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Theorem pntlemq 27539

Description: Lemma for pntlemj 27541. (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 27535	. . . . . . . . 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 1142	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ ℝ⁺)
18	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	pntlemc 27533	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸 ∈ ℝ⁺ ∧ 𝐾 ∈ ℝ⁺ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ⁺)))
19	18	simp2d 1143	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ℝ⁺)
20		pntlem1.j	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (𝑀..^𝑁))
21		elfzoelz 13559	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (𝑀..^𝑁) → 𝐽 ∈ ℤ)
22	20, 21	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ ℤ)
23	22	peano2zd 12580	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 + 1) ∈ ℤ)
24	19, 23	rpexpcld 14154	. . . . . . . 8 ⊢ (𝜑 → (𝐾↑(𝐽 + 1)) ∈ ℝ⁺)
25	17, 24	rpdivcld 12951	. . . . . . 7 ⊢ (𝜑 → (𝑍 / (𝐾↑(𝐽 + 1))) ∈ ℝ⁺)
26	25	rpred 12934	. . . . . 6 ⊢ (𝜑 → (𝑍 / (𝐾↑(𝐽 + 1))) ∈ ℝ)
27	26	flcld 13702	. . . . 5 ⊢ (𝜑 → (⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) ∈ ℤ)
28		1rp 12894	. . . . . . . . . 10 ⊢ 1 ∈ ℝ⁺
29	1, 2, 3, 4, 5, 6	pntlemd 27532	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐿 ∈ ℝ⁺ ∧ 𝐷 ∈ ℝ⁺ ∧ 𝐹 ∈ ℝ⁺))
30	29	simp1d 1142	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ ℝ⁺)
31	18	simp1d 1142	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
32	30, 31	rpmulcld 12950	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿 · 𝐸) ∈ ℝ⁺)
33		rpaddcl 12914	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ⁺ ∧ (𝐿 · 𝐸) ∈ ℝ⁺) → (1 + (𝐿 · 𝐸)) ∈ ℝ⁺)
34	28, 32, 33	sylancr 587	. . . . . . . . 9 ⊢ (𝜑 → (1 + (𝐿 · 𝐸)) ∈ ℝ⁺)
35		pntlem1.v	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ∈ ℝ⁺)
36	34, 35	rpmulcld 12950	. . . . . . . 8 ⊢ (𝜑 → ((1 + (𝐿 · 𝐸)) · 𝑉) ∈ ℝ⁺)
37	17, 36	rpdivcld 12951	. . . . . . 7 ⊢ (𝜑 → (𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉)) ∈ ℝ⁺)
38	37	rpred 12934	. . . . . 6 ⊢ (𝜑 → (𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉)) ∈ ℝ)
39	38	flcld 13702	. . . . 5 ⊢ (𝜑 → (⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))) ∈ ℤ)
40	36	rpred 12934	. . . . . . . 8 ⊢ (𝜑 → ((1 + (𝐿 · 𝐸)) · 𝑉) ∈ ℝ)
41	24	rpred 12934	. . . . . . . 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 494	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐾↑𝐽) < 𝑉 ∧ ((1 + (𝐿 · 𝐸)) · 𝑉) < (𝐾 · (𝐾↑𝐽))))
44	43	simprd 495	. . . . . . . . 9 ⊢ (𝜑 → ((1 + (𝐿 · 𝐸)) · 𝑉) < (𝐾 · (𝐾↑𝐽)))
45	19	rpcnd 12936	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ ℂ)
46	19, 22	rpexpcld 14154	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐾↑𝐽) ∈ ℝ⁺)
47	46	rpcnd 12936	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾↑𝐽) ∈ ℂ)
48	45, 47	mulcomd 11133	. . . . . . . . . 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 27536	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ_≥‘𝑀) ∧ (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁 − 𝑀)))
52	51	simp1d 1142	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℕ)
53		elfzouz 13563	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (𝑀..^𝑁) → 𝐽 ∈ (ℤ_≥‘𝑀))
54	20, 53	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 ∈ (ℤ_≥‘𝑀))
55		eluznn 12816	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝐽 ∈ (ℤ_≥‘𝑀)) → 𝐽 ∈ ℕ)
56	52, 54, 55	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ ℕ)
57	56	nnnn0d 12442	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
58	45, 57	expp1d 14054	. . . . . . . . . 10 ⊢ (𝜑 → (𝐾↑(𝐽 + 1)) = ((𝐾↑𝐽) · 𝐾))
59	48, 58	eqtr4d 2769	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 · (𝐾↑𝐽)) = (𝐾↑(𝐽 + 1)))
60	44, 59	breqtrd 5115	. . . . . . . 8 ⊢ (𝜑 → ((1 + (𝐿 · 𝐸)) · 𝑉) < (𝐾↑(𝐽 + 1)))
61	40, 41, 60	ltled 11261	. . . . . . 7 ⊢ (𝜑 → ((1 + (𝐿 · 𝐸)) · 𝑉) ≤ (𝐾↑(𝐽 + 1)))
62	36, 24, 17	lediv2d 12958	. . . . . . 7 ⊢ (𝜑 → (((1 + (𝐿 · 𝐸)) · 𝑉) ≤ (𝐾↑(𝐽 + 1)) ↔ (𝑍 / (𝐾↑(𝐽 + 1))) ≤ (𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))))
63	61, 62	mpbid 232	. . . . . 6 ⊢ (𝜑 → (𝑍 / (𝐾↑(𝐽 + 1))) ≤ (𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉)))
64		flwordi 13716	. . . . . 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 1373	. . . . 5 ⊢ (𝜑 → (⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) ≤ (⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))))
66		eluz2 12738	. . . . 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 12760	. . . 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 13463	. . . 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 12951	. . . . . . 7 ⊢ (𝜑 → (𝑍 / 𝑉) ∈ ℝ⁺)
72	71	rpred 12934	. . . . . 6 ⊢ (𝜑 → (𝑍 / 𝑉) ∈ ℝ)
73	72	flcld 13702	. . . . 5 ⊢ (𝜑 → (⌊‘(𝑍 / 𝑉)) ∈ ℤ)
74	17, 46	rpdivcld 12951	. . . . . . 7 ⊢ (𝜑 → (𝑍 / (𝐾↑𝐽)) ∈ ℝ⁺)
75	74	rpred 12934	. . . . . 6 ⊢ (𝜑 → (𝑍 / (𝐾↑𝐽)) ∈ ℝ)
76	75	flcld 13702	. . . . 5 ⊢ (𝜑 → (⌊‘(𝑍 / (𝐾↑𝐽))) ∈ ℤ)
77	46	rpred 12934	. . . . . . . 8 ⊢ (𝜑 → (𝐾↑𝐽) ∈ ℝ)
78	35	rpred 12934	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ ℝ)
79	43	simpld 494	. . . . . . . 8 ⊢ (𝜑 → (𝐾↑𝐽) < 𝑉)
80	77, 78, 79	ltled 11261	. . . . . . 7 ⊢ (𝜑 → (𝐾↑𝐽) ≤ 𝑉)
81	46, 35, 17	lediv2d 12958	. . . . . . 7 ⊢ (𝜑 → ((𝐾↑𝐽) ≤ 𝑉 ↔ (𝑍 / 𝑉) ≤ (𝑍 / (𝐾↑𝐽))))
82	80, 81	mpbid 232	. . . . . 6 ⊢ (𝜑 → (𝑍 / 𝑉) ≤ (𝑍 / (𝐾↑𝐽)))
83		flwordi 13716	. . . . . 6 ⊢ (((𝑍 / 𝑉) ∈ ℝ ∧ (𝑍 / (𝐾↑𝐽)) ∈ ℝ ∧ (𝑍 / 𝑉) ≤ (𝑍 / (𝐾↑𝐽))) → (⌊‘(𝑍 / 𝑉)) ≤ (⌊‘(𝑍 / (𝐾↑𝐽))))
84	72, 75, 82, 83	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (⌊‘(𝑍 / 𝑉)) ≤ (⌊‘(𝑍 / (𝐾↑𝐽))))
85		eluz2 12738	. . . . 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 13464	. . . 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 3940	. 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 3983	1 ⊢ (𝜑 → 𝐼 ⊆ 𝑂)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ∀wral 3047 ∃wrex 3056 ⊆ wss 3897 class class class wbr 5089 ↦ cmpt 5170 ‘cfv 6481 (class class class)co 7346 ℝcr 11005 0cc0 11006 1c1 11007 + caddc 11009 · cmul 11011 +∞cpnf 11143 < clt 11146 ≤ cle 11147 − cmin 11344 / cdiv 11774 ℕcn 12125 2c2 12180 3c3 12181 4c4 12182 ℤcz 12468 cdc 12588 ℤ_≥cuz 12732 ℝ⁺crp 12890 (,)cioo 13245 [,)cico 13247 [,]cicc 13248 ...cfz 13407 ..^cfzo 13554 ⌊cfl 13694 ↑cexp 13968 √csqrt 15140 abscabs 15141 expce 15968 eceu 15969 logclog 26490 ψcchp 27030

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-inf2 9531 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-ioo 13249 df-ioc 13250 df-ico 13251 df-icc 13252 df-fz 13408 df-fzo 13555 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14974 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396 df-sum 15594 df-ef 15974 df-e 15975 df-sin 15976 df-cos 15977 df-pi 15979 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-mulg 18981 df-cntz 19229 df-cmn 19694 df-psmet 21283 df-xmet 21284 df-met 21285 df-bl 21286 df-mopn 21287 df-fbas 21288 df-fg 21289 df-cnfld 21292 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22861 df-cld 22934 df-ntr 22935 df-cls 22936 df-nei 23013 df-lp 23051 df-perf 23052 df-cn 23142 df-cnp 23143 df-haus 23230 df-tx 23477 df-hmeo 23670 df-fil 23761 df-fm 23853 df-flim 23854 df-flf 23855 df-xms 24235 df-ms 24236 df-tms 24237 df-cncf 24798 df-limc 25794 df-dv 25795 df-log 26492

This theorem is referenced by: pntlemj 27541

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