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Theorem pntleml 27662

Description: Lemma for pnt 27665. 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 2761	. 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 27645	. . 3 ⊢ (𝜑 → (𝐿 ∈ ℝ⁺ ∧ 𝐷 ∈ ℝ⁺ ∧ 𝐹 ∈ ℝ⁺))
10	9	simp3d 1156	. 2 ⊢ (𝜑 → 𝐹 ∈ ℝ⁺)
11		0m0e0 12329	. . . . 5 ⊢ (0 − 0) = 0
12		simpr 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}) ∧ 𝑟 = 0) → 𝑟 = 0)
13	12	oveq1d 7405	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}) ∧ 𝑟 = 0) → (𝑟↑3) = (0↑3))
14		3nn 12290	. . . . . . . . . 10 ⊢ 3 ∈ ℕ
15		0exp 14103	. . . . . . . . . 10 ⊢ (3 ∈ ℕ → (0↑3) = 0)
16	14, 15	ax-mp 5	. . . . . . . . 9 ⊢ (0↑3) = 0
17	13, 16	eqtrdi 2812	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}) ∧ 𝑟 = 0) → (𝑟↑3) = 0)
18	17	oveq2d 7406	. . . . . . 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 13032	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ ℂ)
20	19	mul01d 11375	. . . . . . . 8 ⊢ (𝜑 → (𝐹 · 0) = 0)
21	20	ad2antrr 736	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑟 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}) ∧ 𝑟 = 0) → (𝐹 · 0) = 0)
22	18, 21	eqtrd 2796	. . . . . 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 7408	. . . . 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 2822	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}) ∧ 𝑟 = 0) → (𝑟 − (𝐹 · (𝑟↑3))) = 𝑟)
25		simplr 778	. . . 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 2861	. . 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 7397	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑠 → (𝑦[,)+∞) = (𝑠[,)+∞))
28	27	raleqdv 3319	. . . . . . . . . 10 ⊢ (𝑦 = 𝑠 → (∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟 ↔ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟))
29	28	cbvrexvw 3240	. . . . . . . . 9 ⊢ (∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟 ↔ ∃𝑠 ∈ ℝ⁺ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)
30		simplrr 787	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑟 ∈ (0[,]𝐴))
31		0re 11176	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
32	2	ad2antrr 736	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝐴 ∈ ℝ⁺)
33	32	rpred 13030	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝐴 ∈ ℝ)
34		elicc2 13408	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑟 ∈ (0[,]𝐴) ↔ (𝑟 ∈ ℝ ∧ 0 ≤ 𝑟 ∧ 𝑟 ≤ 𝐴)))
35	31, 33, 34	sylancr 596	. . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . 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 1154	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑟 ∈ ℝ)
38	10	ad2antrr 736	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝐹 ∈ ℝ⁺)
39	36	simp2d 1155	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 0 ≤ 𝑟)
40		simplrl 786	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑟 ≠ 0)
41	37, 39, 40	ne0gt0d 11313	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 0 < 𝑟)
42	37, 41	elrpd 13027	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑟 ∈ ℝ⁺)
43		3z 12597	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℤ
44		rpexpcl 14086	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟 ∈ ℝ⁺ ∧ 3 ∈ ℤ) → (𝑟↑3) ∈ ℝ⁺)
45	42, 43, 44	sylancl 595	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (𝑟↑3) ∈ ℝ⁺)
46	38, 45	rpmulcld 13046	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (𝐹 · (𝑟↑3)) ∈ ℝ⁺)
47	46	rpred 13030	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (𝐹 · (𝑟↑3)) ∈ ℝ)
48	37, 47	resubcld 11608	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (𝑟 − (𝐹 · (𝑟↑3))) ∈ ℝ)
49	3	ad2antrr 736	. . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝐵 ∈ ℝ⁺)
51	6	ad2antrr 736	. . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . 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 1156	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑟 ≤ 𝐴)
55		eqid 2761	. . . . . . . . . . . . . 14 ⊢ (𝑟 / 𝐷) = (𝑟 / 𝐷)
56		eqid 2761	. . . . . . . . . . . . . 14 ⊢ (exp‘(𝐵 / (𝑟 / 𝐷))) = (exp‘(𝐵 / (𝑟 / 𝐷)))
57		simprl 780	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑠 ∈ ℝ⁺)
58		1rp 12990	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ⁺
59		rpaddcl 13010	. . . . . . . . . . . . . . . 16 ⊢ ((𝑠 ∈ ℝ⁺ ∧ 1 ∈ ℝ⁺) → (𝑠 + 1) ∈ ℝ⁺)
60	57, 58, 59	sylancl 595	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (𝑠 + 1) ∈ ℝ⁺)
61	57	rpge0d 13034	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 0 ≤ 𝑠)
62		1re 11174	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
63	57	rpred 13030	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑠 ∈ ℝ)
64		addge02 11691	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ ℝ ∧ 𝑠 ∈ ℝ) → (0 ≤ 𝑠 ↔ 1 ≤ (𝑠 + 1)))
65	62, 63, 64	sylancr 596	. . . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 1 ≤ (𝑠 + 1))
67	60, 66	jca 519	. . . . . . . . . . . . . 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 13031	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑠 ∈ ℝ^*)
69	63	lep1d 12116	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 𝑠 ≤ (𝑠 + 1))
70		df-ico 13348	. . . . . . . . . . . . . . . . 17 ⊢ [,) = (𝑡 ∈ ℝ^, 𝑟 ∈ ℝ^ ↦ {𝑤 ∈ ℝ^* ∣ (𝑡 ≤ 𝑤 ∧ 𝑤 < 𝑟)})
71		xrletr 13153	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑠 ∈ ℝ^* ∧ (𝑠 + 1) ∈ ℝ^* ∧ 𝑣 ∈ ℝ^*) → ((𝑠 ≤ (𝑠 + 1) ∧ (𝑠 + 1) ≤ 𝑣) → 𝑠 ≤ 𝑣))
72	70, 70, 71	ixxss1 13360	. . . . . . . . . . . . . . . 16 ⊢ ((𝑠 ∈ ℝ^* ∧ 𝑠 ≤ (𝑠 + 1)) → ((𝑠 + 1)[,)+∞) ⊆ (𝑠[,)+∞))
73	68, 69, 72	syl2anc 593	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → ((𝑠 + 1)[,)+∞) ⊆ (𝑠[,)+∞))
74		simprr 782	. . . . . . . . . . . . . . 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 4003	. . . . . . . . . . . . . . 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 27661	. . . . . . . . . . . . 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 12995	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ ℝ⁺ → 𝑤 ∈ ℝ)
79	78	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) ∧ 𝑤 ∈ ℝ⁺) → 𝑤 ∈ ℝ)
80	79	leidd 11746	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) ∧ 𝑤 ∈ ℝ⁺) → 𝑤 ≤ 𝑤)
81		elicopnf 13442	. . . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) ∧ 𝑤 ∈ ℝ⁺) → 𝑤 ∈ (𝑤[,)+∞))
84		fveq2 6861	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = 𝑤 → (𝑅‘𝑣) = (𝑅‘𝑤))
85		id 22	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = 𝑤 → 𝑣 = 𝑤)
86	84, 85	oveq12d 7408	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝑤 → ((𝑅‘𝑣) / 𝑣) = ((𝑅‘𝑤) / 𝑤))
87	86	fveq2d 6865	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝑤 → (abs‘((𝑅‘𝑣) / 𝑣)) = (abs‘((𝑅‘𝑤) / 𝑤)))
88	87	breq1d 5107	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝑤 → ((abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑟 − (𝐹 · (𝑟↑3))) ↔ (abs‘((𝑅‘𝑤) / 𝑤)) ≤ (𝑟 − (𝐹 · (𝑟↑3)))))
89	88	rspcv 3576	. . . . . . . . . . . . . . . 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 27614	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑅:ℝ⁺⟶ℝ
92	91	ffvelcdmi 7058	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ ℝ⁺ → (𝑅‘𝑤) ∈ ℝ)
93		rerpdivcl 13018	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅‘𝑤) ∈ ℝ ∧ 𝑤 ∈ ℝ⁺) → ((𝑅‘𝑤) / 𝑤) ∈ ℝ)
94	92, 93	mpancom 698	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ ℝ⁺ → ((𝑅‘𝑤) / 𝑤) ∈ ℝ)
95	94	adantl 485	. . . . . . . . . . . . . . . . . 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 11203	. . . . . . . . . . . . . . . . 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 15464	. . . . . . . . . . . . . . . 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 15456	. . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . 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 11270	. . . . . . . . . . . . . . . . 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 1486	. . . . . . . . . . . . . . . 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 705	. . . . . . . . . . . . . . 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 3162	. . . . . . . . . . . . 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 13034	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → 0 ≤ (𝐹 · (𝑟↑3)))
107	37, 47	subge02d 11772	. . . . . . . . . . . . . 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 234	. . . . . . . . . . . . 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 11333	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑟 ≠ 0 ∧ 𝑟 ∈ (0[,]𝐴))) ∧ (𝑠 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (𝑠[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟)) → (𝑟 − (𝐹 · (𝑟↑3))) ≤ 𝐴)
110		elicc2 13408	. . . . . . . . . . . . 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 596	. . . . . . . . . . . 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 1355	. . . . . . . . . . 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 519	. . . . . . . . . 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 3163	. . . . . . . . 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 244	. . . . . . . 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 471	. . . . . . 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 457	. . . . . 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 5101	. . . . . . . 8 ⊢ (𝑡 = 𝑟 → ((abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 ↔ (abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟))
119	118	rexralbidv 3227	. . . . . . 7 ⊢ (𝑡 = 𝑟 → (∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 ↔ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑟))
120	119	elrab 3649	. . . . . 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 5101	. . . . . . . . 9 ⊢ (𝑡 = (𝑟 − (𝐹 · (𝑟↑3))) → ((abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 ↔ (abs‘((𝑅‘𝑧) / 𝑧)) ≤ (𝑟 − (𝐹 · (𝑟↑3)))))
122	121	rexralbidv 3227	. . . . . . . 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 6861	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑧 → (𝑅‘𝑣) = (𝑅‘𝑧))
124		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑧 → 𝑣 = 𝑧)
125	123, 124	oveq12d 7408	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑧 → ((𝑅‘𝑣) / 𝑣) = ((𝑅‘𝑧) / 𝑧))
126	125	fveq2d 6865	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑧 → (abs‘((𝑅‘𝑣) / 𝑣)) = (abs‘((𝑅‘𝑧) / 𝑧)))
127	126	breq1d 5107	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑧 → ((abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑟 − (𝐹 · (𝑟↑3))) ↔ (abs‘((𝑅‘𝑧) / 𝑧)) ≤ (𝑟 − (𝐹 · (𝑟↑3)))))
128	127	cbvralvw 3239	. . . . . . . . . 10 ⊢ (∀𝑣 ∈ (𝑤[,)+∞)(abs‘((𝑅‘𝑣) / 𝑣)) ≤ (𝑟 − (𝐹 · (𝑟↑3))) ↔ ∀𝑧 ∈ (𝑤[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ (𝑟 − (𝐹 · (𝑟↑3))))
129		oveq1 7397	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → (𝑤[,)+∞) = (𝑦[,)+∞))
130	129	raleqdv 3319	. . . . . . . . . 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 285	. . . . . . . . 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 3240	. . . . . . . 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 291	. . . . . . 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 3649	. . . . . 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 298	. . . . 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 410	. . . 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 662	. . 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 3043	. 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 27660	1 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ⇝_𝑟 1)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 ∀wral 3075 ∃wrex 3085 {crab 3413 ⊆ wss 3902 class class class wbr 5097 ↦ cmpt 5178 ‘cfv 6515 (class class class)co 7390 ℝcr 11065 0cc0 11066 1c1 11067 + caddc 11069 · cmul 11071 +∞cpnf 11206 ℝ^*cxr 11208 < clt 11209 ≤ cle 11210 − cmin 11407 / cdiv 11837 ℕcn 12203 2c2 12265 3c3 12266 ℤcz 12561 cdc 12681 ℝ⁺crp 12986 (,)cioo 13342 [,)cico 13344 [,]cicc 13345 ↑cexp 14067 abscabs 15251 ⇝_𝑟 crli 15502 expce 16081 ψcchp 27144

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7712 ax-inf2 9589 ax-cnex 11122 ax-resscn 11123 ax-1cn 11124 ax-icn 11125 ax-addcl 11126 ax-addrcl 11127 ax-mulcl 11128 ax-mulrcl 11129 ax-mulcom 11130 ax-addass 11131 ax-mulass 11132 ax-distr 11133 ax-i2m1 11134 ax-1ne0 11135 ax-1rid 11136 ax-rnegex 11137 ax-rrecex 11138 ax-cnre 11139 ax-pre-lttri 11140 ax-pre-lttrn 11141 ax-pre-ltadd 11142 ax-pre-mulgt0 11143 ax-pre-sup 11144 ax-addf 11145

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-iin 4949 df-disj 5065 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-se 5597 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-isom 6524 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7654 df-om 7841 df-1st 7964 df-2nd 7965 df-supp 8134 df-frecs 8255 df-wrecs 8286 df-recs 8335 df-rdg 8374 df-1o 8430 df-2o 8431 df-oadd 8434 df-er 8671 df-map 8803 df-pm 8804 df-ixp 8873 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-fsupp 9301 df-fi 9350 df-sup 9381 df-inf 9382 df-oi 9451 df-dju 9852 df-card 9890 df-pnf 11211 df-mnf 11212 df-xr 11213 df-ltxr 11214 df-le 11215 df-sub 11409 df-neg 11410 df-div 11838 df-nn 12204 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12475 df-xnn0 12548 df-z 12562 df-dec 12682 df-uz 12833 df-q 12943 df-rp 12987 df-xneg 13107 df-xadd 13108 df-xmul 13109 df-ioo 13346 df-ioc 13347 df-ico 13348 df-icc 13349 df-fz 13506 df-fzo 13653 df-fl 13795 df-mod 13873 df-seq 14008 df-exp 14068 df-fac 14280 df-bc 14309 df-hash 14337 df-shft 15073 df-cj 15116 df-re 15117 df-im 15118 df-sqrt 15252 df-abs 15253 df-limsup 15488 df-clim 15505 df-rlim 15506 df-o1 15507 df-lo1 15508 df-sum 15704 df-ef 16087 df-e 16088 df-sin 16089 df-cos 16090 df-tan 16091 df-pi 16092 df-dvds 16277 df-gcd 16519 df-prm 16696 df-pc 16863 df-struct 17173 df-sets 17190 df-slot 17208 df-ndx 17220 df-base 17236 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-hom 17300 df-cco 17301 df-rest 17441 df-topn 17442 df-0g 17460 df-gsum 17461 df-topgen 17462 df-pt 17463 df-prds 17466 df-xrs 17522 df-qtop 17527 df-imas 17528 df-xps 17530 df-mre 17604 df-mrc 17605 df-acs 17607 df-mgm 18664 df-sgrp 18743 df-mnd 18759 df-submnd 18808 df-mulg 19100 df-cntz 19347 df-cmn 19812 df-psmet 21403 df-xmet 21404 df-met 21405 df-bl 21406 df-mopn 21407 df-fbas 21408 df-fg 21409 df-cnfld 21412 df-top 22941 df-topon 22958 df-topsp 22980 df-bases 22993 df-cld 23066 df-ntr 23067 df-cls 23068 df-nei 23145 df-lp 23183 df-perf 23184 df-cn 23274 df-cnp 23275 df-haus 23362 df-cmp 23434 df-tx 23609 df-hmeo 23802 df-fil 23893 df-fm 23985 df-flim 23986 df-flf 23987 df-xms 24367 df-ms 24368 df-tms 24369 df-cncf 24927 df-limc 25915 df-dv 25916 df-ulm 26427 df-log 26608 df-cxp 26609 df-atan 26919 df-em 27044 df-cht 27148 df-vma 27149 df-chp 27150 df-ppi 27151 df-mu 27152

This theorem is referenced by: pnt3 27663

W3C validator