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Theorem pntlemq 27765

Description: Lemma for pntlemj 27767. (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 27761	. . . . . . . . 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 1160	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ ℝ⁺)
18	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	pntlemc 27759	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸 ∈ ℝ⁺ ∧ 𝐾 ∈ ℝ⁺ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ⁺)))
19	18	simp2d 1161	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ℝ⁺)
20		pntlem1.j	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (𝑀..^𝑁))
21		elfzoelz 13683	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (𝑀..^𝑁) → 𝐽 ∈ ℤ)
22	20, 21	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ ℤ)
23	22	peano2zd 12698	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 + 1) ∈ ℤ)
24	19, 23	rpexpcld 14279	. . . . . . . 8 ⊢ (𝜑 → (𝐾↑(𝐽 + 1)) ∈ ℝ⁺)
25	17, 24	rpdivcld 13072	. . . . . . 7 ⊢ (𝜑 → (𝑍 / (𝐾↑(𝐽 + 1))) ∈ ℝ⁺)
26	25	rpred 13055	. . . . . 6 ⊢ (𝜑 → (𝑍 / (𝐾↑(𝐽 + 1))) ∈ ℝ)
27	26	flcld 13827	. . . . 5 ⊢ (𝜑 → (⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) ∈ ℤ)
28		1rp 13015	. . . . . . . . . 10 ⊢ 1 ∈ ℝ⁺
29	1, 2, 3, 4, 5, 6	pntlemd 27758	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐿 ∈ ℝ⁺ ∧ 𝐷 ∈ ℝ⁺ ∧ 𝐹 ∈ ℝ⁺))
30	29	simp1d 1160	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ ℝ⁺)
31	18	simp1d 1160	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
32	30, 31	rpmulcld 13071	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿 · 𝐸) ∈ ℝ⁺)
33		rpaddcl 13035	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ⁺ ∧ (𝐿 · 𝐸) ∈ ℝ⁺) → (1 + (𝐿 · 𝐸)) ∈ ℝ⁺)
34	28, 32, 33	sylancr 598	. . . . . . . . 9 ⊢ (𝜑 → (1 + (𝐿 · 𝐸)) ∈ ℝ⁺)
35		pntlem1.v	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ∈ ℝ⁺)
36	34, 35	rpmulcld 13071	. . . . . . . 8 ⊢ (𝜑 → ((1 + (𝐿 · 𝐸)) · 𝑉) ∈ ℝ⁺)
37	17, 36	rpdivcld 13072	. . . . . . 7 ⊢ (𝜑 → (𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉)) ∈ ℝ⁺)
38	37	rpred 13055	. . . . . 6 ⊢ (𝜑 → (𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉)) ∈ ℝ)
39	38	flcld 13827	. . . . 5 ⊢ (𝜑 → (⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))) ∈ ℤ)
40	36	rpred 13055	. . . . . . . 8 ⊢ (𝜑 → ((1 + (𝐿 · 𝐸)) · 𝑉) ∈ ℝ)
41	24	rpred 13055	. . . . . . . 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 499	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐾↑𝐽) < 𝑉 ∧ ((1 + (𝐿 · 𝐸)) · 𝑉) < (𝐾 · (𝐾↑𝐽))))
44	43	simprd 500	. . . . . . . . 9 ⊢ (𝜑 → ((1 + (𝐿 · 𝐸)) · 𝑉) < (𝐾 · (𝐾↑𝐽)))
45	19	rpcnd 13057	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ ℂ)
46	19, 22	rpexpcld 14279	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐾↑𝐽) ∈ ℝ⁺)
47	46	rpcnd 13057	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾↑𝐽) ∈ ℂ)
48	45, 47	mulcomd 11225	. . . . . . . . . 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 27762	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ_≥‘𝑀) ∧ (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁 − 𝑀)))
52	51	simp1d 1160	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℕ)
53		elfzouz 13688	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (𝑀..^𝑁) → 𝐽 ∈ (ℤ_≥‘𝑀))
54	20, 53	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 ∈ (ℤ_≥‘𝑀))
55		eluznn 12937	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝐽 ∈ (ℤ_≥‘𝑀)) → 𝐽 ∈ ℕ)
56	52, 54, 55	syl2anc 595	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ ℕ)
57	56	nnnn0d 12560	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
58	45, 57	expp1d 14179	. . . . . . . . . 10 ⊢ (𝜑 → (𝐾↑(𝐽 + 1)) = ((𝐾↑𝐽) · 𝐾))
59	48, 58	eqtr4d 2801	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 · (𝐾↑𝐽)) = (𝐾↑(𝐽 + 1)))
60	44, 59	breqtrd 5137	. . . . . . . 8 ⊢ (𝜑 → ((1 + (𝐿 · 𝐸)) · 𝑉) < (𝐾↑(𝐽 + 1)))
61	40, 41, 60	ltled 11353	. . . . . . 7 ⊢ (𝜑 → ((1 + (𝐿 · 𝐸)) · 𝑉) ≤ (𝐾↑(𝐽 + 1)))
62	36, 24, 17	lediv2d 13079	. . . . . . 7 ⊢ (𝜑 → (((1 + (𝐿 · 𝐸)) · 𝑉) ≤ (𝐾↑(𝐽 + 1)) ↔ (𝑍 / (𝐾↑(𝐽 + 1))) ≤ (𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))))
63	61, 62	mpbid 235	. . . . . 6 ⊢ (𝜑 → (𝑍 / (𝐾↑(𝐽 + 1))) ≤ (𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉)))
64		flwordi 13841	. . . . . 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 1398	. . . . 5 ⊢ (𝜑 → (⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) ≤ (⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))))
66		eluz2 12863	. . . . 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 1362	. . . 4 ⊢ (𝜑 → (⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))) ∈ (ℤ_≥‘(⌊‘(𝑍 / (𝐾↑(𝐽 + 1))))))
68		eluzp1p1 12885	. . . 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 13587	. . . 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 19	. . 3 ⊢ (𝜑 → (((⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))) + 1)...(⌊‘(𝑍 / 𝑉))) ⊆ (((⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) + 1)...(⌊‘(𝑍 / 𝑉))))
71	17, 35	rpdivcld 13072	. . . . . . 7 ⊢ (𝜑 → (𝑍 / 𝑉) ∈ ℝ⁺)
72	71	rpred 13055	. . . . . 6 ⊢ (𝜑 → (𝑍 / 𝑉) ∈ ℝ)
73	72	flcld 13827	. . . . 5 ⊢ (𝜑 → (⌊‘(𝑍 / 𝑉)) ∈ ℤ)
74	17, 46	rpdivcld 13072	. . . . . . 7 ⊢ (𝜑 → (𝑍 / (𝐾↑𝐽)) ∈ ℝ⁺)
75	74	rpred 13055	. . . . . 6 ⊢ (𝜑 → (𝑍 / (𝐾↑𝐽)) ∈ ℝ)
76	75	flcld 13827	. . . . 5 ⊢ (𝜑 → (⌊‘(𝑍 / (𝐾↑𝐽))) ∈ ℤ)
77	46	rpred 13055	. . . . . . . 8 ⊢ (𝜑 → (𝐾↑𝐽) ∈ ℝ)
78	35	rpred 13055	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ ℝ)
79	43	simpld 499	. . . . . . . 8 ⊢ (𝜑 → (𝐾↑𝐽) < 𝑉)
80	77, 78, 79	ltled 11353	. . . . . . 7 ⊢ (𝜑 → (𝐾↑𝐽) ≤ 𝑉)
81	46, 35, 17	lediv2d 13079	. . . . . . 7 ⊢ (𝜑 → ((𝐾↑𝐽) ≤ 𝑉 ↔ (𝑍 / 𝑉) ≤ (𝑍 / (𝐾↑𝐽))))
82	80, 81	mpbid 235	. . . . . 6 ⊢ (𝜑 → (𝑍 / 𝑉) ≤ (𝑍 / (𝐾↑𝐽)))
83		flwordi 13841	. . . . . 6 ⊢ (((𝑍 / 𝑉) ∈ ℝ ∧ (𝑍 / (𝐾↑𝐽)) ∈ ℝ ∧ (𝑍 / 𝑉) ≤ (𝑍 / (𝐾↑𝐽))) → (⌊‘(𝑍 / 𝑉)) ≤ (⌊‘(𝑍 / (𝐾↑𝐽))))
84	72, 75, 82, 83	syl3anc 1398	. . . . 5 ⊢ (𝜑 → (⌊‘(𝑍 / 𝑉)) ≤ (⌊‘(𝑍 / (𝐾↑𝐽))))
85		eluz2 12863	. . . . 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 1362	. . . 4 ⊢ (𝜑 → (⌊‘(𝑍 / (𝐾↑𝐽))) ∈ (ℤ_≥‘(⌊‘(𝑍 / 𝑉))))
87		fzss2 13588	. . . 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 18	. . 3 ⊢ (𝜑 → (((⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) + 1)...(⌊‘(𝑍 / 𝑉))) ⊆ (((⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝐽)))))
89	70, 88	sstrd 3947	. 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 3990	1 ⊢ (𝜑 → 𝐼 ⊆ 𝑂)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ∀wral 3079 ∃wrex 3089 ⊆ wss 3905 class class class wbr 5109 ↦ cmpt 5192 ‘cfv 6536 (class class class)co 7410 ℝcr 11094 0cc0 11095 1c1 11096 + caddc 11098 · cmul 11100 +∞cpnf 11235 < clt 11238 ≤ cle 11239 − cmin 11436 / cdiv 11866 ℕcn 12228 2c2 12290 3c3 12291 4c4 12292 ℤcz 12586 cdc 12706 ℤ_≥cuz 12857 ℝ⁺crp 13011 (,)cioo 13367 [,)cico 13369 [,]cicc 13370 ...cfz 13530 ..^cfzo 13678 ⌊cfl 13819 ↑cexp 14093 √csqrt 15280 abscabs 15281 expce 16110 eceu 16111 logclog 26719 ψcchp 27257

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9606 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173 ax-addf 11174

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-7 12303 df-8 12304 df-9 12305 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12858 df-q 12968 df-rp 13012 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13371 df-ioc 13372 df-ico 13373 df-icc 13374 df-fz 13531 df-fzo 13679 df-fl 13821 df-mod 13899 df-seq 14034 df-exp 14094 df-fac 14306 df-bc 14335 df-hash 14363 df-shft 15100 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-limsup 15518 df-clim 15535 df-rlim 15536 df-sum 15734 df-ef 16116 df-e 16117 df-sin 16118 df-cos 16119 df-pi 16121 df-struct 17202 df-sets 17219 df-slot 17237 df-ndx 17249 df-base 17265 df-ress 17286 df-plusg 17318 df-mulr 17319 df-starv 17320 df-sca 17321 df-vsca 17322 df-ip 17323 df-tset 17324 df-ple 17325 df-ds 17327 df-unif 17328 df-hom 17329 df-cco 17330 df-rest 17470 df-topn 17471 df-0g 17489 df-gsum 17490 df-topgen 17491 df-pt 17492 df-prds 17495 df-xrs 17551 df-qtop 17556 df-imas 17557 df-xps 17559 df-mre 17633 df-mrc 17634 df-acs 17636 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-submnd 18837 df-mulg 19129 df-cntz 19382 df-cmn 19847 df-psmet 21514 df-xmet 21515 df-met 21516 df-bl 21517 df-mopn 21518 df-fbas 21519 df-fg 21520 df-cnfld 21523 df-top 23051 df-topon 23068 df-topsp 23090 df-bases 23103 df-cld 23176 df-ntr 23177 df-cls 23178 df-nei 23255 df-lp 23293 df-perf 23294 df-cn 23384 df-cnp 23385 df-haus 23472 df-tx 23719 df-hmeo 23912 df-fil 24003 df-fm 24095 df-flim 24096 df-flf 24097 df-xms 24477 df-ms 24478 df-tms 24479 df-cncf 25037 df-limc 26025 df-dv 26026 df-log 26721

This theorem is referenced by: pntlemj 27767

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