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Theorem pntleml 27670

Description: Lemma for pnt 27673. 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 2735	. 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 27653	. . 3 ⊢ (𝜑 → (𝐿 ∈ ℝ⁺ ∧ 𝐷 ∈ ℝ⁺ ∧ 𝐹 ∈ ℝ⁺))
10	9	simp3d 1143	. 2 ⊢ (𝜑 → 𝐹 ∈ ℝ⁺)
11		0m0e0 12384	. . . . 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 7446	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}) ∧ 𝑟 = 0) → (𝑟↑3) = (0↑3))
14		3nn 12343	. . . . . . . . . 10 ⊢ 3 ∈ ℕ
15		0exp 14135	. . . . . . . . . 10 ⊢ (3 ∈ ℕ → (0↑3) = 0)
16	14, 15	ax-mp 5	. . . . . . . . 9 ⊢ (0↑3) = 0
17	13, 16	eqtrdi 2791	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}) ∧ 𝑟 = 0) → (𝑟↑3) = 0)
18	17	oveq2d 7447	. . . . . . 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 13077	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ ℂ)
20	19	mul01d 11458	. . . . . . . 8 ⊢ (𝜑 → (𝐹 · 0) = 0)
21	20	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑟 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}) ∧ 𝑟 = 0) → (𝐹 · 0) = 0)
22	18, 21	eqtrd 2775	. . . . . 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 7449	. . . . 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 2801	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}) ∧ 𝑟 = 0) → (𝑟 − (𝐹 · (𝑟↑3))) = 𝑟)
25		simplr 769	. . . 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 7438	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑠 → (𝑦[,)+∞) = (𝑠[,)+∞))
28	27	raleqdv 3324	. . . . . . . . . 10 ⊢ (𝑦 = 𝑠 → (∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟 ↔ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟))
29	28	cbvrexvw 3236	. . . . . . . . 9 ⊢ (∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟 ↔ ∃𝑠 ∈ ℝ⁺ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)
30		simplrr 778	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑟 ∈ (0[,]𝐴))
31		0re 11261	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
32	2	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝐴 ∈ ℝ⁺)
33	32	rpred 13075	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝐴 ∈ ℝ)
34		elicc2 13449	. . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑟 ≠ 0)
41	37, 39, 40	ne0gt0d 11396	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 0 < 𝑟)
42	37, 41	elrpd 13072	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑟 ∈ ℝ⁺)
43		3z 12648	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℤ
44		rpexpcl 14118	. . . . . . . . . . . . . . . 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 13091	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (𝐹 · (𝑟↑3)) ∈ ℝ⁺)
47	46	rpred 13075	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (𝐹 · (𝑟↑3)) ∈ ℝ)
48	37, 47	resubcld 11689	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (𝑟 − (𝐹 · (𝑟↑3))) ∈ ℝ)
49	3	ad2antrr 726	. . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝐵 ∈ ℝ⁺)
51	6	ad2antrr 726	. . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . 14 ⊢ (𝑟 / 𝐷) = (𝑟 / 𝐷)
56		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (exp‘(𝐵 / (𝑟 / 𝐷))) = (exp‘(𝐵 / (𝑟 / 𝐷)))
57		simprl 771	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑠 ∈ ℝ⁺)
58		1rp 13036	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ⁺
59		rpaddcl 13055	. . . . . . . . . . . . . . . 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 13079	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 0 ≤ 𝑠)
62		1re 11259	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
63	57	rpred 13075	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑠 ∈ ℝ)
64		addge02 11772	. . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . 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 13076	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑠 ∈ ℝ^*)
69	63	lep1d 12197	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑠 ≤ (𝑠 + 1))
70		df-ico 13390	. . . . . . . . . . . . . . . . 17 ⊢ [,) = (𝑡 ∈ ℝ^, 𝑟 ∈ ℝ^ ↦ {𝑤 ∈ ℝ^* ∣ (𝑡 ≤ 𝑤 ∧ 𝑤 < 𝑟)})
71		xrletr 13197	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑠 ∈ ℝ^* ∧ (𝑠 + 1) ∈ ℝ^* ∧ 𝑣 ∈ ℝ^*) → ((𝑠 ≤ (𝑠 + 1) ∧ (𝑠 + 1) ≤ 𝑣) → 𝑠 ≤ 𝑣))
72	70, 70, 71	ixxss1 13402	. . . . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . 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 4064	. . . . . . . . . . . . . . 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 27669	. . . . . . . . . . . . 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 13041	. . . . . . . . . . . . . . . . . 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 11827	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) ∧ 𝑤 ∈ ℝ⁺) → 𝑤 ≤ 𝑤)
81		elicopnf 13482	. . . . . . . . . . . . . . . . . 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 713	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) ∧ 𝑤 ∈ ℝ⁺) → 𝑤 ∈ (𝑤[,)+∞))
84		fveq2 6907	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = 𝑤 → (𝑅‘𝑣) = (𝑅‘𝑤))
85		id 22	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = 𝑤 → 𝑣 = 𝑤)
86	84, 85	oveq12d 7449	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝑤 → ((𝑅‘𝑣) / 𝑣) = ((𝑅‘𝑤) / 𝑤))
87	86	fveq2d 6911	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝑤 → (abs‘((𝑅‘𝑣) / 𝑣)) = (abs‘((𝑅‘𝑤) / 𝑤)))
88	87	breq1d 5158	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝑤 → ((abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑟 − (𝐹 · (𝑟↑3))) ↔ (abs‘((𝑅‘𝑤) / 𝑤)) ≤ (𝑟 − (𝐹 · (𝑟↑3)))))
89	88	rspcv 3618	. . . . . . . . . . . . . . . 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 27622	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑅:ℝ⁺⟶ℝ
92	91	ffvelcdmi 7103	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ ℝ⁺ → (𝑅‘𝑤) ∈ ℝ)
93		rerpdivcl 13063	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅‘𝑤) ∈ ℝ ∧ 𝑤 ∈ ℝ⁺) → ((𝑅‘𝑤) / 𝑤) ∈ ℝ)
94	92, 93	mpancom 688	. . . . . . . . . . . . . . . . . . 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 11287	. . . . . . . . . . . . . . . . 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 15480	. . . . . . . . . . . . . . . 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 15472	. . . . . . . . . . . . . . . . 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 11353	. . . . . . . . . . . . . . . . 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 695	. . . . . . . . . . . . . . 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 3153	. . . . . . . . . . . . 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 13079	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 0 ≤ (𝐹 · (𝑟↑3)))
107	37, 47	subge02d 11853	. . . . . . . . . . . . . 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 232	. . . . . . . . . . . . 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 11416	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (𝑟 − (𝐹 · (𝑟↑3))) ≤ 𝐴)
110		elicc2 13449	. . . . . . . . . . . . 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 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 3154	. . . . . . . . 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 242	. . . . . . . 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 5152	. . . . . . . 8 ⊢ (𝑡 = 𝑟 → ((abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 ↔ (abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟))
119	118	rexralbidv 3221	. . . . . . 7 ⊢ (𝑡 = 𝑟 → (∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 ↔ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟))
120	119	elrab 3695	. . . . . 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 5152	. . . . . . . . 9 ⊢ (𝑡 = (𝑟 − (𝐹 · (𝑟↑3))) → ((abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 ↔ (abs‘((𝑅‘𝑧) / 𝑧)) ≤ (𝑟 − (𝐹 · (𝑟↑3)))))
122	121	rexralbidv 3221	. . . . . . . 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 6907	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑧 → (𝑅‘𝑣) = (𝑅‘𝑧))
124		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑧 → 𝑣 = 𝑧)
125	123, 124	oveq12d 7449	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑧 → ((𝑅‘𝑣) / 𝑣) = ((𝑅‘𝑧) / 𝑧))
126	125	fveq2d 6911	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑧 → (abs‘((𝑅‘𝑣) / 𝑣)) = (abs‘((𝑅‘𝑧) / 𝑧)))
127	126	breq1d 5158	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑧 → ((abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑟 − (𝐹 · (𝑟↑3))) ↔ (abs‘((𝑅‘𝑧) / 𝑧)) ≤ (𝑟 − (𝐹 · (𝑟↑3)))))
128	127	cbvralvw 3235	. . . . . . . . . 10 ⊢ (∀𝑣 ∈ (𝑤[,)+∞)(abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑟 − (𝐹 · (𝑟↑3))) ↔ ∀𝑧 ∈ (𝑤[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ (𝑟 − (𝐹 · (𝑟↑3))))
129		oveq1 7438	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → (𝑤[,)+∞) = (𝑦[,)+∞))
130	129	raleqdv 3324	. . . . . . . . . 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 3236	. . . . . . . 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 3695	. . . . . 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 652	. . 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 3027	. 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 27668	1 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ⇝_𝑟 1)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 {crab 3433 ⊆ wss 3963 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 +∞cpnf 11290 ℝ^*cxr 11292 < clt 11293 ≤ cle 11294 − cmin 11490 / cdiv 11918 ℕcn 12264 2c2 12319 3c3 12320 ℤcz 12611 cdc 12731 ℝ⁺crp 13032 (,)cioo 13384 [,)cico 13386 [,]cicc 13387 ↑cexp 14099 abscabs 15270 ⇝_𝑟 crli 15518 expce 16094 ψcchp 27151

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-disj 5116 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-xnn0 12598 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-o1 15523 df-lo1 15524 df-sum 15720 df-ef 16100 df-e 16101 df-sin 16102 df-cos 16103 df-tan 16104 df-pi 16105 df-dvds 16288 df-gcd 16529 df-prm 16706 df-pc 16871 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-cmp 23411 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-limc 25916 df-dv 25917 df-ulm 26435 df-log 26613 df-cxp 26614 df-atan 26925 df-em 27051 df-cht 27155 df-vma 27156 df-chp 27157 df-ppi 27158 df-mu 27159

This theorem is referenced by: pnt3 27671

W3C validator