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Theorem pntleml 27448

Description: Lemma for pnt 27451. 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 2724	. 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 27431	. . 3 ⊢ (𝜑 → (𝐿 ∈ ℝ⁺ ∧ 𝐷 ∈ ℝ⁺ ∧ 𝐹 ∈ ℝ⁺))
10	9	simp3d 1141	. 2 ⊢ (𝜑 → 𝐹 ∈ ℝ⁺)
11		0m0e0 12328	. . . . 5 ⊢ (0 − 0) = 0
12		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}) ∧ 𝑟 = 0) → 𝑟 = 0)
13	12	oveq1d 7416	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}) ∧ 𝑟 = 0) → (𝑟↑3) = (0↑3))
14		3nn 12287	. . . . . . . . . 10 ⊢ 3 ∈ ℕ
15		0exp 14059	. . . . . . . . . 10 ⊢ (3 ∈ ℕ → (0↑3) = 0)
16	14, 15	ax-mp 5	. . . . . . . . 9 ⊢ (0↑3) = 0
17	13, 16	eqtrdi 2780	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}) ∧ 𝑟 = 0) → (𝑟↑3) = 0)
18	17	oveq2d 7417	. . . . . . 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 13014	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ ℂ)
20	19	mul01d 11409	. . . . . . . 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 2764	. . . . . 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 7419	. . . . 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 2790	. . . 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 2825	. . 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 7408	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑠 → (𝑦[,)+∞) = (𝑠[,)+∞))
28	27	raleqdv 3317	. . . . . . . . . 10 ⊢ (𝑦 = 𝑠 → (∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟 ↔ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟))
29	28	cbvrexvw 3227	. . . . . . . . 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 11212	. . . . . . . . . . . . . . . 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 13012	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝐴 ∈ ℝ)
34		elicc2 13385	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑟 ∈ (0[,]𝐴) ↔ (𝑟 ∈ ℝ ∧ 0 ≤ 𝑟 ∧ 𝑟 ≤ 𝐴)))
35	31, 33, 34	sylancr 586	. . . . . . . . . . . . . . 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 1139	. . . . . . . . . . . . 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 1140	. . . . . . . . . . . . . . . . . 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 11347	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 0 < 𝑟)
42	37, 41	elrpd 13009	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑟 ∈ ℝ⁺)
43		3z 12591	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℤ
44		rpexpcl 14042	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ∈ ℝ⁺ ∧ 3 ∈ ℤ) → (𝑟↑3) ∈ ℝ⁺)
45	42, 43, 44	sylancl 585	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (𝑟↑3) ∈ ℝ⁺)
46	38, 45	rpmulcld 13028	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (𝐹 · (𝑟↑3)) ∈ ℝ⁺)
47	46	rpred 13012	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (𝐹 · (𝑟↑3)) ∈ ℝ)
48	37, 47	resubcld 11638	. . . . . . . . . . . 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 1141	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑟 ≤ 𝐴)
55		eqid 2724	. . . . . . . . . . . . . 14 ⊢ (𝑟 / 𝐷) = (𝑟 / 𝐷)
56		eqid 2724	. . . . . . . . . . . . . 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 12974	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ⁺
59		rpaddcl 12992	. . . . . . . . . . . . . . . 16 ⊢ ((𝑠 ∈ ℝ⁺ ∧ 1 ∈ ℝ⁺) → (𝑠 + 1) ∈ ℝ⁺)
60	57, 58, 59	sylancl 585	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (𝑠 + 1) ∈ ℝ⁺)
61	57	rpge0d 13016	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 0 ≤ 𝑠)
62		1re 11210	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
63	57	rpred 13012	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑠 ∈ ℝ)
64		addge02 11721	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ ℝ ∧ 𝑠 ∈ ℝ) → (0 ≤ 𝑠 ↔ 1 ≤ (𝑠 + 1)))
65	62, 63, 64	sylancr 586	. . . . . . . . . . . . . . . 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 511	. . . . . . . . . . . . . 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 13013	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑠 ∈ ℝ^*)
69	63	lep1d 12141	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑠 ≤ (𝑠 + 1))
70		df-ico 13326	. . . . . . . . . . . . . . . . 17 ⊢ [,) = (𝑡 ∈ ℝ^, 𝑟 ∈ ℝ^ ↦ {𝑤 ∈ ℝ^* ∣ (𝑡 ≤ 𝑤 ∧ 𝑤 < 𝑟)})
71		xrletr 13133	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑠 ∈ ℝ^* ∧ (𝑠 + 1) ∈ ℝ^* ∧ 𝑣 ∈ ℝ^*) → ((𝑠 ≤ (𝑠 + 1) ∧ (𝑠 + 1) ≤ 𝑣) → 𝑠 ≤ 𝑣))
72	70, 70, 71	ixxss1 13338	. . . . . . . . . . . . . . . 16 ⊢ ((𝑠 ∈ ℝ^* ∧ 𝑠 ≤ (𝑠 + 1)) → ((𝑠 + 1)[,)+∞) ⊆ (𝑠[,)+∞))
73	68, 69, 72	syl2anc 583	. . . . . . . . . . . . . . 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 4042	. . . . . . . . . . . . . . 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 27447	. . . . . . . . . . . . 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 12978	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ ℝ⁺ → 𝑤 ∈ ℝ)
79	78	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) ∧ 𝑤 ∈ ℝ⁺) → 𝑤 ∈ ℝ)
80	79	leidd 11776	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) ∧ 𝑤 ∈ ℝ⁺) → 𝑤 ≤ 𝑤)
81		elicopnf 13418	. . . . . . . . . . . . . . . . . 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 6881	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = 𝑤 → (𝑅‘𝑣) = (𝑅‘𝑤))
85		id 22	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = 𝑤 → 𝑣 = 𝑤)
86	84, 85	oveq12d 7419	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝑤 → ((𝑅‘𝑣) / 𝑣) = ((𝑅‘𝑤) / 𝑤))
87	86	fveq2d 6885	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝑤 → (abs‘((𝑅‘𝑣) / 𝑣)) = (abs‘((𝑅‘𝑤) / 𝑤)))
88	87	breq1d 5148	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝑤 → ((abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑟 − (𝐹 · (𝑟↑3))) ↔ (abs‘((𝑅‘𝑤) / 𝑤)) ≤ (𝑟 − (𝐹 · (𝑟↑3)))))
89	88	rspcv 3600	. . . . . . . . . . . . . . . 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 27400	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑅:ℝ⁺⟶ℝ
92	91	ffvelcdmi 7075	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ ℝ⁺ → (𝑅‘𝑤) ∈ ℝ)
93		rerpdivcl 13000	. . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . 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 11238	. . . . . . . . . . . . . . . . 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 15387	. . . . . . . . . . . . . . . 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 15379	. . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 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 11304	. . . . . . . . . . . . . . . . 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 1462	. . . . . . . . . . . . . . . 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 3147	. . . . . . . . . . . . 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 13016	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 0 ≤ (𝐹 · (𝑟↑3)))
107	37, 47	subge02d 11802	. . . . . . . . . . . . . 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 11367	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (𝑟 − (𝐹 · (𝑟↑3))) ≤ 𝐴)
110		elicc2 13385	. . . . . . . . . . . . 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 586	. . . . . . . . . . . 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 1339	. . . . . . . . . . 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 511	. . . . . . . . . 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 3148	. . . . . . . . 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	biimtrid 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 467	. . . . . . 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 453	. . . . . 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 5142	. . . . . . . 8 ⊢ (𝑡 = 𝑟 → ((abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 ↔ (abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟))
119	118	rexralbidv 3212	. . . . . . 7 ⊢ (𝑡 = 𝑟 → (∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 ↔ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟))
120	119	elrab 3675	. . . . . 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 5142	. . . . . . . . 9 ⊢ (𝑡 = (𝑟 − (𝐹 · (𝑟↑3))) → ((abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 ↔ (abs‘((𝑅‘𝑧) / 𝑧)) ≤ (𝑟 − (𝐹 · (𝑟↑3)))))
122	121	rexralbidv 3212	. . . . . . . 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 6881	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑧 → (𝑅‘𝑣) = (𝑅‘𝑧))
124		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑧 → 𝑣 = 𝑧)
125	123, 124	oveq12d 7419	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑧 → ((𝑅‘𝑣) / 𝑣) = ((𝑅‘𝑧) / 𝑧))
126	125	fveq2d 6885	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑧 → (abs‘((𝑅‘𝑣) / 𝑣)) = (abs‘((𝑅‘𝑧) / 𝑧)))
127	126	breq1d 5148	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑧 → ((abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑟 − (𝐹 · (𝑟↑3))) ↔ (abs‘((𝑅‘𝑧) / 𝑧)) ≤ (𝑟 − (𝐹 · (𝑟↑3)))))
128	127	cbvralvw 3226	. . . . . . . . . 10 ⊢ (∀𝑣 ∈ (𝑤[,)+∞)(abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑟 − (𝐹 · (𝑟↑3))) ↔ ∀𝑧 ∈ (𝑤[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ (𝑟 − (𝐹 · (𝑟↑3))))
129		oveq1 7408	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → (𝑤[,)+∞) = (𝑦[,)+∞))
130	129	raleqdv 3317	. . . . . . . . . 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 283	. . . . . . . . 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 3227	. . . . . . . 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 3675	. . . . . 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 406	. . . 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 3021	. 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 27446	1 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ⇝_𝑟 1)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∀wral 3053 ∃wrex 3062 {crab 3424 ⊆ wss 3940 class class class wbr 5138 ↦ cmpt 5221 ‘cfv 6533 (class class class)co 7401 ℝcr 11104 0cc0 11105 1c1 11106 + caddc 11108 · cmul 11110 +∞cpnf 11241 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 − cmin 11440 / cdiv 11867 ℕcn 12208 2c2 12263 3c3 12264 ℤcz 12554 cdc 12673 ℝ⁺crp 12970 (,)cioo 13320 [,)cico 13322 [,]cicc 13323 ↑cexp 14023 abscabs 15177 ⇝_𝑟 crli 15425 expce 16001 ψcchp 26929

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9631 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 ax-addf 11184

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-disj 5104 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-oadd 8465 df-er 8698 df-map 8817 df-pm 8818 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fsupp 9357 df-fi 9401 df-sup 9432 df-inf 9433 df-oi 9500 df-dju 9891 df-card 9929 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-xnn0 12541 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-o1 15430 df-lo1 15431 df-sum 15629 df-ef 16007 df-e 16008 df-sin 16009 df-cos 16010 df-tan 16011 df-pi 16012 df-dvds 16194 df-gcd 16432 df-prm 16605 df-pc 16766 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18560 df-sgrp 18639 df-mnd 18655 df-submnd 18701 df-mulg 18983 df-cntz 19218 df-cmn 19687 df-psmet 21215 df-xmet 21216 df-met 21217 df-bl 21218 df-mopn 21219 df-fbas 21220 df-fg 21221 df-cnfld 21224 df-top 22706 df-topon 22723 df-topsp 22745 df-bases 22759 df-cld 22833 df-ntr 22834 df-cls 22835 df-nei 22912 df-lp 22950 df-perf 22951 df-cn 23041 df-cnp 23042 df-haus 23129 df-cmp 23201 df-tx 23376 df-hmeo 23569 df-fil 23660 df-fm 23752 df-flim 23753 df-flf 23754 df-xms 24136 df-ms 24137 df-tms 24138 df-cncf 24708 df-limc 25705 df-dv 25706 df-ulm 26218 df-log 26395 df-cxp 26396 df-atan 26703 df-em 26829 df-cht 26933 df-vma 26934 df-chp 26935 df-ppi 26936 df-mu 26937

This theorem is referenced by: pnt3 27449

W3C validator