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 25090

Description: Lemma for pntlemj 25092. (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 25086	. . . . . . . . 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 1066	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ ℝ⁺)
18	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	pntlemc 25084	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸 ∈ ℝ⁺ ∧ 𝐾 ∈ ℝ⁺ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ⁺)))
19	18	simp2d 1067	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ℝ⁺)
20		pntlem1.j	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (𝑀..^𝑁))
21		elfzoelz 12339	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (𝑀..^𝑁) → 𝐽 ∈ ℤ)
22	20, 21	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ ℤ)
23	22	peano2zd 11361	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 + 1) ∈ ℤ)
24	19, 23	rpexpcld 12894	. . . . . . . 8 ⊢ (𝜑 → (𝐾↑(𝐽 + 1)) ∈ ℝ⁺)
25	17, 24	rpdivcld 11765	. . . . . . 7 ⊢ (𝜑 → (𝑍 / (𝐾↑(𝐽 + 1))) ∈ ℝ⁺)
26	25	rpred 11748	. . . . . 6 ⊢ (𝜑 → (𝑍 / (𝐾↑(𝐽 + 1))) ∈ ℝ)
27	26	flcld 12461	. . . . 5 ⊢ (𝜑 → (⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) ∈ ℤ)
28		1rp 11712	. . . . . . . . . 10 ⊢ 1 ∈ ℝ⁺
29	1, 2, 3, 4, 5, 6	pntlemd 25083	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐿 ∈ ℝ⁺ ∧ 𝐷 ∈ ℝ⁺ ∧ 𝐹 ∈ ℝ⁺))
30	29	simp1d 1066	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ ℝ⁺)
31	18	simp1d 1066	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
32	30, 31	rpmulcld 11764	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿 · 𝐸) ∈ ℝ⁺)
33		rpaddcl 11730	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ⁺ ∧ (𝐿 · 𝐸) ∈ ℝ⁺) → (1 + (𝐿 · 𝐸)) ∈ ℝ⁺)
34	28, 32, 33	sylancr 694	. . . . . . . . 9 ⊢ (𝜑 → (1 + (𝐿 · 𝐸)) ∈ ℝ⁺)
35		pntlem1.v	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ∈ ℝ⁺)
36	34, 35	rpmulcld 11764	. . . . . . . 8 ⊢ (𝜑 → ((1 + (𝐿 · 𝐸)) · 𝑉) ∈ ℝ⁺)
37	17, 36	rpdivcld 11765	. . . . . . 7 ⊢ (𝜑 → (𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉)) ∈ ℝ⁺)
38	37	rpred 11748	. . . . . 6 ⊢ (𝜑 → (𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉)) ∈ ℝ)
39	38	flcld 12461	. . . . 5 ⊢ (𝜑 → (⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))) ∈ ℤ)
40	36	rpred 11748	. . . . . . . 8 ⊢ (𝜑 → ((1 + (𝐿 · 𝐸)) · 𝑉) ∈ ℝ)
41	24	rpred 11748	. . . . . . . 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 474	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐾↑𝐽) < 𝑉 ∧ ((1 + (𝐿 · 𝐸)) · 𝑉) < (𝐾 · (𝐾↑𝐽))))
44	43	simprd 478	. . . . . . . . 9 ⊢ (𝜑 → ((1 + (𝐿 · 𝐸)) · 𝑉) < (𝐾 · (𝐾↑𝐽)))
45	19	rpcnd 11750	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ ℂ)
46	19, 22	rpexpcld 12894	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐾↑𝐽) ∈ ℝ⁺)
47	46	rpcnd 11750	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾↑𝐽) ∈ ℂ)
48	45, 47	mulcomd 9940	. . . . . . . . . 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 25087	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ_≥‘𝑀) ∧ (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁 − 𝑀)))
52	51	simp1d 1066	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℕ)
53		elfzouz 12343	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ (𝑀..^𝑁) → 𝐽 ∈ (ℤ_≥‘𝑀))
54	20, 53	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 ∈ (ℤ_≥‘𝑀))
55		eluznn 11634	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝐽 ∈ (ℤ_≥‘𝑀)) → 𝐽 ∈ ℕ)
56	52, 54, 55	syl2anc 691	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ ℕ)
57	56	nnnn0d 11228	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
58	45, 57	expp1d 12871	. . . . . . . . . 10 ⊢ (𝜑 → (𝐾↑(𝐽 + 1)) = ((𝐾↑𝐽) · 𝐾))
59	48, 58	eqtr4d 2647	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 · (𝐾↑𝐽)) = (𝐾↑(𝐽 + 1)))
60	44, 59	breqtrd 4609	. . . . . . . 8 ⊢ (𝜑 → ((1 + (𝐿 · 𝐸)) · 𝑉) < (𝐾↑(𝐽 + 1)))
61	40, 41, 60	ltled 10064	. . . . . . 7 ⊢ (𝜑 → ((1 + (𝐿 · 𝐸)) · 𝑉) ≤ (𝐾↑(𝐽 + 1)))
62	36, 24, 17	lediv2d 11772	. . . . . . 7 ⊢ (𝜑 → (((1 + (𝐿 · 𝐸)) · 𝑉) ≤ (𝐾↑(𝐽 + 1)) ↔ (𝑍 / (𝐾↑(𝐽 + 1))) ≤ (𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))))
63	61, 62	mpbid 221	. . . . . 6 ⊢ (𝜑 → (𝑍 / (𝐾↑(𝐽 + 1))) ≤ (𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉)))
64		flwordi 12475	. . . . . 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 1318	. . . . 5 ⊢ (𝜑 → (⌊‘(𝑍 / (𝐾↑(𝐽 + 1)))) ≤ (⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))))
66		eluz2 11569	. . . . 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 1239	. . . 4 ⊢ (𝜑 → (⌊‘(𝑍 / ((1 + (𝐿 · 𝐸)) · 𝑉))) ∈ (ℤ_≥‘(⌊‘(𝑍 / (𝐾↑(𝐽 + 1))))))
68		eluzp1p1 11589	. . . 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 12251	. . . 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 11765	. . . . . . 7 ⊢ (𝜑 → (𝑍 / 𝑉) ∈ ℝ⁺)
72	71	rpred 11748	. . . . . 6 ⊢ (𝜑 → (𝑍 / 𝑉) ∈ ℝ)
73	72	flcld 12461	. . . . 5 ⊢ (𝜑 → (⌊‘(𝑍 / 𝑉)) ∈ ℤ)
74	17, 46	rpdivcld 11765	. . . . . . 7 ⊢ (𝜑 → (𝑍 / (𝐾↑𝐽)) ∈ ℝ⁺)
75	74	rpred 11748	. . . . . 6 ⊢ (𝜑 → (𝑍 / (𝐾↑𝐽)) ∈ ℝ)
76	75	flcld 12461	. . . . 5 ⊢ (𝜑 → (⌊‘(𝑍 / (𝐾↑𝐽))) ∈ ℤ)
77	46	rpred 11748	. . . . . . . 8 ⊢ (𝜑 → (𝐾↑𝐽) ∈ ℝ)
78	35	rpred 11748	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ ℝ)
79	43	simpld 474	. . . . . . . 8 ⊢ (𝜑 → (𝐾↑𝐽) < 𝑉)
80	77, 78, 79	ltled 10064	. . . . . . 7 ⊢ (𝜑 → (𝐾↑𝐽) ≤ 𝑉)
81	46, 35, 17	lediv2d 11772	. . . . . . 7 ⊢ (𝜑 → ((𝐾↑𝐽) ≤ 𝑉 ↔ (𝑍 / 𝑉) ≤ (𝑍 / (𝐾↑𝐽))))
82	80, 81	mpbid 221	. . . . . 6 ⊢ (𝜑 → (𝑍 / 𝑉) ≤ (𝑍 / (𝐾↑𝐽)))
83		flwordi 12475	. . . . . 6 ⊢ (((𝑍 / 𝑉) ∈ ℝ ∧ (𝑍 / (𝐾↑𝐽)) ∈ ℝ ∧ (𝑍 / 𝑉) ≤ (𝑍 / (𝐾↑𝐽))) → (⌊‘(𝑍 / 𝑉)) ≤ (⌊‘(𝑍 / (𝐾↑𝐽))))
84	72, 75, 82, 83	syl3anc 1318	. . . . 5 ⊢ (𝜑 → (⌊‘(𝑍 / 𝑉)) ≤ (⌊‘(𝑍 / (𝐾↑𝐽))))
85		eluz2 11569	. . . . 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 1239	. . . 4 ⊢ (𝜑 → (⌊‘(𝑍 / (𝐾↑𝐽))) ∈ (ℤ_≥‘(⌊‘(𝑍 / 𝑉))))
87		fzss2 12252	. . . 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 3578	. 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 3609	1 ⊢ (𝜑 → 𝐼 ⊆ 𝑂)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 ⊆ wss 3540 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 +∞cpnf 9950 < clt 9953 ≤ cle 9954 − cmin 10145 / cdiv 10563 ℕcn 10897 2c2 10947 3c3 10948 4c4 10949 ℤcz 11254 cdc 11369 ℤ_≥cuz 11563 ℝ⁺crp 11708 (,)cioo 12046 [,)cico 12048 [,]cicc 12049 ...cfz 12197 ..^cfzo 12334 ⌊cfl 12453 ↑cexp 12722 √csqrt 13821 abscabs 13822 expce 14631 eceu 14632 logclog 24105 ψcchp 24619

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-e 14638 df-sin 14639 df-cos 14640 df-pi 14642 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-limc 23436 df-dv 23437 df-log 24107

This theorem is referenced by: pntlemj 25092

W3C validator