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Theorem pntleml 26759

Description: Lemma for pnt 26762. Equation 10.6.35 in [Shapiro], p. 436. (Contributed by Mario Carneiro, 14-Apr-2016.)

**Hypotheses**
Ref	Expression
pntlem3.r	⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
pntlem3.a	⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
pntlem3.A	⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ (abs‘((𝑅‘𝑥) / 𝑥)) ≤ 𝐴)
pntlemp.b	⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
pntlemp.l	⊢ (𝜑 → 𝐿 ∈ (0(,)1))
pntlemp.d	⊢ 𝐷 = (𝐴 + 1)
pntlemp.f	⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2)))
pntlemp.K	⊢ (𝜑 → ∀𝑒 ∈ (0(,)1)∃𝑥 ∈ ℝ⁺ ∀𝑘 ∈ ((exp‘(𝐵 / 𝑒))[,)+∞)∀𝑦 ∈ (𝑥(,)+∞)∃𝑧 ∈ ℝ⁺ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝑒)) · 𝑧) < (𝑘 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝑒)) · 𝑧))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝑒))

**Assertion**
Ref	Expression
pntleml	⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ⇝_𝑟 1)

Distinct variable groups: 𝑥,𝑦,𝑧,𝐴 𝑒,𝑎,𝑘,𝑢,𝑥,𝑦,𝑧,𝐷 𝑦,𝐹,𝑧 𝑅,𝑒,𝑘,𝑢,𝑥,𝑦,𝑧 𝑒,𝐿,𝑘,𝑢,𝑥,𝑦,𝑧 𝜑,𝑥,𝑦 𝐵,𝑒,𝑘,𝑥,𝑦,𝑧 𝜑,𝑧 Allowed substitution hints: 𝜑(𝑢,𝑒,𝑘,𝑎) 𝐴(𝑢,𝑒,𝑘,𝑎) 𝐵(𝑢,𝑎) 𝑅(𝑎) 𝐹(𝑥,𝑢,𝑒,𝑘,𝑎) 𝐿(𝑎)

Proof of Theorem pntleml Dummy variables 𝑠 𝑟 𝑡 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pntlem3.r	. 2 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
2		pntlem3.a	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
3		pntlem3.A	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ (abs‘((𝑅‘𝑥) / 𝑥)) ≤ 𝐴)
4		eqid 2738	. 2 ⊢ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡} = {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}
5		pntlemp.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
6		pntlemp.l	. . . 4 ⊢ (𝜑 → 𝐿 ∈ (0(,)1))
7		pntlemp.d	. . . 4 ⊢ 𝐷 = (𝐴 + 1)
8		pntlemp.f	. . . 4 ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2)))
9	1, 2, 5, 6, 7, 8	pntlemd 26742	. . 3 ⊢ (𝜑 → (𝐿 ∈ ℝ⁺ ∧ 𝐷 ∈ ℝ⁺ ∧ 𝐹 ∈ ℝ⁺))
10	9	simp3d 1143	. 2 ⊢ (𝜑 → 𝐹 ∈ ℝ⁺)
11		0m0e0 12093	. . . . 5 ⊢ (0 − 0) = 0
12		simpr 485	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}) ∧ 𝑟 = 0) → 𝑟 = 0)
13	12	oveq1d 7290	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}) ∧ 𝑟 = 0) → (𝑟↑3) = (0↑3))
14		3nn 12052	. . . . . . . . . 10 ⊢ 3 ∈ ℕ
15		0exp 13818	. . . . . . . . . 10 ⊢ (3 ∈ ℕ → (0↑3) = 0)
16	14, 15	ax-mp 5	. . . . . . . . 9 ⊢ (0↑3) = 0
17	13, 16	eqtrdi 2794	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}) ∧ 𝑟 = 0) → (𝑟↑3) = 0)
18	17	oveq2d 7291	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑟 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}) ∧ 𝑟 = 0) → (𝐹 · (𝑟↑3)) = (𝐹 · 0))
19	10	rpcnd 12774	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ ℂ)
20	19	mul01d 11174	. . . . . . . 8 ⊢ (𝜑 → (𝐹 · 0) = 0)
21	20	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑟 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}) ∧ 𝑟 = 0) → (𝐹 · 0) = 0)
22	18, 21	eqtrd 2778	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}) ∧ 𝑟 = 0) → (𝐹 · (𝑟↑3)) = 0)
23	12, 22	oveq12d 7293	. . . . 5 ⊢ (((𝜑 ∧ 𝑟 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}) ∧ 𝑟 = 0) → (𝑟 − (𝐹 · (𝑟↑3))) = (0 − 0))
24	11, 23, 12	3eqtr4a 2804	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}) ∧ 𝑟 = 0) → (𝑟 − (𝐹 · (𝑟↑3))) = 𝑟)
25		simplr 766	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}) ∧ 𝑟 = 0) → 𝑟 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡})
26	24, 25	eqeltrd 2839	. . 3 ⊢ (((𝜑 ∧ 𝑟 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}) ∧ 𝑟 = 0) → (𝑟 − (𝐹 · (𝑟↑3))) ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡})
27		oveq1 7282	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑠 → (𝑦[,)+∞) = (𝑠[,)+∞))
28	27	raleqdv 3348	. . . . . . . . . 10 ⊢ (𝑦 = 𝑠 → (∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟 ↔ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟))
29	28	cbvrexvw 3384	. . . . . . . . 9 ⊢ (∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟 ↔ ∃𝑠 ∈ ℝ⁺ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)
30		simplrr 775	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑟 ∈ (0[,]𝐴))
31		0re 10977	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
32	2	ad2antrr 723	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝐴 ∈ ℝ⁺)
33	32	rpred 12772	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝐴 ∈ ℝ)
34		elicc2 13144	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑟 ∈ (0[,]𝐴) ↔ (𝑟 ∈ ℝ ∧ 0 ≤ 𝑟 ∧ 𝑟 ≤ 𝐴)))
35	31, 33, 34	sylancr 587	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (𝑟 ∈ (0[,]𝐴) ↔ (𝑟 ∈ ℝ ∧ 0 ≤ 𝑟 ∧ 𝑟 ≤ 𝐴)))
36	30, 35	mpbid 231	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (𝑟 ∈ ℝ ∧ 0 ≤ 𝑟 ∧ 𝑟 ≤ 𝐴))
37	36	simp1d 1141	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑟 ∈ ℝ)
38	10	ad2antrr 723	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝐹 ∈ ℝ⁺)
39	36	simp2d 1142	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 0 ≤ 𝑟)
40		simplrl 774	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑟 ≠ 0)
41	37, 39, 40	ne0gt0d 11112	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 0 < 𝑟)
42	37, 41	elrpd 12769	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑟 ∈ ℝ⁺)
43		3z 12353	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℤ
44		rpexpcl 13801	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ∈ ℝ⁺ ∧ 3 ∈ ℤ) → (𝑟↑3) ∈ ℝ⁺)
45	42, 43, 44	sylancl 586	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (𝑟↑3) ∈ ℝ⁺)
46	38, 45	rpmulcld 12788	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (𝐹 · (𝑟↑3)) ∈ ℝ⁺)
47	46	rpred 12772	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (𝐹 · (𝑟↑3)) ∈ ℝ)
48	37, 47	resubcld 11403	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (𝑟 − (𝐹 · (𝑟↑3))) ∈ ℝ)
49	3	ad2antrr 723	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → ∀𝑥 ∈ ℝ⁺ (abs‘((𝑅‘𝑥) / 𝑥)) ≤ 𝐴)
50	5	ad2antrr 723	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝐵 ∈ ℝ⁺)
51	6	ad2antrr 723	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝐿 ∈ (0(,)1))
52		pntlemp.K	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑒 ∈ (0(,)1)∃𝑥 ∈ ℝ⁺ ∀𝑘 ∈ ((exp‘(𝐵 / 𝑒))[,)+∞)∀𝑦 ∈ (𝑥(,)+∞)∃𝑧 ∈ ℝ⁺ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝑒)) · 𝑧) < (𝑘 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝑒)) · 𝑧))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝑒))
53	52	ad2antrr 723	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → ∀𝑒 ∈ (0(,)1)∃𝑥 ∈ ℝ⁺ ∀𝑘 ∈ ((exp‘(𝐵 / 𝑒))[,)+∞)∀𝑦 ∈ (𝑥(,)+∞)∃𝑧 ∈ ℝ⁺ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝑒)) · 𝑧) < (𝑘 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝑒)) · 𝑧))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝑒))
54	36	simp3d 1143	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑟 ≤ 𝐴)
55		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (𝑟 / 𝐷) = (𝑟 / 𝐷)
56		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (exp‘(𝐵 / (𝑟 / 𝐷))) = (exp‘(𝐵 / (𝑟 / 𝐷)))
57		simprl 768	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑠 ∈ ℝ⁺)
58		1rp 12734	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ⁺
59		rpaddcl 12752	. . . . . . . . . . . . . . . 16 ⊢ ((𝑠 ∈ ℝ⁺ ∧ 1 ∈ ℝ⁺) → (𝑠 + 1) ∈ ℝ⁺)
60	57, 58, 59	sylancl 586	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (𝑠 + 1) ∈ ℝ⁺)
61	57	rpge0d 12776	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 0 ≤ 𝑠)
62		1re 10975	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
63	57	rpred 12772	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑠 ∈ ℝ)
64		addge02 11486	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ ℝ ∧ 𝑠 ∈ ℝ) → (0 ≤ 𝑠 ↔ 1 ≤ (𝑠 + 1)))
65	62, 63, 64	sylancr 587	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (0 ≤ 𝑠 ↔ 1 ≤ (𝑠 + 1)))
66	61, 65	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 1 ≤ (𝑠 + 1))
67	60, 66	jca 512	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → ((𝑠 + 1) ∈ ℝ⁺ ∧ 1 ≤ (𝑠 + 1)))
68	57	rpxrd 12773	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑠 ∈ ℝ^*)
69	63	lep1d 11906	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑠 ≤ (𝑠 + 1))
70		df-ico 13085	. . . . . . . . . . . . . . . . 17 ⊢ [,) = (𝑡 ∈ ℝ^, 𝑟 ∈ ℝ^ ↦ {𝑤 ∈ ℝ^* ∣ (𝑡 ≤ 𝑤 ∧ 𝑤 < 𝑟)})
71		xrletr 12892	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑠 ∈ ℝ^* ∧ (𝑠 + 1) ∈ ℝ^* ∧ 𝑣 ∈ ℝ^*) → ((𝑠 ≤ (𝑠 + 1) ∧ (𝑠 + 1) ≤ 𝑣) → 𝑠 ≤ 𝑣))
72	70, 70, 71	ixxss1 13097	. . . . . . . . . . . . . . . 16 ⊢ ((𝑠 ∈ ℝ^* ∧ 𝑠 ≤ (𝑠 + 1)) → ((𝑠 + 1)[,)+∞) ⊆ (𝑠[,)+∞))
73	68, 69, 72	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → ((𝑠 + 1)[,)+∞) ⊆ (𝑠[,)+∞))
74		simprr 770	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)
75		ssralv 3987	. . . . . . . . . . . . . . 15 ⊢ (((𝑠 + 1)[,)+∞) ⊆ (𝑠[,)+∞) → (∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟 → ∀𝑧 ∈ ((𝑠 + 1)[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟))
76	73, 74, 75	sylc 65	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → ∀𝑧 ∈ ((𝑠 + 1)[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)
77	1, 32, 49, 50, 51, 7, 8, 53, 42, 54, 55, 56, 67, 76	pntlemp 26758	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → ∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ (𝑤[,)+∞)(abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑟 − (𝐹 · (𝑟↑3))))
78		rpre 12738	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ ℝ⁺ → 𝑤 ∈ ℝ)
79	78	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) ∧ 𝑤 ∈ ℝ⁺) → 𝑤 ∈ ℝ)
80	79	leidd 11541	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) ∧ 𝑤 ∈ ℝ⁺) → 𝑤 ≤ 𝑤)
81		elicopnf 13177	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ ℝ → (𝑤 ∈ (𝑤[,)+∞) ↔ (𝑤 ∈ ℝ ∧ 𝑤 ≤ 𝑤)))
82	79, 81	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) ∧ 𝑤 ∈ ℝ⁺) → (𝑤 ∈ (𝑤[,)+∞) ↔ (𝑤 ∈ ℝ ∧ 𝑤 ≤ 𝑤)))
83	79, 80, 82	mpbir2and 710	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) ∧ 𝑤 ∈ ℝ⁺) → 𝑤 ∈ (𝑤[,)+∞))
84		fveq2 6774	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = 𝑤 → (𝑅‘𝑣) = (𝑅‘𝑤))
85		id 22	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = 𝑤 → 𝑣 = 𝑤)
86	84, 85	oveq12d 7293	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝑤 → ((𝑅‘𝑣) / 𝑣) = ((𝑅‘𝑤) / 𝑤))
87	86	fveq2d 6778	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝑤 → (abs‘((𝑅‘𝑣) / 𝑣)) = (abs‘((𝑅‘𝑤) / 𝑤)))
88	87	breq1d 5084	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝑤 → ((abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑟 − (𝐹 · (𝑟↑3))) ↔ (abs‘((𝑅‘𝑤) / 𝑤)) ≤ (𝑟 − (𝐹 · (𝑟↑3)))))
89	88	rspcv 3557	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ (𝑤[,)+∞) → (∀𝑣 ∈ (𝑤[,)+∞)(abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑟 − (𝐹 · (𝑟↑3))) → (abs‘((𝑅‘𝑤) / 𝑤)) ≤ (𝑟 − (𝐹 · (𝑟↑3)))))
90	83, 89	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) ∧ 𝑤 ∈ ℝ⁺) → (∀𝑣 ∈ (𝑤[,)+∞)(abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑟 − (𝐹 · (𝑟↑3))) → (abs‘((𝑅‘𝑤) / 𝑤)) ≤ (𝑟 − (𝐹 · (𝑟↑3)))))
91	1	pntrf 26711	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑅:ℝ⁺⟶ℝ
92	91	ffvelrni 6960	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ ℝ⁺ → (𝑅‘𝑤) ∈ ℝ)
93		rerpdivcl 12760	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅‘𝑤) ∈ ℝ ∧ 𝑤 ∈ ℝ⁺) → ((𝑅‘𝑤) / 𝑤) ∈ ℝ)
94	92, 93	mpancom 685	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ ℝ⁺ → ((𝑅‘𝑤) / 𝑤) ∈ ℝ)
95	94	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) ∧ 𝑤 ∈ ℝ⁺) → ((𝑅‘𝑤) / 𝑤) ∈ ℝ)
96	95	recnd 11003	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) ∧ 𝑤 ∈ ℝ⁺) → ((𝑅‘𝑤) / 𝑤) ∈ ℂ)
97	96	absge0d 15156	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) ∧ 𝑤 ∈ ℝ⁺) → 0 ≤ (abs‘((𝑅‘𝑤) / 𝑤)))
98	96	abscld 15148	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) ∧ 𝑤 ∈ ℝ⁺) → (abs‘((𝑅‘𝑤) / 𝑤)) ∈ ℝ)
99	48	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) ∧ 𝑤 ∈ ℝ⁺) → (𝑟 − (𝐹 · (𝑟↑3))) ∈ ℝ)
100		letr 11069	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∈ ℝ ∧ (abs‘((𝑅‘𝑤) / 𝑤)) ∈ ℝ ∧ (𝑟 − (𝐹 · (𝑟↑3))) ∈ ℝ) → ((0 ≤ (abs‘((𝑅‘𝑤) / 𝑤)) ∧ (abs‘((𝑅‘𝑤) / 𝑤)) ≤ (𝑟 − (𝐹 · (𝑟↑3)))) → 0 ≤ (𝑟 − (𝐹 · (𝑟↑3)))))
101	31, 98, 99, 100	mp3an2i 1465	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) ∧ 𝑤 ∈ ℝ⁺) → ((0 ≤ (abs‘((𝑅‘𝑤) / 𝑤)) ∧ (abs‘((𝑅‘𝑤) / 𝑤)) ≤ (𝑟 − (𝐹 · (𝑟↑3)))) → 0 ≤ (𝑟 − (𝐹 · (𝑟↑3)))))
102	97, 101	mpand 692	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) ∧ 𝑤 ∈ ℝ⁺) → ((abs‘((𝑅‘𝑤) / 𝑤)) ≤ (𝑟 − (𝐹 · (𝑟↑3))) → 0 ≤ (𝑟 − (𝐹 · (𝑟↑3)))))
103	90, 102	syld 47	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) ∧ 𝑤 ∈ ℝ⁺) → (∀𝑣 ∈ (𝑤[,)+∞)(abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑟 − (𝐹 · (𝑟↑3))) → 0 ≤ (𝑟 − (𝐹 · (𝑟↑3)))))
104	103	rexlimdva 3213	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ (𝑤[,)+∞)(abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑟 − (𝐹 · (𝑟↑3))) → 0 ≤ (𝑟 − (𝐹 · (𝑟↑3)))))
105	77, 104	mpd 15	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 0 ≤ (𝑟 − (𝐹 · (𝑟↑3))))
106	46	rpge0d 12776	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 0 ≤ (𝐹 · (𝑟↑3)))
107	37, 47	subge02d 11567	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (0 ≤ (𝐹 · (𝑟↑3)) ↔ (𝑟 − (𝐹 · (𝑟↑3))) ≤ 𝑟))
108	106, 107	mpbid 231	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (𝑟 − (𝐹 · (𝑟↑3))) ≤ 𝑟)
109	48, 37, 33, 108, 54	letrd 11132	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (𝑟 − (𝐹 · (𝑟↑3))) ≤ 𝐴)
110		elicc2 13144	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝑟 − (𝐹 · (𝑟↑3))) ∈ (0[,]𝐴) ↔ ((𝑟 − (𝐹 · (𝑟↑3))) ∈ ℝ ∧ 0 ≤ (𝑟 − (𝐹 · (𝑟↑3))) ∧ (𝑟 − (𝐹 · (𝑟↑3))) ≤ 𝐴)))
111	31, 33, 110	sylancr 587	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → ((𝑟 − (𝐹 · (𝑟↑3))) ∈ (0[,]𝐴) ↔ ((𝑟 − (𝐹 · (𝑟↑3))) ∈ ℝ ∧ 0 ≤ (𝑟 − (𝐹 · (𝑟↑3))) ∧ (𝑟 − (𝐹 · (𝑟↑3))) ≤ 𝐴)))
112	48, 105, 109, 111	mpbir3and 1341	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (𝑟 − (𝐹 · (𝑟↑3))) ∈ (0[,]𝐴))
113	112, 77	jca 512	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → ((𝑟 − (𝐹 · (𝑟↑3))) ∈ (0[,]𝐴) ∧ ∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ (𝑤[,)+∞)(abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑟 − (𝐹 · (𝑟↑3)))))
114	113	rexlimdvaa 3214	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) → (∃𝑠 ∈ ℝ⁺ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟 → ((𝑟 − (𝐹 · (𝑟↑3))) ∈ (0[,]𝐴) ∧ ∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ (𝑤[,)+∞)(abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑟 − (𝐹 · (𝑟↑3))))))
115	29, 114	syl5bi 241	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) → (∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟 → ((𝑟 − (𝐹 · (𝑟↑3))) ∈ (0[,]𝐴) ∧ ∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ (𝑤[,)+∞)(abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑟 − (𝐹 · (𝑟↑3))))))
116	115	anassrs 468	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑟 ≠ 0) ∧ 𝑟 ∈ (0[,]𝐴)) → (∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟 → ((𝑟 − (𝐹 · (𝑟↑3))) ∈ (0[,]𝐴) ∧ ∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ (𝑤[,)+∞)(abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑟 − (𝐹 · (𝑟↑3))))))
117	116	expimpd 454	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ≠ 0) → ((𝑟 ∈ (0[,]𝐴) ∧ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟) → ((𝑟 − (𝐹 · (𝑟↑3))) ∈ (0[,]𝐴) ∧ ∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ (𝑤[,)+∞)(abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑟 − (𝐹 · (𝑟↑3))))))
118		breq2 5078	. . . . . . . 8 ⊢ (𝑡 = 𝑟 → ((abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 ↔ (abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟))
119	118	rexralbidv 3230	. . . . . . 7 ⊢ (𝑡 = 𝑟 → (∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 ↔ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟))
120	119	elrab 3624	. . . . . 6 ⊢ (𝑟 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡} ↔ (𝑟 ∈ (0[,]𝐴) ∧ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟))
121		breq2 5078	. . . . . . . . 9 ⊢ (𝑡 = (𝑟 − (𝐹 · (𝑟↑3))) → ((abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 ↔ (abs‘((𝑅‘𝑧) / 𝑧)) ≤ (𝑟 − (𝐹 · (𝑟↑3)))))
122	121	rexralbidv 3230	. . . . . . . 8 ⊢ (𝑡 = (𝑟 − (𝐹 · (𝑟↑3))) → (∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 ↔ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ (𝑟 − (𝐹 · (𝑟↑3)))))
123		fveq2 6774	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑧 → (𝑅‘𝑣) = (𝑅‘𝑧))
124		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑧 → 𝑣 = 𝑧)
125	123, 124	oveq12d 7293	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑧 → ((𝑅‘𝑣) / 𝑣) = ((𝑅‘𝑧) / 𝑧))
126	125	fveq2d 6778	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑧 → (abs‘((𝑅‘𝑣) / 𝑣)) = (abs‘((𝑅‘𝑧) / 𝑧)))
127	126	breq1d 5084	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑧 → ((abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑟 − (𝐹 · (𝑟↑3))) ↔ (abs‘((𝑅‘𝑧) / 𝑧)) ≤ (𝑟 − (𝐹 · (𝑟↑3)))))
128	127	cbvralvw 3383	. . . . . . . . . 10 ⊢ (∀𝑣 ∈ (𝑤[,)+∞)(abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑟 − (𝐹 · (𝑟↑3))) ↔ ∀𝑧 ∈ (𝑤[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ (𝑟 − (𝐹 · (𝑟↑3))))
129		oveq1 7282	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → (𝑤[,)+∞) = (𝑦[,)+∞))
130	129	raleqdv 3348	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (∀𝑧 ∈ (𝑤[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ (𝑟 − (𝐹 · (𝑟↑3))) ↔ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ (𝑟 − (𝐹 · (𝑟↑3)))))
131	128, 130	bitrid 282	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (∀𝑣 ∈ (𝑤[,)+∞)(abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑟 − (𝐹 · (𝑟↑3))) ↔ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ (𝑟 − (𝐹 · (𝑟↑3)))))
132	131	cbvrexvw 3384	. . . . . . . 8 ⊢ (∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ (𝑤[,)+∞)(abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑟 − (𝐹 · (𝑟↑3))) ↔ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ (𝑟 − (𝐹 · (𝑟↑3))))
133	122, 132	bitr4di 289	. . . . . . 7 ⊢ (𝑡 = (𝑟 − (𝐹 · (𝑟↑3))) → (∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 ↔ ∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ (𝑤[,)+∞)(abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑟 − (𝐹 · (𝑟↑3)))))
134	133	elrab 3624	. . . . . 6 ⊢ ((𝑟 − (𝐹 · (𝑟↑3))) ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡} ↔ ((𝑟 − (𝐹 · (𝑟↑3))) ∈ (0[,]𝐴) ∧ ∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ (𝑤[,)+∞)(abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑟 − (𝐹 · (𝑟↑3)))))
135	117, 120, 134	3imtr4g 296	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ≠ 0) → (𝑟 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡} → (𝑟 − (𝐹 · (𝑟↑3))) ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}))
136	135	imp 407	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ≠ 0) ∧ 𝑟 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}) → (𝑟 − (𝐹 · (𝑟↑3))) ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡})
137	136	an32s 649	. . 3 ⊢ (((𝜑 ∧ 𝑟 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}) ∧ 𝑟 ≠ 0) → (𝑟 − (𝐹 · (𝑟↑3))) ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡})
138	26, 137	pm2.61dane 3032	. 2 ⊢ ((𝜑 ∧ 𝑟 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}) → (𝑟 − (𝐹 · (𝑟↑3))) ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡})
139	1, 2, 3, 4, 10, 138	pntlem3 26757	1 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ⇝_𝑟 1)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3065 {crab 3068 ⊆ wss 3887 class class class wbr 5074 ↦ cmpt 5157 ‘cfv 6433 (class class class)co 7275 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 +∞cpnf 11006 ℝ^*cxr 11008 < clt 11009 ≤ cle 11010 − cmin 11205 / cdiv 11632 ℕcn 11973 2c2 12028 3c3 12029 ℤcz 12319 cdc 12437 ℝ⁺crp 12730 (,)cioo 13079 [,)cico 13081 [,]cicc 13082 ↑cexp 13782 abscabs 14945 ⇝_𝑟 crli 15194 expce 15771 ψcchp 26242

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-disj 5040 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-dju 9659 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-xnn0 12306 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-fac 13988 df-bc 14017 df-hash 14045 df-shft 14778 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-limsup 15180 df-clim 15197 df-rlim 15198 df-o1 15199 df-lo1 15200 df-sum 15398 df-ef 15777 df-e 15778 df-sin 15779 df-cos 15780 df-tan 15781 df-pi 15782 df-dvds 15964 df-gcd 16202 df-prm 16377 df-pc 16538 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-cmp 22538 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041 df-limc 25030 df-dv 25031 df-ulm 25536 df-log 25712 df-cxp 25713 df-atan 26017 df-em 26142 df-cht 26246 df-vma 26247 df-chp 26248 df-ppi 26249 df-mu 26250

This theorem is referenced by: pnt3 26760

W3C validator