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Theorem pntlemq 27104

Description: Lemma for pntlemj 27106. (Contributed by Mario Carneiro, 7-Jun-2016.)

**Hypotheses**
Ref	Expression
pntlem1.r	⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
pntlem1.a	⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
pntlem1.b	⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
pntlem1.l	⊢ (𝜑 → 𝐿 ∈ (0(,)1))
pntlem1.d	⊢ 𝐷 = (𝐴 + 1)
pntlem1.f	⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2)))
pntlem1.u	⊢ (𝜑 → 𝑈 ∈ ℝ⁺)
pntlem1.u2	⊢ (𝜑 → 𝑈 ≤ 𝐴)
pntlem1.e	⊢ 𝐸 = (𝑈 / 𝐷)
pntlem1.k	⊢ 𝐾 = (exp‘(𝐵 / 𝐸))
pntlem1.y	⊢ (𝜑 → (𝑌 ∈ ℝ⁺ ∧ 1 ≤ 𝑌))
pntlem1.x	⊢ (𝜑 → (𝑋 ∈ ℝ⁺ ∧ 𝑌 < 𝑋))
pntlem1.c	⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
pntlem1.w	⊢ 𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))
pntlem1.z	⊢ (𝜑 → 𝑍 ∈ (𝑊[,)+∞))
pntlem1.m	⊢ 𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)
pntlem1.n	⊢ 𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2))
pntlem1.U	⊢ (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑈)
pntlem1.K	⊢ (𝜑 → ∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ⁺ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝐸))
pntlem1.o	⊢ 𝑂 = (((⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝐽))))
pntlem1.v	⊢ (𝜑 → 𝑉 ∈ ℝ⁺)
pntlem1.V	⊢ (𝜑 → (((𝐾↑𝐽) < 𝑉 ∧ ((1 + (𝐿 · 𝐸)) · 𝑉) < (𝐾 · (𝐾↑𝐽))) ∧ ∀𝑢 ∈ (𝑉[,]((1 + (𝐿 · 𝐸)) · 𝑉))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝐸))
pntlem1.j	⊢ (𝜑 → 𝐽 ∈ (𝑀..^𝑁))
pntlem1.i	⊢ 𝐼 = (((⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))) + 1)...(⌊‘(𝑍 / 𝑉)))

**Assertion**
Ref	Expression
pntlemq	⊢ (𝜑 → 𝐼 ⊆ 𝑂)

Distinct variable groups: 𝑧,𝐶 𝑦,𝑧,𝐽 𝑦,𝑢,𝑧,𝐿 𝑦,𝐾,𝑧 𝑧,𝑀 𝑧,𝑂 𝑧,𝑁 𝑢,𝑅,𝑦,𝑧 𝑢,𝑉 𝑧,𝑈 𝑧,𝑊 𝑦,𝑋,𝑧 𝑧,𝑌 𝑢,𝑎,𝑦,𝑧,𝐸 𝑢,𝑍,𝑧 Allowed substitution hints: 𝜑(𝑦,𝑧,𝑢,𝑎) 𝐴(𝑦,𝑧,𝑢,𝑎) 𝐵(𝑦,𝑧,𝑢,𝑎) 𝐶(𝑦,𝑢,𝑎) 𝐷(𝑦,𝑧,𝑢,𝑎) 𝑅(𝑎) 𝑈(𝑦,𝑢,𝑎) 𝐹(𝑦,𝑧,𝑢,𝑎) 𝐼(𝑦,𝑧,𝑢,𝑎) 𝐽(𝑢,𝑎) 𝐾(𝑢,𝑎) 𝐿(𝑎) 𝑀(𝑦,𝑢,𝑎) 𝑁(𝑦,𝑢,𝑎) 𝑂(𝑦,𝑢,𝑎) 𝑉(𝑦,𝑧,𝑎) 𝑊(𝑦,𝑢,𝑎) 𝑋(𝑢,𝑎) 𝑌(𝑦,𝑢,𝑎) 𝑍(𝑦,𝑎)

**Proof of Theorem pntlemq**
Step	Hyp	Ref	Expression
1		pntlem1.r	. . . . . . . . . 10 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
2		pntlem1.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
3		pntlem1.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
4		pntlem1.l	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ (0(,)1))
5		pntlem1.d	. . . . . . . . . 10 ⊢ 𝐷 = (𝐴 + 1)
6		pntlem1.f	. . . . . . . . . 10 ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2)))
7		pntlem1.u	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ ℝ⁺)
8		pntlem1.u2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ≤ 𝐴)
9		pntlem1.e	. . . . . . . . . 10 ⊢ 𝐸 = (𝑈 / 𝐷)
10		pntlem1.k	. . . . . . . . . 10 ⊢ 𝐾 = (exp‘(𝐵 / 𝐸))
11		pntlem1.y	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌 ∈ ℝ⁺ ∧ 1 ≤ 𝑌))
12		pntlem1.x	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 ∈ ℝ⁺ ∧ 𝑌 < 𝑋))
13		pntlem1.c	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
14		pntlem1.w	. . . . . . . . . 10 ⊢ 𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))
15		pntlem1.z	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ (𝑊[,)+∞))
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15	pntlemb 27100	. . . . . . . . 9 ⊢ (𝜑 → (𝑍 ∈ ℝ⁺ ∧ (1 < 𝑍 ∧ e ≤ (√‘𝑍) ∧ (√‘𝑍) ≤ (𝑍 / 𝑌)) ∧ ((4 / (𝐿 · 𝐸)) ≤ (√‘𝑍) ∧ (((log‘𝑋) / (log‘𝐾)) + 2) ≤ (((log‘𝑍) / (log‘𝐾)) / 4) ∧ ((𝑈 · 3) + 𝐶) ≤ (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)))))
17	16	simp1d 1143	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ ℝ⁺)
18	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	pntlemc 27098	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸 ∈ ℝ⁺ ∧ 𝐾 ∈ ℝ⁺ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ⁺)))
19	18	simp2d 1144	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ℝ⁺)
20		pntlem1.j	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (𝑀..^𝑁))
21		elfzoelz 13632	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (𝑀..^𝑁) → 𝐽 ∈ ℤ)
22	20, 21	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ ℤ)
23	22	peano2zd 12669	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 + 1) ∈ ℤ)
24	19, 23	rpexpcld 14210	. . . . . . . 8 ⊢ (𝜑 → (𝐾↑(𝐽 + 1)) ∈ ℝ⁺)
25	17, 24	rpdivcld 13033	. . . . . . 7 ⊢ (𝜑 → (𝑍 / (𝐾↑(𝐽 + 1))) ∈ ℝ⁺)
26	25	rpred 13016	. . . . . 6 ⊢ (𝜑 → (𝑍 / (𝐾↑(𝐽 + 1))) ∈ ℝ)
27	26	flcld 13763	. . . . 5 ⊢ (𝜑 → (⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) ∈ ℤ)
28		1rp 12978	. . . . . . . . . 10 ⊢ 1 ∈ ℝ⁺
29	1, 2, 3, 4, 5, 6	pntlemd 27097	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐿 ∈ ℝ⁺ ∧ 𝐷 ∈ ℝ⁺ ∧ 𝐹 ∈ ℝ⁺))
30	29	simp1d 1143	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ ℝ⁺)
31	18	simp1d 1143	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
32	30, 31	rpmulcld 13032	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿 · 𝐸) ∈ ℝ⁺)
33		rpaddcl 12996	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ⁺ ∧ (𝐿 · 𝐸) ∈ ℝ⁺) → (1 + (𝐿 · 𝐸)) ∈ ℝ⁺)
34	28, 32, 33	sylancr 588	. . . . . . . . 9 ⊢ (𝜑 → (1 + (𝐿 · 𝐸)) ∈ ℝ⁺)
35		pntlem1.v	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ∈ ℝ⁺)
36	34, 35	rpmulcld 13032	. . . . . . . 8 ⊢ (𝜑 → ((1 + (𝐿 · 𝐸)) · 𝑉) ∈ ℝ⁺)
37	17, 36	rpdivcld 13033	. . . . . . 7 ⊢ (𝜑 → (𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉)) ∈ ℝ⁺)
38	37	rpred 13016	. . . . . 6 ⊢ (𝜑 → (𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉)) ∈ ℝ)
39	38	flcld 13763	. . . . 5 ⊢ (𝜑 → (⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))) ∈ ℤ)
40	36	rpred 13016	. . . . . . . 8 ⊢ (𝜑 → ((1 + (𝐿 · 𝐸)) · 𝑉) ∈ ℝ)
41	24	rpred 13016	. . . . . . . 8 ⊢ (𝜑 → (𝐾↑(𝐽 + 1)) ∈ ℝ)
42		pntlem1.V	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐾↑𝐽) < 𝑉 ∧ ((1 + (𝐿 · 𝐸)) · 𝑉) < (𝐾 · (𝐾↑𝐽))) ∧ ∀𝑢 ∈ (𝑉[,]((1 + (𝐿 · 𝐸)) · 𝑉))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝐸))
43	42	simpld 496	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐾↑𝐽) < 𝑉 ∧ ((1 + (𝐿 · 𝐸)) · 𝑉) < (𝐾 · (𝐾↑𝐽))))
44	43	simprd 497	. . . . . . . . 9 ⊢ (𝜑 → ((1 + (𝐿 · 𝐸)) · 𝑉) < (𝐾 · (𝐾↑𝐽)))
45	19	rpcnd 13018	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ ℂ)
46	19, 22	rpexpcld 14210	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐾↑𝐽) ∈ ℝ⁺)
47	46	rpcnd 13018	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾↑𝐽) ∈ ℂ)
48	45, 47	mulcomd 11235	. . . . . . . . . 10 ⊢ (𝜑 → (𝐾 · (𝐾↑𝐽)) = ((𝐾↑𝐽) · 𝐾))
49		pntlem1.m	. . . . . . . . . . . . . . 15 ⊢ 𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)
50		pntlem1.n	. . . . . . . . . . . . . . 15 ⊢ 𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2))
51	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 49, 50	pntlemg 27101	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ_≥‘𝑀) ∧ (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁 − 𝑀)))
52	51	simp1d 1143	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℕ)
53		elfzouz 13636	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (𝑀..^𝑁) → 𝐽 ∈ (ℤ_≥‘𝑀))
54	20, 53	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 ∈ (ℤ_≥‘𝑀))
55		eluznn 12902	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝐽 ∈ (ℤ_≥‘𝑀)) → 𝐽 ∈ ℕ)
56	52, 54, 55	syl2anc 585	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ ℕ)
57	56	nnnn0d 12532	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
58	45, 57	expp1d 14112	. . . . . . . . . 10 ⊢ (𝜑 → (𝐾↑(𝐽 + 1)) = ((𝐾↑𝐽) · 𝐾))
59	48, 58	eqtr4d 2776	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 · (𝐾↑𝐽)) = (𝐾↑(𝐽 + 1)))
60	44, 59	breqtrd 5175	. . . . . . . 8 ⊢ (𝜑 → ((1 + (𝐿 · 𝐸)) · 𝑉) < (𝐾↑(𝐽 + 1)))
61	40, 41, 60	ltled 11362	. . . . . . 7 ⊢ (𝜑 → ((1 + (𝐿 · 𝐸)) · 𝑉) ≤ (𝐾↑(𝐽 + 1)))
62	36, 24, 17	lediv2d 13040	. . . . . . 7 ⊢ (𝜑 → (((1 + (𝐿 · 𝐸)) · 𝑉) ≤ (𝐾↑(𝐽 + 1)) ↔ (𝑍 / (𝐾↑(𝐽 + 1))) ≤ (𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))))
63	61, 62	mpbid 231	. . . . . 6 ⊢ (𝜑 → (𝑍 / (𝐾↑(𝐽 + 1))) ≤ (𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉)))
64		flwordi 13777	. . . . . 6 ⊢ (((𝑍 / (𝐾↑(𝐽 + 1))) ∈ ℝ ∧ (𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉)) ∈ ℝ ∧ (𝑍 / (𝐾↑(𝐽 + 1))) ≤ (𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))) → (⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) ≤ (⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))))
65	26, 38, 63, 64	syl3anc 1372	. . . . 5 ⊢ (𝜑 → (⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) ≤ (⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))))
66		eluz2 12828	. . . . 5 ⊢ ((⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))) ∈ (ℤ_≥‘(⌊‘(𝑍 / (𝐾↑(𝐽 + 1))))) ↔ ((⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) ∈ ℤ ∧ (⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))) ∈ ℤ ∧ (⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) ≤ (⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉)))))
67	27, 39, 65, 66	syl3anbrc 1344	. . . 4 ⊢ (𝜑 → (⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))) ∈ (ℤ_≥‘(⌊‘(𝑍 / (𝐾↑(𝐽 + 1))))))
68		eluzp1p1 12850	. . . 4 ⊢ ((⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))) ∈ (ℤ_≥‘(⌊‘(𝑍 / (𝐾↑(𝐽 + 1))))) → ((⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))) + 1) ∈ (ℤ_≥‘((⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) + 1)))
69		fzss1 13540	. . . 4 ⊢ (((⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))) + 1) ∈ (ℤ_≥‘((⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) + 1)) → (((⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))) + 1)...(⌊‘(𝑍 / 𝑉))) ⊆ (((⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) + 1)...(⌊‘(𝑍 / 𝑉))))
70	67, 68, 69	3syl 18	. . 3 ⊢ (𝜑 → (((⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))) + 1)...(⌊‘(𝑍 / 𝑉))) ⊆ (((⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) + 1)...(⌊‘(𝑍 / 𝑉))))
71	17, 35	rpdivcld 13033	. . . . . . 7 ⊢ (𝜑 → (𝑍 / 𝑉) ∈ ℝ⁺)
72	71	rpred 13016	. . . . . 6 ⊢ (𝜑 → (𝑍 / 𝑉) ∈ ℝ)
73	72	flcld 13763	. . . . 5 ⊢ (𝜑 → (⌊‘(𝑍 / 𝑉)) ∈ ℤ)
74	17, 46	rpdivcld 13033	. . . . . . 7 ⊢ (𝜑 → (𝑍 / (𝐾↑𝐽)) ∈ ℝ⁺)
75	74	rpred 13016	. . . . . 6 ⊢ (𝜑 → (𝑍 / (𝐾↑𝐽)) ∈ ℝ)
76	75	flcld 13763	. . . . 5 ⊢ (𝜑 → (⌊‘(𝑍 / (𝐾↑𝐽))) ∈ ℤ)
77	46	rpred 13016	. . . . . . . 8 ⊢ (𝜑 → (𝐾↑𝐽) ∈ ℝ)
78	35	rpred 13016	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ ℝ)
79	43	simpld 496	. . . . . . . 8 ⊢ (𝜑 → (𝐾↑𝐽) < 𝑉)
80	77, 78, 79	ltled 11362	. . . . . . 7 ⊢ (𝜑 → (𝐾↑𝐽) ≤ 𝑉)
81	46, 35, 17	lediv2d 13040	. . . . . . 7 ⊢ (𝜑 → ((𝐾↑𝐽) ≤ 𝑉 ↔ (𝑍 / 𝑉) ≤ (𝑍 / (𝐾↑𝐽))))
82	80, 81	mpbid 231	. . . . . 6 ⊢ (𝜑 → (𝑍 / 𝑉) ≤ (𝑍 / (𝐾↑𝐽)))
83		flwordi 13777	. . . . . 6 ⊢ (((𝑍 / 𝑉) ∈ ℝ ∧ (𝑍 / (𝐾↑𝐽)) ∈ ℝ ∧ (𝑍 / 𝑉) ≤ (𝑍 / (𝐾↑𝐽))) → (⌊‘(𝑍 / 𝑉)) ≤ (⌊‘(𝑍 / (𝐾↑𝐽))))
84	72, 75, 82, 83	syl3anc 1372	. . . . 5 ⊢ (𝜑 → (⌊‘(𝑍 / 𝑉)) ≤ (⌊‘(𝑍 / (𝐾↑𝐽))))
85		eluz2 12828	. . . . 5 ⊢ ((⌊‘(𝑍 / (𝐾↑𝐽))) ∈ (ℤ_≥‘(⌊‘(𝑍 / 𝑉))) ↔ ((⌊‘(𝑍 / 𝑉)) ∈ ℤ ∧ (⌊‘(𝑍 / (𝐾↑𝐽))) ∈ ℤ ∧ (⌊‘(𝑍 / 𝑉)) ≤ (⌊‘(𝑍 / (𝐾↑𝐽)))))
86	73, 76, 84, 85	syl3anbrc 1344	. . . 4 ⊢ (𝜑 → (⌊‘(𝑍 / (𝐾↑𝐽))) ∈ (ℤ_≥‘(⌊‘(𝑍 / 𝑉))))
87		fzss2 13541	. . . 4 ⊢ ((⌊‘(𝑍 / (𝐾↑𝐽))) ∈ (ℤ_≥‘(⌊‘(𝑍 / 𝑉))) → (((⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) + 1)...(⌊‘(𝑍 / 𝑉))) ⊆ (((⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝐽)))))
88	86, 87	syl 17	. . 3 ⊢ (𝜑 → (((⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) + 1)...(⌊‘(𝑍 / 𝑉))) ⊆ (((⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝐽)))))
89	70, 88	sstrd 3993	. 2 ⊢ (𝜑 → (((⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))) + 1)...(⌊‘(𝑍 / 𝑉))) ⊆ (((⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝐽)))))
90		pntlem1.i	. 2 ⊢ 𝐼 = (((⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))) + 1)...(⌊‘(𝑍 / 𝑉)))
91		pntlem1.o	. 2 ⊢ 𝑂 = (((⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝐽))))
92	89, 90, 91	3sstr4g 4028	1 ⊢ (𝜑 → 𝐼 ⊆ 𝑂)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 ⊆ wss 3949 class class class wbr 5149 ↦ cmpt 5232 ‘cfv 6544 (class class class)co 7409 ℝcr 11109 0cc0 11110 1c1 11111 + caddc 11113 · cmul 11115 +∞cpnf 11245 < clt 11248 ≤ cle 11249 − cmin 11444 / cdiv 11871 ℕcn 12212 2c2 12267 3c3 12268 4c4 12269 ℤcz 12558 cdc 12677 ℤ_≥cuz 12822 ℝ⁺crp 12974 (,)cioo 13324 [,)cico 13326 [,]cicc 13327 ...cfz 13484 ..^cfzo 13627 ⌊cfl 13755 ↑cexp 14027 √csqrt 15180 abscabs 15181 expce 16005 eceu 16006 logclog 26063 ψcchp 26597

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ioc 13329 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-fac 14234 df-bc 14263 df-hash 14291 df-shft 15014 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-limsup 15415 df-clim 15432 df-rlim 15433 df-sum 15633 df-ef 16011 df-e 16012 df-sin 16013 df-cos 16014 df-pi 16016 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17368 df-topn 17369 df-0g 17387 df-gsum 17388 df-topgen 17389 df-pt 17390 df-prds 17393 df-xrs 17448 df-qtop 17453 df-imas 17454 df-xps 17456 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-mulg 18951 df-cntz 19181 df-cmn 19650 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-nei 22602 df-lp 22640 df-perf 22641 df-cn 22731 df-cnp 22732 df-haus 22819 df-tx 23066 df-hmeo 23259 df-fil 23350 df-fm 23442 df-flim 23443 df-flf 23444 df-xms 23826 df-ms 23827 df-tms 23828 df-cncf 24394 df-limc 25383 df-dv 25384 df-log 26065

This theorem is referenced by: pntlemj 27106

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