13799	. . . . . . . . . . . . . . 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 12971	. . . . . . . . . . . . . . 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 20716	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ 𝐴 ∧ (⌊‘(𝑘 / (𝑀↑𝑛))) ∈ ℚ) → (𝐹‘(⌊‘(𝑘 / (𝑀↑𝑛)))) ∈ ℝ)
108	83, 106, 107	syl2anc 582	. . . . . . . . . . . . 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 11276	. . . . . . . . . . . 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 11275	. . . . . . . . . . 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 12260	. . . . . . . . . . . . 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 12544	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ)
113	112	ad2antlr 725	. . . . . . . . . . . . . 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 12260	. . . . . . . . . . . . 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 11276	. . . . . . . . . . . 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 724	. . . . . . . . . . . . 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 12545	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ₀)
118	117	ad2antlr 725	. . . . . . . . . . . . 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 14163	. . . . . . . . . . . 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 11276	. . . . . . . . . . 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 12979	. . . . . . . . . . . . . . 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 12984	. . . . . . . . . . . . . 14 ⊢ (((𝑀↑𝑛) ∈ ℚ ∧ (⌊‘(𝑘 / (𝑀↑𝑛))) ∈ ℚ) → ((𝑀↑𝑛) · (⌊‘(𝑘 / (𝑀↑𝑛)))) ∈ ℚ)
124	122, 106, 123	syl2anc 582	. . . . . . . . . . . . 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 12978	. . . . . . . . . . . . . . 15 ⊢ ℚ ∈ V
126		cnfldadd 21302	. . . . . . . . . . . . . . . 16 ⊢ + = (+_g‘ℂ_fld)
127	51, 126	ressplusg 17274	. . . . . . . . . . . . . . 15 ⊢ (ℚ ∈ V → + = (+_g‘𝑄))
128	125, 127	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ + = (+_g‘𝑄)
129	50, 61, 128	abvtri 20722	. . . . . . . . . . . . 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 1368	. . . . . . . . . . . 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 13049	. . . . . . . . . . . . . . . 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 13872	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℝ ∧ (𝑀↑𝑛) ∈ ℝ⁺) → (𝑘 mod (𝑀↑𝑛)) = (𝑘 − ((𝑀↑𝑛) · (⌊‘(𝑘 / (𝑀↑𝑛))))))
133	102, 131, 132	syl2anc 582	. . . . . . . . . . . . . . 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 7434	. . . . . . . . . . . . . 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 11274	. . . . . . . . . . . . . . 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 12980	. . . . . . . . . . . . . . . 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 11607	. . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . 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 6900	. . . . . . . . . . . 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 21304	. . . . . . . . . . . . . . . . . 18 ⊢ · = (._r‘ℂ_fld)
142	51, 141	ressmulr 17291	. . . . . . . . . . . . . . . . 17 ⊢ (ℚ ∈ V → · = (._r‘𝑄))
143	125, 142	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ · = (._r‘𝑄)
144	50, 61, 143	abvmul 20721	. . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . 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 27604	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑀 ∈ ℚ ∧ 𝑛 ∈ ℕ₀) → (𝐹‘(𝑀↑𝑛)) = ((𝐹‘𝑀)↑𝑛))
147	83, 99, 91, 146	syl3anc 1368	. . . . . . . . . . . . . . 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 7434	. . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . 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 7435	. . . . . . . . . . . 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 5180	. . . . . . . . . . 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 14163	. . . . . . . . . . . . 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 11276	. . . . . . . . . . . 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 12514	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℝ)
155	154	ad2antlr 725	. . . . . . . . . . . . . . . 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 11276	. . . . . . . . . . . . . . 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 11276	. . . . . . . . . . . . . 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 11276	. . . . . . . . . . . . . 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		fveq2 6896	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = (𝑘 mod (𝑀↑𝑛)) → (𝐹‘𝑗) = (𝐹‘(𝑘 mod (𝑀↑𝑛))))
160	159	breq1d 5159	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = (𝑘 mod (𝑀↑𝑛)) → ((𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)) ↔ (𝐹‘(𝑘 mod (𝑀↑𝑛))) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛))))
161		simprr 771	. . . . . . . . . . . . . . 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		zmodfz 13894	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℤ ∧ (𝑀↑𝑛) ∈ ℕ) → (𝑘 mod (𝑀↑𝑛)) ∈ (0...((𝑀↑𝑛) − 1)))
163	85, 92, 162	syl2anc 582	. . . . . . . . . . . . . . 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)))
164	160, 161, 163	rspcdva 3607	. . . . . . . . . . . . . 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)))
165	111, 101	remulcld 11276	. . . . . . . . . . . . . . 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)
166	101	recnd 11274	. . . . . . . . . . . . . . . . 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)
167	108	recnd 11274	. . . . . . . . . . . . . . . . 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)
168	166, 167	mulcomd 11267	. . . . . . . . . . . . . . . 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)))
169	50, 61	abvge0 20717	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑀 ∈ ℚ) → 0 ≤ (𝐹‘𝑀))
170	83, 99, 169	syl2anc 582	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 0 ≤ (𝐹‘𝑀))
171	100, 91, 170	expge0d 14164	. . . . . . . . . . . . . . . . 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))
172	104	zred 12699	. . . . . . . . . . . . . . . . . 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)
173		elfzle1 13539	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) → 0 ≤ 𝑘)
174	173	ad2antrl 726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 0 ≤ 𝑘)
175	92	nnred 12260	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑀↑𝑛) ∈ ℝ)
176	92	nngt0d 12294	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 0 < (𝑀↑𝑛))
177		divge0 12116	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑘 ∈ ℝ ∧ 0 ≤ 𝑘) ∧ ((𝑀↑𝑛) ∈ ℝ ∧ 0 < (𝑀↑𝑛))) → 0 ≤ (𝑘 / (𝑀↑𝑛)))
178	102, 174, 175, 176, 177	syl22anc 837	. . . . . . . . . . . . . . . . . . . 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)))
179		flge0nn0 13821	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑘 / (𝑀↑𝑛)) ∈ ℝ ∧ 0 ≤ (𝑘 / (𝑀↑𝑛))) → (⌊‘(𝑘 / (𝑀↑𝑛))) ∈ ℕ₀)
180	103, 178, 179	syl2anc 582	. . . . . . . . . . . . . . . . . . 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)
181	51, 50	qabvle 27603	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹 ∈ 𝐴 ∧ (⌊‘(𝑘 / (𝑀↑𝑛))) ∈ ℕ₀) → (𝐹‘(⌊‘(𝑘 / (𝑀↑𝑛)))) ≤ (⌊‘(𝑘 / (𝑀↑𝑛))))
182	83, 180, 181	syl2anc 582	. . . . . . . . . . . . . . . . . 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))))
183		simprl 769	. . . . . . . . . . . . . . . . . . . . . . . 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)))
184		0z 12602	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 0 ∈ ℤ
185	90, 118	nnexpcld 14243	. . . . . . . . . . . . . . . . . . . . . . . . . 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)
186	185	nnzd 12618	. . . . . . . . . . . . . . . . . . . . . . . . 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)
187		elfzm11 13607	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((0 ∈ ℤ ∧ (𝑀↑(𝑛 + 1)) ∈ ℤ) → (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ↔ (𝑘 ∈ ℤ ∧ 0 ≤ 𝑘 ∧ 𝑘 < (𝑀↑(𝑛 + 1)))))
188	184, 186, 187	sylancr 585	. . . . . . . . . . . . . . . . . . . . . . . 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)))))
189	183, 188	mpbid 231	. . . . . . . . . . . . . . . . . . . . . . 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))))
190	189	simp3d 1141	. . . . . . . . . . . . . . . . . . . . . 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)))
191	90	nncnd 12261	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 𝑀 ∈ ℂ)
192	191, 91	expp1d 14147	. . . . . . . . . . . . . . . . . . . . . 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))
193	190, 192	breqtrd 5175	. . . . . . . . . . . . . . . . . . . . 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))
194		ltdivmul 12122	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ ((𝑀↑𝑛) ∈ ℝ ∧ 0 < (𝑀↑𝑛))) → ((𝑘 / (𝑀↑𝑛)) < 𝑀 ↔ 𝑘 < ((𝑀↑𝑛) · 𝑀)))
195	102, 111, 175, 176, 194	syl112anc 1371	. . . . . . . . . . . . . . . . . . . . 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)))
196	193, 195	mpbird 256	. . . . . . . . . . . . . . . . . . . 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)
197	90	nnzd 12618	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 𝑀 ∈ ℤ)
198		fllt 13807	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑘 / (𝑀↑𝑛)) ∈ ℝ ∧ 𝑀 ∈ ℤ) → ((𝑘 / (𝑀↑𝑛)) < 𝑀 ↔ (⌊‘(𝑘 / (𝑀↑𝑛))) < 𝑀))
199	103, 197, 198	syl2anc 582	. . . . . . . . . . . . . . . . . . . 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))
200	196, 199	mpbid 231	. . . . . . . . . . . . . . . . . . 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)
201	172, 111, 200	ltled 11394	. . . . . . . . . . . . . . . . . 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)
202	108, 172, 111, 182, 201	letrd 11403	. . . . . . . . . . . . . . . . 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)
203	108, 111, 101, 171, 202	lemul1ad 12186	. . . . . . . . . . . . . . . 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)))
204	168, 203	eqbrtrd 5171	. . . . . . . . . . . . . . 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)))
205	90	nnnn0d 12565	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 𝑀 ∈ ℕ₀)
206	205	nn0ge0d 12568	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 0 ≤ 𝑀)
207		max1 13199	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹‘𝑀) ∈ ℝ ∧ 1 ∈ ℝ) → (𝐹‘𝑀) ≤ if((𝐹‘𝑀) ≤ 1, 1, (𝐹‘𝑀)))
208	100, 58, 207	sylancl 584	. . . . . . . . . . . . . . . . . 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)))
209	208, 57	breqtrrdi 5191	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝐹‘𝑀) ≤ 𝑇)
210		leexp1a 14175	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹‘𝑀) ∈ ℝ ∧ 𝑇 ∈ ℝ ∧ 𝑛 ∈ ℕ₀) ∧ (0 ≤ (𝐹‘𝑀) ∧ (𝐹‘𝑀) ≤ 𝑇)) → ((𝐹‘𝑀)↑𝑛) ≤ (𝑇↑𝑛))
211	100, 116, 91, 170, 209, 210	syl32anc 1375	. . . . . . . . . . . . . . . 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))
212	101, 152, 111, 206, 211	lemul2ad 12187	. . . . . . . . . . . . . . 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)))
213	109, 165, 158, 204, 212	letrd 11403	. . . . . . . . . . . . . 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)))
214	98, 109, 157, 158, 164, 213	le2addd 11865	. . . . . . . . . . . . 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))))
215		nn0cn 12515	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℂ)
216	215	ad2antlr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 𝑛 ∈ ℂ)
217		1cnd 11241	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 1 ∈ ℂ)
218	191, 216, 217	adddid 11270	. . . . . . . . . . . . . . . 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)))
219	191	mulridd 11263	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑀 · 1) = 𝑀)
220	219	oveq2d 7435	. . . . . . . . . . . . . . . 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))
221	218, 220	eqtrd 2765	. . . . . . . . . . . . . . 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))
222	221	oveq1d 7434	. . . . . . . . . . . . . 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)))
223	191, 216	mulcld 11266	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑀 · 𝑛) ∈ ℂ)
224	152	recnd 11274	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → (𝑇↑𝑛) ∈ ℂ)
225	223, 191, 224	adddird 11271	. . . . . . . . . . . . . 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))))
226	222, 225	eqtrd 2765	. . . . . . . . . . . . 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))))
227	214, 226	breqtrrd 5177	. . . . . . . . . . . 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)))
228		max2 13201	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑀) ∈ ℝ ∧ 1 ∈ ℝ) → 1 ≤ if((𝐹‘𝑀) ≤ 1, 1, (𝐹‘𝑀)))
229	100, 58, 228	sylancl 584	. . . . . . . . . . . . . . 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)))
230	229, 57	breqtrrdi 5191	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 1 ≤ 𝑇)
231		nn0z 12616	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℤ)
232	231	ad2antlr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ₀) ∧ (𝑘 ∈ (0...((𝑀↑(𝑛 + 1)) − 1)) ∧ ∀𝑗 ∈ (0...((𝑀↑𝑛) − 1))(𝐹‘𝑗) ≤ ((𝑀 · 𝑛) · (𝑇↑𝑛)))) → 𝑛 ∈ ℤ)
233		uzid 12870	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ_≥‘𝑛))
234	232, 233	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))
235		peano2uz 12918	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (ℤ_≥‘𝑛) → (𝑛 + 1) ∈ (ℤ_≥‘𝑛))
236	234, 235	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))
237	116, 230, 236	leexp2ad 14252	. . . . . . . . . . . . 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)))
238	90, 113	nnmulcld 12298	. . . . . . . . . . . . . . 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)
239	238	nngt0d 12294	. . . . . . . . . . . . . 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)))
240		lemul2 12100	. . . . . . . . . . . . . 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)))))
241	152, 119, 115, 239, 240	syl112anc 1371	. . . . . . . . . . . . 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)))))
242	237, 241	mpbid 231	. . . . . . . . . . . 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))))
243	110, 153, 120, 227, 242	letrd 11403	. . . . . . . . . . 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))))
244	89, 110, 120, 151, 243	letrd 11403	. . . . . . . . . 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))))
245	244	expr 455	. . . . . . . . 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)))))
246	245	ralrimdva 3143	. . . . . . . 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)))))
247	82, 246	biimtrid 241	. . . . . . 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)))))
248	247	expcom 412	. . . . . 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))))))
249	248	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))))))
250	9, 18, 27, 36, 79, 249	nn0ind 12690	. . . 4 ⊢ (𝑋 ∈ ℕ₀ → (𝜑 → ∀𝑘 ∈ (0...((𝑀↑𝑋) − 1))(𝐹‘𝑘) ≤ ((𝑀 · 𝑋) · (𝑇↑𝑋))))
251	250	impcom 406	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ ℕ₀) → ∀𝑘 ∈ (0...((𝑀↑𝑋) − 1))(𝐹‘𝑘) ≤ ((𝑀 · 𝑋) · (𝑇↑𝑋)))
252		fveq2 6896	. . . . 5 ⊢ (𝑘 = 𝑌 → (𝐹‘𝑘) = (𝐹‘𝑌))
253	252	breq1d 5159	. . . 4 ⊢ (𝑘 = 𝑌 → ((𝐹‘𝑘) ≤ ((𝑀 · 𝑋) · (𝑇↑𝑋)) ↔ (𝐹‘𝑌) ≤ ((𝑀 · 𝑋) · (𝑇↑𝑋))))
254	253	rspccv 3603	. . 3 ⊢ (∀𝑘 ∈ (0...((𝑀↑𝑋) − 1))(𝐹‘𝑘) ≤ ((𝑀 · 𝑋) · (𝑇↑𝑋)) → (𝑌 ∈ (0...((𝑀↑𝑋) − 1)) → (𝐹‘𝑌) ≤ ((𝑀 · 𝑋) · (𝑇↑𝑋))))
255	251, 254	syl 17	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ ℕ₀) → (𝑌 ∈ (0...((𝑀↑𝑋) − 1)) → (𝐹‘𝑌) ≤ ((𝑀 · 𝑋) · (𝑇↑𝑋))))
256	255	3impia 1114	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 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3050 Vcvv 3461 ifcif 4530 class class class wbr 5149 ↦ cmpt 5232 ‘cfv 6549 (class class class)co 7419 ℂcc 11138 ℝcr 11139 0cc0 11140 1c1 11141 + caddc 11143 · cmul 11145 < clt 11280 ≤ cle 11281 − cmin 11476 -cneg 11477 / cdiv 11903 ℕcn 12245 2c2 12300 ℕ₀cn0 12505 ℤcz 12591 ℤ_≥cuz 12855 ℚcq 12965 ℝ⁺crp 13009 ...cfz 13519 ⌊cfl 13791 mod cmo 13870 ↑cexp 14062 ℙcprime 16645 pCnt cpc 16808 ↾_s cress 17212 +_gcplusg 17236 ._rcmulr 17237 AbsValcabv 20708 ℂ_fldccnfld 21296 logclog 26533

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 ax-addf 11219 ax-mulf 11220

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9467 df-inf 9468 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-q 12966 df-rp 13010 df-ico 13365 df-fz 13520 df-fl 13793 df-mod 13871 df-seq 14003 df-exp 14063 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-starv 17251 df-tset 17255 df-ple 17256 df-ds 17258 df-unif 17259 df-0g 17426 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-grp 18901 df-minusg 18902 df-subg 19086 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-cring 20188 df-oppr 20285 df-dvdsr 20308 df-unit 20309 df-invr 20339 df-dvr 20352 df-subrng 20495 df-subrg 20520 df-drng 20638 df-abv 20709 df-cnfld 21297

This theorem is referenced by: ostth2lem3 27613

W3C validator