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Theorem pntleml 25752

Description: Lemma for pnt 25755. 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 2777	. 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 25735	. . 3 ⊢ (𝜑 → (𝐿 ∈ ℝ⁺ ∧ 𝐷 ∈ ℝ⁺ ∧ 𝐹 ∈ ℝ⁺))
10	9	simp3d 1135	. 2 ⊢ (𝜑 → 𝐹 ∈ ℝ⁺)
11		0m0e0 11502	. . . . 5 ⊢ (0 − 0) = 0
12		simpr 479	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}) ∧ 𝑟 = 0) → 𝑟 = 0)
13	12	oveq1d 6937	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}) ∧ 𝑟 = 0) → (𝑟↑3) = (0↑3))
14		3nn 11454	. . . . . . . . . 10 ⊢ 3 ∈ ℕ
15		0exp 13213	. . . . . . . . . 10 ⊢ (3 ∈ ℕ → (0↑3) = 0)
16	14, 15	ax-mp 5	. . . . . . . . 9 ⊢ (0↑3) = 0
17	13, 16	syl6eq 2829	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}) ∧ 𝑟 = 0) → (𝑟↑3) = 0)
18	17	oveq2d 6938	. . . . . . 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 12183	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ ℂ)
20	19	mul01d 10575	. . . . . . . 8 ⊢ (𝜑 → (𝐹 · 0) = 0)
21	20	ad2antrr 716	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑟 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}) ∧ 𝑟 = 0) → (𝐹 · 0) = 0)
22	18, 21	eqtrd 2813	. . . . . 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 6940	. . . . 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 2839	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}) ∧ 𝑟 = 0) → (𝑟 − (𝐹 · (𝑟↑3))) = 𝑟)
25		simplr 759	. . . 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 2858	. . 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 6929	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑠 → (𝑦[,)+∞) = (𝑠[,)+∞))
28	27	raleqdv 3339	. . . . . . . . . 10 ⊢ (𝑦 = 𝑠 → (∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟 ↔ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟))
29	28	cbvrexv 3367	. . . . . . . . 9 ⊢ (∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟 ↔ ∃𝑠 ∈ ℝ⁺ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)
30		simplrr 768	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑟 ∈ (0[,]𝐴))
31		0re 10378	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
32	2	ad2antrr 716	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝐴 ∈ ℝ⁺)
33	32	rpred 12181	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝐴 ∈ ℝ)
34		elicc2 12550	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑟 ∈ (0[,]𝐴) ↔ (𝑟 ∈ ℝ ∧ 0 ≤ 𝑟 ∧ 𝑟 ≤ 𝐴)))
35	31, 33, 34	sylancr 581	. . . . . . . . . . . . . . 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 224	. . . . . . . . . . . . . 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 1133	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑟 ∈ ℝ)
38	10	ad2antrr 716	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝐹 ∈ ℝ⁺)
39	36	simp2d 1134	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 0 ≤ 𝑟)
40		simplrl 767	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑟 ≠ 0)
41	37, 39, 40	ne0gt0d 10513	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 0 < 𝑟)
42	37, 41	elrpd 12178	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑟 ∈ ℝ⁺)
43		3z 11762	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℤ
44		rpexpcl 13197	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ∈ ℝ⁺ ∧ 3 ∈ ℤ) → (𝑟↑3) ∈ ℝ⁺)
45	42, 43, 44	sylancl 580	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (𝑟↑3) ∈ ℝ⁺)
46	38, 45	rpmulcld 12197	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (𝐹 · (𝑟↑3)) ∈ ℝ⁺)
47	46	rpred 12181	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (𝐹 · (𝑟↑3)) ∈ ℝ)
48	37, 47	resubcld 10803	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (𝑟 − (𝐹 · (𝑟↑3))) ∈ ℝ)
49	3	ad2antrr 716	. . . . . . . . . . . . . 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 716	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝐵 ∈ ℝ⁺)
51	6	ad2antrr 716	. . . . . . . . . . . . . 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 716	. . . . . . . . . . . . . 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 1135	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑟 ≤ 𝐴)
55		eqid 2777	. . . . . . . . . . . . . 14 ⊢ (𝑟 / 𝐷) = (𝑟 / 𝐷)
56		eqid 2777	. . . . . . . . . . . . . 14 ⊢ (exp‘(𝐵 / (𝑟 / 𝐷))) = (exp‘(𝐵 / (𝑟 / 𝐷)))
57		simprl 761	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑠 ∈ ℝ⁺)
58		1rp 12141	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ⁺
59		rpaddcl 12161	. . . . . . . . . . . . . . . 16 ⊢ ((𝑠 ∈ ℝ⁺ ∧ 1 ∈ ℝ⁺) → (𝑠 + 1) ∈ ℝ⁺)
60	57, 58, 59	sylancl 580	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (𝑠 + 1) ∈ ℝ⁺)
61	57	rpge0d 12185	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 0 ≤ 𝑠)
62		1re 10376	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
63	57	rpred 12181	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑠 ∈ ℝ)
64		addge02 10886	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ ℝ ∧ 𝑠 ∈ ℝ) → (0 ≤ 𝑠 ↔ 1 ≤ (𝑠 + 1)))
65	62, 63, 64	sylancr 581	. . . . . . . . . . . . . . . 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 224	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 1 ≤ (𝑠 + 1))
67	60, 66	jca 507	. . . . . . . . . . . . . 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 12182	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑠 ∈ ℝ^*)
69	63	lep1d 11309	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑠 ≤ (𝑠 + 1))
70		df-ico 12493	. . . . . . . . . . . . . . . . 17 ⊢ [,) = (𝑡 ∈ ℝ^, 𝑟 ∈ ℝ^ ↦ {𝑤 ∈ ℝ^* ∣ (𝑡 ≤ 𝑤 ∧ 𝑤 < 𝑟)})
71		xrletr 12301	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑠 ∈ ℝ^* ∧ (𝑠 + 1) ∈ ℝ^* ∧ 𝑣 ∈ ℝ^*) → ((𝑠 ≤ (𝑠 + 1) ∧ (𝑠 + 1) ≤ 𝑣) → 𝑠 ≤ 𝑣))
72	70, 70, 71	ixxss1 12505	. . . . . . . . . . . . . . . 16 ⊢ ((𝑠 ∈ ℝ^* ∧ 𝑠 ≤ (𝑠 + 1)) → ((𝑠 + 1)[,)+∞) ⊆ (𝑠[,)+∞))
73	68, 69, 72	syl2anc 579	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → ((𝑠 + 1)[,)+∞) ⊆ (𝑠[,)+∞))
74		simprr 763	. . . . . . . . . . . . . . 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 3884	. . . . . . . . . . . . . . 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 25751	. . . . . . . . . . . . 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 12145	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ ℝ⁺ → 𝑤 ∈ ℝ)
79	78	adantl 475	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) ∧ 𝑤 ∈ ℝ⁺) → 𝑤 ∈ ℝ)
80	79	leidd 10941	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) ∧ 𝑤 ∈ ℝ⁺) → 𝑤 ≤ 𝑤)
81		elicopnf 12582	. . . . . . . . . . . . . . . . . 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 703	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) ∧ 𝑤 ∈ ℝ⁺) → 𝑤 ∈ (𝑤[,)+∞))
84		fveq2 6446	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = 𝑤 → (𝑅‘𝑣) = (𝑅‘𝑤))
85		id 22	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = 𝑤 → 𝑣 = 𝑤)
86	84, 85	oveq12d 6940	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝑤 → ((𝑅‘𝑣) / 𝑣) = ((𝑅‘𝑤) / 𝑤))
87	86	fveq2d 6450	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝑤 → (abs‘((𝑅‘𝑣) / 𝑣)) = (abs‘((𝑅‘𝑤) / 𝑤)))
88	87	breq1d 4896	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝑤 → ((abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑟 − (𝐹 · (𝑟↑3))) ↔ (abs‘((𝑅‘𝑤) / 𝑤)) ≤ (𝑟 − (𝐹 · (𝑟↑3)))))
89	88	rspcv 3506	. . . . . . . . . . . . . . . 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 25704	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑅:ℝ⁺⟶ℝ
92	91	ffvelrni 6622	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ ℝ⁺ → (𝑅‘𝑤) ∈ ℝ)
93		rerpdivcl 12169	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅‘𝑤) ∈ ℝ ∧ 𝑤 ∈ ℝ⁺) → ((𝑅‘𝑤) / 𝑤) ∈ ℝ)
94	92, 93	mpancom 678	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ ℝ⁺ → ((𝑅‘𝑤) / 𝑤) ∈ ℝ)
95	94	adantl 475	. . . . . . . . . . . . . . . . . 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 10405	. . . . . . . . . . . . . . . . 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 14591	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) ∧ 𝑤 ∈ ℝ⁺) → 0 ≤ (abs‘((𝑅‘𝑤) / 𝑤)))
98		0red 10380	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) ∧ 𝑤 ∈ ℝ⁺) → 0 ∈ ℝ)
99	96	abscld 14583	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) ∧ 𝑤 ∈ ℝ⁺) → (abs‘((𝑅‘𝑤) / 𝑤)) ∈ ℝ)
100	48	adantr 474	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) ∧ 𝑤 ∈ ℝ⁺) → (𝑟 − (𝐹 · (𝑟↑3))) ∈ ℝ)
101		letr 10470	. . . . . . . . . . . . . . . . 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)))))
102	98, 99, 100, 101	syl3anc 1439	. . . . . . . . . . . . . . . 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)))))
103	97, 102	mpand 685	. . . . . . . . . . . . . . 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)))))
104	90, 103	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)))))
105	104	rexlimdva 3212	. . . . . . . . . . . . 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)))))
106	77, 105	mpd 15	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 0 ≤ (𝑟 − (𝐹 · (𝑟↑3))))
107	46	rpge0d 12185	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 0 ≤ (𝐹 · (𝑟↑3)))
108	37, 47	subge02d 10967	. . . . . . . . . . . . . 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))
109	107, 108	mpbid 224	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (𝑟 − (𝐹 · (𝑟↑3))) ≤ 𝑟)
110	48, 37, 33, 109, 54	letrd 10533	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (𝑟 − (𝐹 · (𝑟↑3))) ≤ 𝐴)
111		elicc2 12550	. . . . . . . . . . . . 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)))
112	31, 33, 111	sylancr 581	. . . . . . . . . . . 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)))
113	48, 106, 110, 112	mpbir3and 1399	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (𝑟 − (𝐹 · (𝑟↑3))) ∈ (0[,]𝐴))
114	113, 77	jca 507	. . . . . . . . . 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)))))
115	114	rexlimdvaa 3213	. . . . . . . . 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))))))
116	29, 115	syl5bi 234	. . . . . . . 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))))))
117	116	anassrs 461	. . . . . . 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))))))
118	117	expimpd 447	. . . . . 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))))))
119		breq2 4890	. . . . . . . 8 ⊢ (𝑡 = 𝑟 → ((abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 ↔ (abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟))
120	119	rexralbidv 3242	. . . . . . 7 ⊢ (𝑡 = 𝑟 → (∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 ↔ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟))
121	120	elrab 3571	. . . . . 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))
122		breq2 4890	. . . . . . . . 9 ⊢ (𝑡 = (𝑟 − (𝐹 · (𝑟↑3))) → ((abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 ↔ (abs‘((𝑅‘𝑧) / 𝑧)) ≤ (𝑟 − (𝐹 · (𝑟↑3)))))
123	122	rexralbidv 3242	. . . . . . . 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)))))
124		fveq2 6446	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑧 → (𝑅‘𝑣) = (𝑅‘𝑧))
125		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑧 → 𝑣 = 𝑧)
126	124, 125	oveq12d 6940	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑧 → ((𝑅‘𝑣) / 𝑣) = ((𝑅‘𝑧) / 𝑧))
127	126	fveq2d 6450	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑧 → (abs‘((𝑅‘𝑣) / 𝑣)) = (abs‘((𝑅‘𝑧) / 𝑧)))
128	127	breq1d 4896	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑧 → ((abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑟 − (𝐹 · (𝑟↑3))) ↔ (abs‘((𝑅‘𝑧) / 𝑧)) ≤ (𝑟 − (𝐹 · (𝑟↑3)))))
129	128	cbvralv 3366	. . . . . . . . . 10 ⊢ (∀𝑣 ∈ (𝑤[,)+∞)(abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑟 − (𝐹 · (𝑟↑3))) ↔ ∀𝑧 ∈ (𝑤[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ (𝑟 − (𝐹 · (𝑟↑3))))
130		oveq1 6929	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → (𝑤[,)+∞) = (𝑦[,)+∞))
131	130	raleqdv 3339	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (∀𝑧 ∈ (𝑤[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ (𝑟 − (𝐹 · (𝑟↑3))) ↔ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ (𝑟 − (𝐹 · (𝑟↑3)))))
132	129, 131	syl5bb 275	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (∀𝑣 ∈ (𝑤[,)+∞)(abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑟 − (𝐹 · (𝑟↑3))) ↔ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ (𝑟 − (𝐹 · (𝑟↑3)))))
133	132	cbvrexv 3367	. . . . . . . 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))))
134	123, 133	syl6bbr 281	. . . . . . 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)))))
135	134	elrab 3571	. . . . . 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)))))
136	118, 121, 135	3imtr4g 288	. . . . 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}))
137	136	imp 397	. . . 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})
138	137	an32s 642	. . 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})
139	26, 138	pm2.61dane 3056	. 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})
140	1, 2, 3, 4, 10, 139	pntlem3 25750	1 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ⇝_𝑟 1)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2106 ≠ wne 2968 ∀wral 3089 ∃wrex 3090 {crab 3093 ⊆ wss 3791 class class class wbr 4886 ↦ cmpt 4965 ‘cfv 6135 (class class class)co 6922 ℝcr 10271 0cc0 10272 1c1 10273 + caddc 10275 · cmul 10277 +∞cpnf 10408 ℝ^*cxr 10410 < clt 10411 ≤ cle 10412 − cmin 10606 / cdiv 11032 ℕcn 11374 2c2 11430 3c3 11431 ℤcz 11728 cdc 11845 ℝ⁺crp 12137 (,)cioo 12487 [,)cico 12489 [,]cicc 12490 ↑cexp 13178 abscabs 14381 ⇝_𝑟 crli 14624 expce 15194 ψcchp 25271

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 ax-mulf 10352

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-disj 4855 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-fi 8605 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-xnn0 11715 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-ioo 12491 df-ioc 12492 df-ico 12493 df-icc 12494 df-fz 12644 df-fzo 12785 df-fl 12912 df-mod 12988 df-seq 13120 df-exp 13179 df-fac 13379 df-bc 13408 df-hash 13436 df-shft 14214 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-limsup 14610 df-clim 14627 df-rlim 14628 df-o1 14629 df-lo1 14630 df-sum 14825 df-ef 15200 df-e 15201 df-sin 15202 df-cos 15203 df-pi 15205 df-dvds 15388 df-gcd 15623 df-prm 15791 df-pc 15946 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-hom 16362 df-cco 16363 df-rest 16469 df-topn 16470 df-0g 16488 df-gsum 16489 df-topgen 16490 df-pt 16491 df-prds 16494 df-xrs 16548 df-qtop 16553 df-imas 16554 df-xps 16556 df-mre 16632 df-mrc 16633 df-acs 16635 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-mulg 17928 df-cntz 18133 df-cmn 18581 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-fbas 20139 df-fg 20140 df-cnfld 20143 df-top 21106 df-topon 21123 df-topsp 21145 df-bases 21158 df-cld 21231 df-ntr 21232 df-cls 21233 df-nei 21310 df-lp 21348 df-perf 21349 df-cn 21439 df-cnp 21440 df-haus 21527 df-cmp 21599 df-tx 21774 df-hmeo 21967 df-fil 22058 df-fm 22150 df-flim 22151 df-flf 22152 df-xms 22533 df-ms 22534 df-tms 22535 df-cncf 23089 df-limc 24067 df-dv 24068 df-log 24740 df-cxp 24741 df-em 25171 df-cht 25275 df-vma 25276 df-chp 25277 df-ppi 25278 df-mu 25279

This theorem is referenced by: pnt3 25753

W3C validator