pntlemq - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > pntlemq		Structured version Visualization version GIF version

Theorem pntlemq 25290

Description: Lemma for pntlemj 25292. (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 25286	. . . . . . . . 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 1073	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ ℝ⁺)
18	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	pntlemc 25284	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸 ∈ ℝ⁺ ∧ 𝐾 ∈ ℝ⁺ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ⁺)))
19	18	simp2d 1074	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ℝ⁺)
20		pntlem1.j	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (𝑀..^𝑁))
21		elfzoelz 12470	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (𝑀..^𝑁) → 𝐽 ∈ ℤ)
22	20, 21	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ ℤ)
23	22	peano2zd 11485	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 + 1) ∈ ℤ)
24	19, 23	rpexpcld 13032	. . . . . . . 8 ⊢ (𝜑 → (𝐾↑(𝐽 + 1)) ∈ ℝ⁺)
25	17, 24	rpdivcld 11889	. . . . . . 7 ⊢ (𝜑 → (𝑍 / (𝐾↑(𝐽 + 1))) ∈ ℝ⁺)
26	25	rpred 11872	. . . . . 6 ⊢ (𝜑 → (𝑍 / (𝐾↑(𝐽 + 1))) ∈ ℝ)
27	26	flcld 12599	. . . . 5 ⊢ (𝜑 → (⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) ∈ ℤ)
28		1rp 11836	. . . . . . . . . 10 ⊢ 1 ∈ ℝ⁺
29	1, 2, 3, 4, 5, 6	pntlemd 25283	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐿 ∈ ℝ⁺ ∧ 𝐷 ∈ ℝ⁺ ∧ 𝐹 ∈ ℝ⁺))
30	29	simp1d 1073	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ ℝ⁺)
31	18	simp1d 1073	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
32	30, 31	rpmulcld 11888	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿 · 𝐸) ∈ ℝ⁺)
33		rpaddcl 11854	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ⁺ ∧ (𝐿 · 𝐸) ∈ ℝ⁺) → (1 + (𝐿 · 𝐸)) ∈ ℝ⁺)
34	28, 32, 33	sylancr 695	. . . . . . . . 9 ⊢ (𝜑 → (1 + (𝐿 · 𝐸)) ∈ ℝ⁺)
35		pntlem1.v	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ∈ ℝ⁺)
36	34, 35	rpmulcld 11888	. . . . . . . 8 ⊢ (𝜑 → ((1 + (𝐿 · 𝐸)) · 𝑉) ∈ ℝ⁺)
37	17, 36	rpdivcld 11889	. . . . . . 7 ⊢ (𝜑 → (𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉)) ∈ ℝ⁺)
38	37	rpred 11872	. . . . . 6 ⊢ (𝜑 → (𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉)) ∈ ℝ)
39	38	flcld 12599	. . . . 5 ⊢ (𝜑 → (⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))) ∈ ℤ)
40	36	rpred 11872	. . . . . . . 8 ⊢ (𝜑 → ((1 + (𝐿 · 𝐸)) · 𝑉) ∈ ℝ)
41	24	rpred 11872	. . . . . . . 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 475	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐾↑𝐽) < 𝑉 ∧ ((1 + (𝐿 · 𝐸)) · 𝑉) < (𝐾 · (𝐾↑𝐽))))
44	43	simprd 479	. . . . . . . . 9 ⊢ (𝜑 → ((1 + (𝐿 · 𝐸)) · 𝑉) < (𝐾 · (𝐾↑𝐽)))
45	19	rpcnd 11874	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ ℂ)
46	19, 22	rpexpcld 13032	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐾↑𝐽) ∈ ℝ⁺)
47	46	rpcnd 11874	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾↑𝐽) ∈ ℂ)
48	45, 47	mulcomd 10061	. . . . . . . . . 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 25287	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ_≥‘𝑀) ∧ (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁 − 𝑀)))
52	51	simp1d 1073	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℕ)
53		elfzouz 12474	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (𝑀..^𝑁) → 𝐽 ∈ (ℤ_≥‘𝑀))
54	20, 53	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 ∈ (ℤ_≥‘𝑀))
55		eluznn 11758	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝐽 ∈ (ℤ_≥‘𝑀)) → 𝐽 ∈ ℕ)
56	52, 54, 55	syl2anc 693	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ ℕ)
57	56	nnnn0d 11351	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
58	45, 57	expp1d 13009	. . . . . . . . . 10 ⊢ (𝜑 → (𝐾↑(𝐽 + 1)) = ((𝐾↑𝐽) · 𝐾))
59	48, 58	eqtr4d 2659	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 · (𝐾↑𝐽)) = (𝐾↑(𝐽 + 1)))
60	44, 59	breqtrd 4679	. . . . . . . 8 ⊢ (𝜑 → ((1 + (𝐿 · 𝐸)) · 𝑉) < (𝐾↑(𝐽 + 1)))
61	40, 41, 60	ltled 10185	. . . . . . 7 ⊢ (𝜑 → ((1 + (𝐿 · 𝐸)) · 𝑉) ≤ (𝐾↑(𝐽 + 1)))
62	36, 24, 17	lediv2d 11896	. . . . . . 7 ⊢ (𝜑 → (((1 + (𝐿 · 𝐸)) · 𝑉) ≤ (𝐾↑(𝐽 + 1)) ↔ (𝑍 / (𝐾↑(𝐽 + 1))) ≤ (𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))))
63	61, 62	mpbid 222	. . . . . 6 ⊢ (𝜑 → (𝑍 / (𝐾↑(𝐽 + 1))) ≤ (𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉)))
64		flwordi 12613	. . . . . 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 1326	. . . . 5 ⊢ (𝜑 → (⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) ≤ (⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))))
66		eluz2 11693	. . . . 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 1246	. . . 4 ⊢ (𝜑 → (⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))) ∈ (ℤ_≥‘(⌊‘(𝑍 / (𝐾↑(𝐽 + 1))))))
68		eluzp1p1 11713	. . . 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 12380	. . . 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 11889	. . . . . . 7 ⊢ (𝜑 → (𝑍 / 𝑉) ∈ ℝ⁺)
72	71	rpred 11872	. . . . . 6 ⊢ (𝜑 → (𝑍 / 𝑉) ∈ ℝ)
73	72	flcld 12599	. . . . 5 ⊢ (𝜑 → (⌊‘(𝑍 / 𝑉)) ∈ ℤ)
74	17, 46	rpdivcld 11889	. . . . . . 7 ⊢ (𝜑 → (𝑍 / (𝐾↑𝐽)) ∈ ℝ⁺)
75	74	rpred 11872	. . . . . 6 ⊢ (𝜑 → (𝑍 / (𝐾↑𝐽)) ∈ ℝ)
76	75	flcld 12599	. . . . 5 ⊢ (𝜑 → (⌊‘(𝑍 / (𝐾↑𝐽))) ∈ ℤ)
77	46	rpred 11872	. . . . . . . 8 ⊢ (𝜑 → (𝐾↑𝐽) ∈ ℝ)
78	35	rpred 11872	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ ℝ)
79	43	simpld 475	. . . . . . . 8 ⊢ (𝜑 → (𝐾↑𝐽) < 𝑉)
80	77, 78, 79	ltled 10185	. . . . . . 7 ⊢ (𝜑 → (𝐾↑𝐽) ≤ 𝑉)
81	46, 35, 17	lediv2d 11896	. . . . . . 7 ⊢ (𝜑 → ((𝐾↑𝐽) ≤ 𝑉 ↔ (𝑍 / 𝑉) ≤ (𝑍 / (𝐾↑𝐽))))
82	80, 81	mpbid 222	. . . . . 6 ⊢ (𝜑 → (𝑍 / 𝑉) ≤ (𝑍 / (𝐾↑𝐽)))
83		flwordi 12613	. . . . . 6 ⊢ (((𝑍 / 𝑉) ∈ ℝ ∧ (𝑍 / (𝐾↑𝐽)) ∈ ℝ ∧ (𝑍 / 𝑉) ≤ (𝑍 / (𝐾↑𝐽))) → (⌊‘(𝑍 / 𝑉)) ≤ (⌊‘(𝑍 / (𝐾↑𝐽))))
84	72, 75, 82, 83	syl3anc 1326	. . . . 5 ⊢ (𝜑 → (⌊‘(𝑍 / 𝑉)) ≤ (⌊‘(𝑍 / (𝐾↑𝐽))))
85		eluz2 11693	. . . . 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 1246	. . . 4 ⊢ (𝜑 → (⌊‘(𝑍 / (𝐾↑𝐽))) ∈ (ℤ_≥‘(⌊‘(𝑍 / 𝑉))))
87		fzss2 12381	. . . 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 3613	. 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 3646	1 ⊢ (𝜑 → 𝐼 ⊆ 𝑂)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 ⊆ wss 3574 class class class wbr 4653 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 +∞cpnf 10071 < clt 10074 ≤ cle 10075 − cmin 10266 / cdiv 10684 ℕcn 11020 2c2 11070 3c3 11071 4c4 11072 ℤcz 11377 cdc 11493 ℤ_≥cuz 11687 ℝ⁺crp 11832 (,)cioo 12175 [,)cico 12177 [,]cicc 12178 ...cfz 12326 ..^cfzo 12465 ⌊cfl 12591 ↑cexp 12860 √csqrt 13973 abscabs 13974 expce 14792 eceu 14793 logclog 24301 ψcchp 24819

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-inf2 8538 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-fal 1489 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-iin 4523 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-se 5074 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-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 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-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-e 14799 df-sin 14800 df-cos 14801 df-pi 14803 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-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631 df-log 24303

This theorem is referenced by: pntlemj 25292

W3C validator