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Theorem pntlem3 27588

Description: Lemma for pnt 27593. Equation 10.6.35 in [Shapiro], p. 436. (Contributed by Mario Carneiro, 8-Apr-2016.) (Proof shortened by AV, 27-Sep-2020.)

**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)
pntlem3.1	⊢ 𝑇 = {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}
pntlem3.2	⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
pntlem3.3	⊢ ((𝜑 ∧ 𝑢 ∈ 𝑇) → (𝑢 − (𝐶 · (𝑢↑3))) ∈ 𝑇)

**Assertion**
Ref	Expression
pntlem3	⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ⇝_𝑟 1)

Distinct variable groups: 𝑥,𝑡,𝑦,𝑧,𝐴 𝑢,𝑎,𝑥,𝑦,𝑧 𝑢,𝐶 𝑢,𝑡,𝑅,𝑥,𝑦,𝑧 𝑡,𝑎 𝑢,𝑇,𝑥 𝜑,𝑡,𝑥,𝑦,𝑢,𝑧 Allowed substitution hints: 𝜑(𝑎) 𝐴(𝑢,𝑎) 𝐶(𝑥,𝑦,𝑧,𝑡,𝑎) 𝑅(𝑎) 𝑇(𝑦,𝑧,𝑡,𝑎)

Proof of Theorem pntlem3 Dummy variables 𝑠 𝑤 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rpssre 12925	. . . 4 ⊢ ℝ⁺ ⊆ ℝ
2		eqid 2737	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
3	2	subcn 24823	. . . . . . . . . . . 12 ⊢ − ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
4	3	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) → − ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld)))
5		ssid 3958	. . . . . . . . . . . . 13 ⊢ ℂ ⊆ ℂ
6		cncfmptid 24874	. . . . . . . . . . . . 13 ⊢ ((ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑝 ∈ ℂ ↦ 𝑝) ∈ (ℂ–cn→ℂ))
7	5, 5, 6	mp2an 693	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ ℂ ↦ 𝑝) ∈ (ℂ–cn→ℂ)
8	7	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) → (𝑝 ∈ ℂ ↦ 𝑝) ∈ (ℂ–cn→ℂ))
9		pntlem3.2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
10	9	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) → 𝐶 ∈ ℝ⁺)
11	10	rpcnd 12963	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) → 𝐶 ∈ ℂ)
12	5	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) → ℂ ⊆ ℂ)
13		cncfmptc 24873	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℂ ∧ ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑝 ∈ ℂ ↦ 𝐶) ∈ (ℂ–cn→ℂ))
14	11, 12, 12, 13	syl3anc 1374	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) → (𝑝 ∈ ℂ ↦ 𝐶) ∈ (ℂ–cn→ℂ))
15		3nn0 12431	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℕ₀
16	2	expcn 24831	. . . . . . . . . . . . . 14 ⊢ (3 ∈ ℕ₀ → (𝑝 ∈ ℂ ↦ (𝑝↑3)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
17	15, 16	mp1i 13	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) → (𝑝 ∈ ℂ ↦ (𝑝↑3)) ∈ ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld)))
18	2	cncfcn1 24872	. . . . . . . . . . . . 13 ⊢ (ℂ–cn→ℂ) = ((TopOpen‘ℂ_fld) Cn (TopOpen‘ℂ_fld))
19	17, 18	eleqtrrdi 2848	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) → (𝑝 ∈ ℂ ↦ (𝑝↑3)) ∈ (ℂ–cn→ℂ))
20	14, 19	mulcncf 25414	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) → (𝑝 ∈ ℂ ↦ (𝐶 · (𝑝↑3))) ∈ (ℂ–cn→ℂ))
21	2, 4, 8, 20	cncfmpt2f 24876	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) → (𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3)))) ∈ (ℂ–cn→ℂ))
22		pntlem3.1	. . . . . . . . . . . . . . 15 ⊢ 𝑇 = {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}
23	22	ssrab3 4036	. . . . . . . . . . . . . 14 ⊢ 𝑇 ⊆ (0[,]𝐴)
24		0re 11146	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ
25		pntlem3.a	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
26	25	rpred 12961	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ ℝ)
27		iccssre 13357	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0[,]𝐴) ⊆ ℝ)
28	24, 26, 27	sylancr 588	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0[,]𝐴) ⊆ ℝ)
29	23, 28	sstrid 3947	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑇 ⊆ ℝ)
30		0xr 11191	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ^*
31	25	rpxrd 12962	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
32	25	rpge0d 12965	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ≤ 𝐴)
33		ubicc2 13393	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) → 𝐴 ∈ (0[,]𝐴))
34	30, 31, 32, 33	mp3an2i 1469	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ (0[,]𝐴))
35		1rp 12921	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ⁺
36		fveq2 6842	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑧 → (𝑅‘𝑥) = (𝑅‘𝑧))
37		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
38	36, 37	oveq12d 7386	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑧 → ((𝑅‘𝑥) / 𝑥) = ((𝑅‘𝑧) / 𝑧))
39	38	fveq2d 6846	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑧 → (abs‘((𝑅‘𝑥) / 𝑥)) = (abs‘((𝑅‘𝑧) / 𝑧)))
40	39	breq1d 5110	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑧 → ((abs‘((𝑅‘𝑥) / 𝑥)) ≤ 𝐴 ↔ (abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝐴))
41		pntlem3.A	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ (abs‘((𝑅‘𝑥) / 𝑥)) ≤ 𝐴)
42	41	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ (1[,)+∞)) → ∀𝑥 ∈ ℝ⁺ (abs‘((𝑅‘𝑥) / 𝑥)) ≤ 𝐴)
43		1re 11144	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℝ
44		elicopnf 13373	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1 ∈ ℝ → (𝑧 ∈ (1[,)+∞) ↔ (𝑧 ∈ ℝ ∧ 1 ≤ 𝑧)))
45	43, 44	mp1i 13	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑧 ∈ (1[,)+∞) ↔ (𝑧 ∈ ℝ ∧ 1 ≤ 𝑧)))
46	45	simprbda 498	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ (1[,)+∞)) → 𝑧 ∈ ℝ)
47		0red 11147	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑧 ∈ (1[,)+∞)) → 0 ∈ ℝ)
48	43	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑧 ∈ (1[,)+∞)) → 1 ∈ ℝ)
49		0lt1 11671	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 < 1
50	49	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑧 ∈ (1[,)+∞)) → 0 < 1)
51	45	simplbda 499	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑧 ∈ (1[,)+∞)) → 1 ≤ 𝑧)
52	47, 48, 46, 50, 51	ltletrd 11305	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ (1[,)+∞)) → 0 < 𝑧)
53	46, 52	elrpd 12958	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ (1[,)+∞)) → 𝑧 ∈ ℝ⁺)
54	40, 42, 53	rspcdva 3579	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ (1[,)+∞)) → (abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝐴)
55	54	ralrimiva 3130	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑧 ∈ (1[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝐴)
56		oveq1 7375	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 1 → (𝑦[,)+∞) = (1[,)+∞))
57	56	raleqdv 3298	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 1 → (∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝐴 ↔ ∀𝑧 ∈ (1[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝐴))
58	57	rspcev 3578	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℝ⁺ ∧ ∀𝑧 ∈ (1[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝐴) → ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝐴)
59	35, 55, 58	sylancr 588	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝐴)
60		breq2 5104	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝐴 → ((abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 ↔ (abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝐴))
61	60	rexralbidv 3204	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝐴 → (∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 ↔ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝐴))
62	61, 22	elrab2 3651	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ 𝑇 ↔ (𝐴 ∈ (0[,]𝐴) ∧ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝐴))
63	34, 59, 62	sylanbrc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ 𝑇)
64	63	ne0d 4296	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑇 ≠ ∅)
65		elicc2 13339	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑡 ∈ (0[,]𝐴) ↔ (𝑡 ∈ ℝ ∧ 0 ≤ 𝑡 ∧ 𝑡 ≤ 𝐴)))
66	24, 26, 65	sylancr 588	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑡 ∈ (0[,]𝐴) ↔ (𝑡 ∈ ℝ ∧ 0 ≤ 𝑡 ∧ 𝑡 ≤ 𝐴)))
67	66	biimpa 476	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 ∈ (0[,]𝐴)) → (𝑡 ∈ ℝ ∧ 0 ≤ 𝑡 ∧ 𝑡 ≤ 𝐴))
68	67	simp2d 1144	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑡 ∈ (0[,]𝐴)) → 0 ≤ 𝑡)
69	68	a1d 25	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑡 ∈ (0[,]𝐴)) → (∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 → 0 ≤ 𝑡))
70	69	ralrimiva 3130	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑡 ∈ (0[,]𝐴)(∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 → 0 ≤ 𝑡))
71	22	raleqi 3296	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑤 ∈ 𝑇 0 ≤ 𝑤 ↔ ∀𝑤 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}0 ≤ 𝑤)
72		breq2 5104	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑡 → (0 ≤ 𝑤 ↔ 0 ≤ 𝑡))
73	72	ralrab2 3658	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑤 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}0 ≤ 𝑤 ↔ ∀𝑡 ∈ (0[,]𝐴)(∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 → 0 ≤ 𝑡))
74	71, 73	bitri 275	. . . . . . . . . . . . . . 15 ⊢ (∀𝑤 ∈ 𝑇 0 ≤ 𝑤 ↔ ∀𝑡 ∈ (0[,]𝐴)(∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 → 0 ≤ 𝑡))
75	70, 74	sylibr 234	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑤 ∈ 𝑇 0 ≤ 𝑤)
76		breq1 5103	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 0 → (𝑥 ≤ 𝑤 ↔ 0 ≤ 𝑤))
77	76	ralbidv 3161	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 0 → (∀𝑤 ∈ 𝑇 𝑥 ≤ 𝑤 ↔ ∀𝑤 ∈ 𝑇 0 ≤ 𝑤))
78	77	rspcev 3578	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ ∧ ∀𝑤 ∈ 𝑇 0 ≤ 𝑤) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑇 𝑥 ≤ 𝑤)
79	24, 75, 78	sylancr 588	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑇 𝑥 ≤ 𝑤)
80		infrecl 12136	. . . . . . . . . . . . 13 ⊢ ((𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑇 𝑥 ≤ 𝑤) → inf(𝑇, ℝ, < ) ∈ ℝ)
81	29, 64, 79, 80	syl3anc 1374	. . . . . . . . . . . 12 ⊢ (𝜑 → inf(𝑇, ℝ, < ) ∈ ℝ)
82	81	recnd 11172	. . . . . . . . . . 11 ⊢ (𝜑 → inf(𝑇, ℝ, < ) ∈ ℂ)
83	82	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) → inf(𝑇, ℝ, < ) ∈ ℂ)
84		elrp 12919	. . . . . . . . . . . . . 14 ⊢ (inf(𝑇, ℝ, < ) ∈ ℝ⁺ ↔ (inf(𝑇, ℝ, < ) ∈ ℝ ∧ 0 < inf(𝑇, ℝ, < )))
85	84	biimpri 228	. . . . . . . . . . . . 13 ⊢ ((inf(𝑇, ℝ, < ) ∈ ℝ ∧ 0 < inf(𝑇, ℝ, < )) → inf(𝑇, ℝ, < ) ∈ ℝ⁺)
86	81, 85	sylan 581	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) → inf(𝑇, ℝ, < ) ∈ ℝ⁺)
87		3z 12536	. . . . . . . . . . . 12 ⊢ 3 ∈ ℤ
88		rpexpcl 14015	. . . . . . . . . . . 12 ⊢ ((inf(𝑇, ℝ, < ) ∈ ℝ⁺ ∧ 3 ∈ ℤ) → (inf(𝑇, ℝ, < )↑3) ∈ ℝ⁺)
89	86, 87, 88	sylancl 587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) → (inf(𝑇, ℝ, < )↑3) ∈ ℝ⁺)
90	10, 89	rpmulcld 12977	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) → (𝐶 · (inf(𝑇, ℝ, < )↑3)) ∈ ℝ⁺)
91		cncfi 24855	. . . . . . . . . 10 ⊢ (((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3)))) ∈ (ℂ–cn→ℂ) ∧ inf(𝑇, ℝ, < ) ∈ ℂ ∧ (𝐶 · (inf(𝑇, ℝ, < )↑3)) ∈ ℝ⁺) → ∃𝑠 ∈ ℝ⁺ ∀𝑢 ∈ ℂ ((abs‘(𝑢 − inf(𝑇, ℝ, < ))) < 𝑠 → (abs‘(((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘𝑢) − ((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘inf(𝑇, ℝ, < )))) < (𝐶 · (inf(𝑇, ℝ, < )↑3))))
92	21, 83, 90, 91	syl3anc 1374	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) → ∃𝑠 ∈ ℝ⁺ ∀𝑢 ∈ ℂ ((abs‘(𝑢 − inf(𝑇, ℝ, < ))) < 𝑠 → (abs‘(((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘𝑢) − ((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘inf(𝑇, ℝ, < )))) < (𝐶 · (inf(𝑇, ℝ, < )↑3))))
93	81	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) → inf(𝑇, ℝ, < ) ∈ ℝ)
94		rphalfcl 12946	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ ℝ⁺ → (𝑠 / 2) ∈ ℝ⁺)
95	94	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) → (𝑠 / 2) ∈ ℝ⁺)
96	93, 95	ltaddrpd 12994	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) → inf(𝑇, ℝ, < ) < (inf(𝑇, ℝ, < ) + (𝑠 / 2)))
97	95	rpred 12961	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) → (𝑠 / 2) ∈ ℝ)
98	93, 97	readdcld 11173	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) → (inf(𝑇, ℝ, < ) + (𝑠 / 2)) ∈ ℝ)
99	93, 98	ltnled 11292	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) → (inf(𝑇, ℝ, < ) < (inf(𝑇, ℝ, < ) + (𝑠 / 2)) ↔ ¬ (inf(𝑇, ℝ, < ) + (𝑠 / 2)) ≤ inf(𝑇, ℝ, < )))
100	96, 99	mpbid 232	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) → ¬ (inf(𝑇, ℝ, < ) + (𝑠 / 2)) ≤ inf(𝑇, ℝ, < ))
101		ax-resscn 11095	. . . . . . . . . . . . . . 15 ⊢ ℝ ⊆ ℂ
102	29, 101	sstrdi 3948	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑇 ⊆ ℂ)
103	102	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) → 𝑇 ⊆ ℂ)
104		ssralv 4004	. . . . . . . . . . . . 13 ⊢ (𝑇 ⊆ ℂ → (∀𝑢 ∈ ℂ ((abs‘(𝑢 − inf(𝑇, ℝ, < ))) < 𝑠 → (abs‘(((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘𝑢) − ((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘inf(𝑇, ℝ, < )))) < (𝐶 · (inf(𝑇, ℝ, < )↑3))) → ∀𝑢 ∈ 𝑇 ((abs‘(𝑢 − inf(𝑇, ℝ, < ))) < 𝑠 → (abs‘(((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘𝑢) − ((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘inf(𝑇, ℝ, < )))) < (𝐶 · (inf(𝑇, ℝ, < )↑3)))))
105	103, 104	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) → (∀𝑢 ∈ ℂ ((abs‘(𝑢 − inf(𝑇, ℝ, < ))) < 𝑠 → (abs‘(((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘𝑢) − ((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘inf(𝑇, ℝ, < )))) < (𝐶 · (inf(𝑇, ℝ, < )↑3))) → ∀𝑢 ∈ 𝑇 ((abs‘(𝑢 − inf(𝑇, ℝ, < ))) < 𝑠 → (abs‘(((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘𝑢) − ((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘inf(𝑇, ℝ, < )))) < (𝐶 · (inf(𝑇, ℝ, < )↑3)))))
106	29	ad2antrr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) → 𝑇 ⊆ ℝ)
107	106	sselda 3935	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → 𝑢 ∈ ℝ)
108	98	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → (inf(𝑇, ℝ, < ) + (𝑠 / 2)) ∈ ℝ)
109	107, 108	ltnled 11292	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → (𝑢 < (inf(𝑇, ℝ, < ) + (𝑠 / 2)) ↔ ¬ (inf(𝑇, ℝ, < ) + (𝑠 / 2)) ≤ 𝑢))
110	81	ad3antrrr 731	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → inf(𝑇, ℝ, < ) ∈ ℝ)
111	97	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → (𝑠 / 2) ∈ ℝ)
112	110, 111	resubcld 11577	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → (inf(𝑇, ℝ, < ) − (𝑠 / 2)) ∈ ℝ)
113	93, 95	ltsubrpd 12993	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) → (inf(𝑇, ℝ, < ) − (𝑠 / 2)) < inf(𝑇, ℝ, < ))
114	113	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → (inf(𝑇, ℝ, < ) − (𝑠 / 2)) < inf(𝑇, ℝ, < ))
115	29	ad3antrrr 731	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → 𝑇 ⊆ ℝ)
116	79	ad3antrrr 731	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑇 𝑥 ≤ 𝑤)
117		simpr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → 𝑢 ∈ 𝑇)
118		infrelb 12139	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑇 ⊆ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑇 𝑥 ≤ 𝑤 ∧ 𝑢 ∈ 𝑇) → inf(𝑇, ℝ, < ) ≤ 𝑢)
119	115, 116, 117, 118	syl3anc 1374	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → inf(𝑇, ℝ, < ) ≤ 𝑢)
120	112, 110, 107, 114, 119	ltletrd 11305	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → (inf(𝑇, ℝ, < ) − (𝑠 / 2)) < 𝑢)
121	107, 110, 111	absdifltd 15371	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → ((abs‘(𝑢 − inf(𝑇, ℝ, < ))) < (𝑠 / 2) ↔ ((inf(𝑇, ℝ, < ) − (𝑠 / 2)) < 𝑢 ∧ 𝑢 < (inf(𝑇, ℝ, < ) + (𝑠 / 2)))))
122	121	biimprd 248	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → (((inf(𝑇, ℝ, < ) − (𝑠 / 2)) < 𝑢 ∧ 𝑢 < (inf(𝑇, ℝ, < ) + (𝑠 / 2))) → (abs‘(𝑢 − inf(𝑇, ℝ, < ))) < (𝑠 / 2)))
123	120, 122	mpand 696	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → (𝑢 < (inf(𝑇, ℝ, < ) + (𝑠 / 2)) → (abs‘(𝑢 − inf(𝑇, ℝ, < ))) < (𝑠 / 2)))
124		rphalflt 12948	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 ∈ ℝ⁺ → (𝑠 / 2) < 𝑠)
125	124	ad2antlr 728	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → (𝑠 / 2) < 𝑠)
126	107, 110	resubcld 11577	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → (𝑢 − inf(𝑇, ℝ, < )) ∈ ℝ)
127	126	recnd 11172	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → (𝑢 − inf(𝑇, ℝ, < )) ∈ ℂ)
128	127	abscld 15374	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → (abs‘(𝑢 − inf(𝑇, ℝ, < ))) ∈ ℝ)
129		rpre 12926	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 ∈ ℝ⁺ → 𝑠 ∈ ℝ)
130	129	ad2antlr 728	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → 𝑠 ∈ ℝ)
131		lttr 11221	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((abs‘(𝑢 − inf(𝑇, ℝ, < ))) ∈ ℝ ∧ (𝑠 / 2) ∈ ℝ ∧ 𝑠 ∈ ℝ) → (((abs‘(𝑢 − inf(𝑇, ℝ, < ))) < (𝑠 / 2) ∧ (𝑠 / 2) < 𝑠) → (abs‘(𝑢 − inf(𝑇, ℝ, < ))) < 𝑠))
132	128, 111, 130, 131	syl3anc 1374	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → (((abs‘(𝑢 − inf(𝑇, ℝ, < ))) < (𝑠 / 2) ∧ (𝑠 / 2) < 𝑠) → (abs‘(𝑢 − inf(𝑇, ℝ, < ))) < 𝑠))
133	125, 132	mpan2d 695	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → ((abs‘(𝑢 − inf(𝑇, ℝ, < ))) < (𝑠 / 2) → (abs‘(𝑢 − inf(𝑇, ℝ, < ))) < 𝑠))
134	123, 133	syld 47	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → (𝑢 < (inf(𝑇, ℝ, < ) + (𝑠 / 2)) → (abs‘(𝑢 − inf(𝑇, ℝ, < ))) < 𝑠))
135	109, 134	sylbird 260	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → (¬ (inf(𝑇, ℝ, < ) + (𝑠 / 2)) ≤ 𝑢 → (abs‘(𝑢 − inf(𝑇, ℝ, < ))) < 𝑠))
136	135	con1d 145	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → (¬ (abs‘(𝑢 − inf(𝑇, ℝ, < ))) < 𝑠 → (inf(𝑇, ℝ, < ) + (𝑠 / 2)) ≤ 𝑢))
137	107	recnd 11172	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → 𝑢 ∈ ℂ)
138		id 22	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 = 𝑢 → 𝑝 = 𝑢)
139		oveq1 7375	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑝 = 𝑢 → (𝑝↑3) = (𝑢↑3))
140	139	oveq2d 7384	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 = 𝑢 → (𝐶 · (𝑝↑3)) = (𝐶 · (𝑢↑3)))
141	138, 140	oveq12d 7386	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 = 𝑢 → (𝑝 − (𝐶 · (𝑝↑3))) = (𝑢 − (𝐶 · (𝑢↑3))))
142		eqid 2737	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3)))) = (𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))
143		ovex 7401	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 − (𝐶 · (𝑢↑3))) ∈ V
144	141, 142, 143	fvmpt 6949	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 ∈ ℂ → ((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘𝑢) = (𝑢 − (𝐶 · (𝑢↑3))))
145	137, 144	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → ((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘𝑢) = (𝑢 − (𝐶 · (𝑢↑3))))
146	83	ad2antrr 727	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → inf(𝑇, ℝ, < ) ∈ ℂ)
147		id 22	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 = inf(𝑇, ℝ, < ) → 𝑝 = inf(𝑇, ℝ, < ))
148		oveq1 7375	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑝 = inf(𝑇, ℝ, < ) → (𝑝↑3) = (inf(𝑇, ℝ, < )↑3))
149	148	oveq2d 7384	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 = inf(𝑇, ℝ, < ) → (𝐶 · (𝑝↑3)) = (𝐶 · (inf(𝑇, ℝ, < )↑3)))
150	147, 149	oveq12d 7386	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 = inf(𝑇, ℝ, < ) → (𝑝 − (𝐶 · (𝑝↑3))) = (inf(𝑇, ℝ, < ) − (𝐶 · (inf(𝑇, ℝ, < )↑3))))
151		ovex 7401	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (inf(𝑇, ℝ, < ) − (𝐶 · (inf(𝑇, ℝ, < )↑3))) ∈ V
152	150, 142, 151	fvmpt 6949	. . . . . . . . . . . . . . . . . . . 20 ⊢ (inf(𝑇, ℝ, < ) ∈ ℂ → ((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘inf(𝑇, ℝ, < )) = (inf(𝑇, ℝ, < ) − (𝐶 · (inf(𝑇, ℝ, < )↑3))))
153	146, 152	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → ((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘inf(𝑇, ℝ, < )) = (inf(𝑇, ℝ, < ) − (𝐶 · (inf(𝑇, ℝ, < )↑3))))
154	145, 153	oveq12d 7386	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → (((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘𝑢) − ((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘inf(𝑇, ℝ, < ))) = ((𝑢 − (𝐶 · (𝑢↑3))) − (inf(𝑇, ℝ, < ) − (𝐶 · (inf(𝑇, ℝ, < )↑3)))))
155	154	fveq2d 6846	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → (abs‘(((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘𝑢) − ((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘inf(𝑇, ℝ, < )))) = (abs‘((𝑢 − (𝐶 · (𝑢↑3))) − (inf(𝑇, ℝ, < ) − (𝐶 · (inf(𝑇, ℝ, < )↑3))))))
156	155	breq1d 5110	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → ((abs‘(((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘𝑢) − ((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘inf(𝑇, ℝ, < )))) < (𝐶 · (inf(𝑇, ℝ, < )↑3)) ↔ (abs‘((𝑢 − (𝐶 · (𝑢↑3))) − (inf(𝑇, ℝ, < ) − (𝐶 · (inf(𝑇, ℝ, < )↑3))))) < (𝐶 · (inf(𝑇, ℝ, < )↑3))))
157	9	rpred 12961	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐶 ∈ ℝ)
158	157	ad3antrrr 731	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → 𝐶 ∈ ℝ)
159		reexpcl 14013	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑢 ∈ ℝ ∧ 3 ∈ ℕ₀) → (𝑢↑3) ∈ ℝ)
160	107, 15, 159	sylancl 587	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → (𝑢↑3) ∈ ℝ)
161	158, 160	remulcld 11174	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → (𝐶 · (𝑢↑3)) ∈ ℝ)
162	107, 161	resubcld 11577	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → (𝑢 − (𝐶 · (𝑢↑3))) ∈ ℝ)
163	15	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → 3 ∈ ℕ₀)
164	110, 163	reexpcld 14098	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → (inf(𝑇, ℝ, < )↑3) ∈ ℝ)
165	158, 164	remulcld 11174	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → (𝐶 · (inf(𝑇, ℝ, < )↑3)) ∈ ℝ)
166	110, 165	resubcld 11577	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → (inf(𝑇, ℝ, < ) − (𝐶 · (inf(𝑇, ℝ, < )↑3))) ∈ ℝ)
167	162, 166, 165	absdifltd 15371	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → ((abs‘((𝑢 − (𝐶 · (𝑢↑3))) − (inf(𝑇, ℝ, < ) − (𝐶 · (inf(𝑇, ℝ, < )↑3))))) < (𝐶 · (inf(𝑇, ℝ, < )↑3)) ↔ (((inf(𝑇, ℝ, < ) − (𝐶 · (inf(𝑇, ℝ, < )↑3))) − (𝐶 · (inf(𝑇, ℝ, < )↑3))) < (𝑢 − (𝐶 · (𝑢↑3))) ∧ (𝑢 − (𝐶 · (𝑢↑3))) < ((inf(𝑇, ℝ, < ) − (𝐶 · (inf(𝑇, ℝ, < )↑3))) + (𝐶 · (inf(𝑇, ℝ, < )↑3))))))
168	165	recnd 11172	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → (𝐶 · (inf(𝑇, ℝ, < )↑3)) ∈ ℂ)
169	146, 168	npcand 11508	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → ((inf(𝑇, ℝ, < ) − (𝐶 · (inf(𝑇, ℝ, < )↑3))) + (𝐶 · (inf(𝑇, ℝ, < )↑3))) = inf(𝑇, ℝ, < ))
170	169	breq2d 5112	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → ((𝑢 − (𝐶 · (𝑢↑3))) < ((inf(𝑇, ℝ, < ) − (𝐶 · (inf(𝑇, ℝ, < )↑3))) + (𝐶 · (inf(𝑇, ℝ, < )↑3))) ↔ (𝑢 − (𝐶 · (𝑢↑3))) < inf(𝑇, ℝ, < )))
171		pntlem3.3	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑇) → (𝑢 − (𝐶 · (𝑢↑3))) ∈ 𝑇)
172	171	ad4ant14 753	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → (𝑢 − (𝐶 · (𝑢↑3))) ∈ 𝑇)
173		infrelb 12139	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑇 ⊆ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑇 𝑥 ≤ 𝑤 ∧ (𝑢 − (𝐶 · (𝑢↑3))) ∈ 𝑇) → inf(𝑇, ℝ, < ) ≤ (𝑢 − (𝐶 · (𝑢↑3))))
174	115, 116, 172, 173	syl3anc 1374	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → inf(𝑇, ℝ, < ) ≤ (𝑢 − (𝐶 · (𝑢↑3))))
175	110, 162, 174	lensymd 11296	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → ¬ (𝑢 − (𝐶 · (𝑢↑3))) < inf(𝑇, ℝ, < ))
176	175	pm2.21d 121	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → ((𝑢 − (𝐶 · (𝑢↑3))) < inf(𝑇, ℝ, < ) → (inf(𝑇, ℝ, < ) + (𝑠 / 2)) ≤ 𝑢))
177	170, 176	sylbid 240	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → ((𝑢 − (𝐶 · (𝑢↑3))) < ((inf(𝑇, ℝ, < ) − (𝐶 · (inf(𝑇, ℝ, < )↑3))) + (𝐶 · (inf(𝑇, ℝ, < )↑3))) → (inf(𝑇, ℝ, < ) + (𝑠 / 2)) ≤ 𝑢))
178	177	adantld 490	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → ((((inf(𝑇, ℝ, < ) − (𝐶 · (inf(𝑇, ℝ, < )↑3))) − (𝐶 · (inf(𝑇, ℝ, < )↑3))) < (𝑢 − (𝐶 · (𝑢↑3))) ∧ (𝑢 − (𝐶 · (𝑢↑3))) < ((inf(𝑇, ℝ, < ) − (𝐶 · (inf(𝑇, ℝ, < )↑3))) + (𝐶 · (inf(𝑇, ℝ, < )↑3)))) → (inf(𝑇, ℝ, < ) + (𝑠 / 2)) ≤ 𝑢))
179	167, 178	sylbid 240	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → ((abs‘((𝑢 − (𝐶 · (𝑢↑3))) − (inf(𝑇, ℝ, < ) − (𝐶 · (inf(𝑇, ℝ, < )↑3))))) < (𝐶 · (inf(𝑇, ℝ, < )↑3)) → (inf(𝑇, ℝ, < ) + (𝑠 / 2)) ≤ 𝑢))
180	156, 179	sylbid 240	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → ((abs‘(((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘𝑢) − ((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘inf(𝑇, ℝ, < )))) < (𝐶 · (inf(𝑇, ℝ, < )↑3)) → (inf(𝑇, ℝ, < ) + (𝑠 / 2)) ≤ 𝑢))
181	136, 180	jad 187	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑢 ∈ 𝑇) → (((abs‘(𝑢 − inf(𝑇, ℝ, < ))) < 𝑠 → (abs‘(((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘𝑢) − ((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘inf(𝑇, ℝ, < )))) < (𝐶 · (inf(𝑇, ℝ, < )↑3))) → (inf(𝑇, ℝ, < ) + (𝑠 / 2)) ≤ 𝑢))
182	181	ralimdva 3150	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) → (∀𝑢 ∈ 𝑇 ((abs‘(𝑢 − inf(𝑇, ℝ, < ))) < 𝑠 → (abs‘(((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘𝑢) − ((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘inf(𝑇, ℝ, < )))) < (𝐶 · (inf(𝑇, ℝ, < )↑3))) → ∀𝑢 ∈ 𝑇 (inf(𝑇, ℝ, < ) + (𝑠 / 2)) ≤ 𝑢))
183	64	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) → 𝑇 ≠ ∅)
184	79	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑇 𝑥 ≤ 𝑤)
185		infregelb 12138	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑇 𝑥 ≤ 𝑤) ∧ (inf(𝑇, ℝ, < ) + (𝑠 / 2)) ∈ ℝ) → ((inf(𝑇, ℝ, < ) + (𝑠 / 2)) ≤ inf(𝑇, ℝ, < ) ↔ ∀𝑢 ∈ 𝑇 (inf(𝑇, ℝ, < ) + (𝑠 / 2)) ≤ 𝑢))
186	106, 183, 184, 98, 185	syl31anc 1376	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) → ((inf(𝑇, ℝ, < ) + (𝑠 / 2)) ≤ inf(𝑇, ℝ, < ) ↔ ∀𝑢 ∈ 𝑇 (inf(𝑇, ℝ, < ) + (𝑠 / 2)) ≤ 𝑢))
187	182, 186	sylibrd 259	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) → (∀𝑢 ∈ 𝑇 ((abs‘(𝑢 − inf(𝑇, ℝ, < ))) < 𝑠 → (abs‘(((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘𝑢) − ((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘inf(𝑇, ℝ, < )))) < (𝐶 · (inf(𝑇, ℝ, < )↑3))) → (inf(𝑇, ℝ, < ) + (𝑠 / 2)) ≤ inf(𝑇, ℝ, < )))
188	105, 187	syld 47	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) → (∀𝑢 ∈ ℂ ((abs‘(𝑢 − inf(𝑇, ℝ, < ))) < 𝑠 → (abs‘(((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘𝑢) − ((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘inf(𝑇, ℝ, < )))) < (𝐶 · (inf(𝑇, ℝ, < )↑3))) → (inf(𝑇, ℝ, < ) + (𝑠 / 2)) ≤ inf(𝑇, ℝ, < )))
189	100, 188	mtod 198	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) ∧ 𝑠 ∈ ℝ⁺) → ¬ ∀𝑢 ∈ ℂ ((abs‘(𝑢 − inf(𝑇, ℝ, < ))) < 𝑠 → (abs‘(((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘𝑢) − ((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘inf(𝑇, ℝ, < )))) < (𝐶 · (inf(𝑇, ℝ, < )↑3))))
190	189	nrexdv 3133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < inf(𝑇, ℝ, < )) → ¬ ∃𝑠 ∈ ℝ⁺ ∀𝑢 ∈ ℂ ((abs‘(𝑢 − inf(𝑇, ℝ, < ))) < 𝑠 → (abs‘(((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘𝑢) − ((𝑝 ∈ ℂ ↦ (𝑝 − (𝐶 · (𝑝↑3))))‘inf(𝑇, ℝ, < )))) < (𝐶 · (inf(𝑇, ℝ, < )↑3))))
191	92, 190	pm2.65da 817	. . . . . . . 8 ⊢ (𝜑 → ¬ 0 < inf(𝑇, ℝ, < ))
192	191	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ⁺) → ¬ 0 < inf(𝑇, ℝ, < ))
193	29	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ⁺) → 𝑇 ⊆ ℝ)
194	64	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ⁺) → 𝑇 ≠ ∅)
195	79	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ⁺) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑇 𝑥 ≤ 𝑤)
196	129	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ⁺) → 𝑠 ∈ ℝ)
197		infregelb 12138	. . . . . . . . . 10 ⊢ (((𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑇 𝑥 ≤ 𝑤) ∧ 𝑠 ∈ ℝ) → (𝑠 ≤ inf(𝑇, ℝ, < ) ↔ ∀𝑤 ∈ 𝑇 𝑠 ≤ 𝑤))
198	193, 194, 195, 196, 197	syl31anc 1376	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ⁺) → (𝑠 ≤ inf(𝑇, ℝ, < ) ↔ ∀𝑤 ∈ 𝑇 𝑠 ≤ 𝑤))
199	22	raleqi 3296	. . . . . . . . . 10 ⊢ (∀𝑤 ∈ 𝑇 𝑠 ≤ 𝑤 ↔ ∀𝑤 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}𝑠 ≤ 𝑤)
200		breq2 5104	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑡 → (𝑠 ≤ 𝑤 ↔ 𝑠 ≤ 𝑡))
201	200	ralrab2 3658	. . . . . . . . . 10 ⊢ (∀𝑤 ∈ {𝑡 ∈ (0[,]𝐴) ∣ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡}𝑠 ≤ 𝑤 ↔ ∀𝑡 ∈ (0[,]𝐴)(∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 → 𝑠 ≤ 𝑡))
202	199, 201	bitri 275	. . . . . . . . 9 ⊢ (∀𝑤 ∈ 𝑇 𝑠 ≤ 𝑤 ↔ ∀𝑡 ∈ (0[,]𝐴)(∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 → 𝑠 ≤ 𝑡))
203	198, 202	bitrdi 287	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ⁺) → (𝑠 ≤ inf(𝑇, ℝ, < ) ↔ ∀𝑡 ∈ (0[,]𝐴)(∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 → 𝑠 ≤ 𝑡)))
204		rpgt0 12930	. . . . . . . . . 10 ⊢ (𝑠 ∈ ℝ⁺ → 0 < 𝑠)
205	204	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ⁺) → 0 < 𝑠)
206	81	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ⁺) → inf(𝑇, ℝ, < ) ∈ ℝ)
207		ltletr 11237	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ 𝑠 ∈ ℝ ∧ inf(𝑇, ℝ, < ) ∈ ℝ) → ((0 < 𝑠 ∧ 𝑠 ≤ inf(𝑇, ℝ, < )) → 0 < inf(𝑇, ℝ, < )))
208	24, 196, 206, 207	mp3an2i 1469	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ⁺) → ((0 < 𝑠 ∧ 𝑠 ≤ inf(𝑇, ℝ, < )) → 0 < inf(𝑇, ℝ, < )))
209	205, 208	mpand 696	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ⁺) → (𝑠 ≤ inf(𝑇, ℝ, < ) → 0 < inf(𝑇, ℝ, < )))
210	203, 209	sylbird 260	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ⁺) → (∀𝑡 ∈ (0[,]𝐴)(∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 → 𝑠 ≤ 𝑡) → 0 < inf(𝑇, ℝ, < )))
211	192, 210	mtod 198	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ⁺) → ¬ ∀𝑡 ∈ (0[,]𝐴)(∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 → 𝑠 ≤ 𝑡))
212		rexanali 3092	. . . . . 6 ⊢ (∃𝑡 ∈ (0[,]𝐴)(∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 ∧ ¬ 𝑠 ≤ 𝑡) ↔ ¬ ∀𝑡 ∈ (0[,]𝐴)(∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 → 𝑠 ≤ 𝑡))
213	211, 212	sylibr 234	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ⁺) → ∃𝑡 ∈ (0[,]𝐴)(∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 ∧ ¬ 𝑠 ≤ 𝑡))
214		fveq2 6842	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑥 → (𝑅‘𝑧) = (𝑅‘𝑥))
215		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑥 → 𝑧 = 𝑥)
216	214, 215	oveq12d 7386	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → ((𝑅‘𝑧) / 𝑧) = ((𝑅‘𝑥) / 𝑥))
217	216	fveq2d 6846	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → (abs‘((𝑅‘𝑧) / 𝑧)) = (abs‘((𝑅‘𝑥) / 𝑥)))
218	217	breq1d 5110	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → ((abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 ↔ (abs‘((𝑅‘𝑥) / 𝑥)) ≤ 𝑡))
219	218	cbvralvw 3216	. . . . . . . . . . 11 ⊢ (∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 ↔ ∀𝑥 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑥) / 𝑥)) ≤ 𝑡)
220		rpre 12926	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
221	220	ad2antll 730	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑦 ≤ 𝑥 ∧ 𝑥 ∈ ℝ⁺)) → 𝑥 ∈ ℝ)
222		simprl 771	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑦 ≤ 𝑥 ∧ 𝑥 ∈ ℝ⁺)) → 𝑦 ≤ 𝑥)
223		simplr 769	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑦 ≤ 𝑥 ∧ 𝑥 ∈ ℝ⁺)) → 𝑦 ∈ ℝ⁺)
224	223	rpred 12961	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑦 ≤ 𝑥 ∧ 𝑥 ∈ ℝ⁺)) → 𝑦 ∈ ℝ)
225		elicopnf 13373	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℝ → (𝑥 ∈ (𝑦[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 𝑦 ≤ 𝑥)))
226	224, 225	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑦 ≤ 𝑥 ∧ 𝑥 ∈ ℝ⁺)) → (𝑥 ∈ (𝑦[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 𝑦 ≤ 𝑥)))
227	221, 222, 226	mpbir2and 714	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑦 ≤ 𝑥 ∧ 𝑥 ∈ ℝ⁺)) → 𝑥 ∈ (𝑦[,)+∞))
228		pntlem3.r	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
229	228	pntrval 27541	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℝ⁺ → (𝑅‘𝑥) = ((ψ‘𝑥) − 𝑥))
230	229	ad2antll 730	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑦 ≤ 𝑥 ∧ 𝑥 ∈ ℝ⁺)) → (𝑅‘𝑥) = ((ψ‘𝑥) − 𝑥))
231	230	oveq1d 7383	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑦 ≤ 𝑥 ∧ 𝑥 ∈ ℝ⁺)) → ((𝑅‘𝑥) / 𝑥) = (((ψ‘𝑥) − 𝑥) / 𝑥))
232		chpcl 27102	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
233	221, 232	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑦 ≤ 𝑥 ∧ 𝑥 ∈ ℝ⁺)) → (ψ‘𝑥) ∈ ℝ)
234	233	recnd 11172	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑦 ≤ 𝑥 ∧ 𝑥 ∈ ℝ⁺)) → (ψ‘𝑥) ∈ ℂ)
235		rpcn 12928	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
236	235	ad2antll 730	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑦 ≤ 𝑥 ∧ 𝑥 ∈ ℝ⁺)) → 𝑥 ∈ ℂ)
237		rpne0 12934	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
238	237	ad2antll 730	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑦 ≤ 𝑥 ∧ 𝑥 ∈ ℝ⁺)) → 𝑥 ≠ 0)
239	234, 236, 236, 238	divsubdird 11968	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑦 ≤ 𝑥 ∧ 𝑥 ∈ ℝ⁺)) → (((ψ‘𝑥) − 𝑥) / 𝑥) = (((ψ‘𝑥) / 𝑥) − (𝑥 / 𝑥)))
240	236, 238	dividd 11927	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑦 ≤ 𝑥 ∧ 𝑥 ∈ ℝ⁺)) → (𝑥 / 𝑥) = 1)
241	240	oveq2d 7384	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑦 ≤ 𝑥 ∧ 𝑥 ∈ ℝ⁺)) → (((ψ‘𝑥) / 𝑥) − (𝑥 / 𝑥)) = (((ψ‘𝑥) / 𝑥) − 1))
242	231, 239, 241	3eqtrrd 2777	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑦 ≤ 𝑥 ∧ 𝑥 ∈ ℝ⁺)) → (((ψ‘𝑥) / 𝑥) − 1) = ((𝑅‘𝑥) / 𝑥))
243	242	fveq2d 6846	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑦 ≤ 𝑥 ∧ 𝑥 ∈ ℝ⁺)) → (abs‘(((ψ‘𝑥) / 𝑥) − 1)) = (abs‘((𝑅‘𝑥) / 𝑥)))
244	243	breq1d 5110	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑦 ≤ 𝑥 ∧ 𝑥 ∈ ℝ⁺)) → ((abs‘(((ψ‘𝑥) / 𝑥) − 1)) ≤ 𝑡 ↔ (abs‘((𝑅‘𝑥) / 𝑥)) ≤ 𝑡))
245		simprr 773	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) → ¬ 𝑠 ≤ 𝑡)
246	245	ad2antrr 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑦 ≤ 𝑥 ∧ 𝑥 ∈ ℝ⁺)) → ¬ 𝑠 ≤ 𝑡)
247	28	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) → (0[,]𝐴) ⊆ ℝ)
248	247	ad2antrr 727	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑦 ≤ 𝑥 ∧ 𝑥 ∈ ℝ⁺)) → (0[,]𝐴) ⊆ ℝ)
249		simplrl 777	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) → 𝑡 ∈ (0[,]𝐴))
250	249	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑦 ≤ 𝑥 ∧ 𝑥 ∈ ℝ⁺)) → 𝑡 ∈ (0[,]𝐴))
251	248, 250	sseldd 3936	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑦 ≤ 𝑥 ∧ 𝑥 ∈ ℝ⁺)) → 𝑡 ∈ ℝ)
252		simp-4r 784	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑦 ≤ 𝑥 ∧ 𝑥 ∈ ℝ⁺)) → 𝑠 ∈ ℝ⁺)
253	252	rpred 12961	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑦 ≤ 𝑥 ∧ 𝑥 ∈ ℝ⁺)) → 𝑠 ∈ ℝ)
254	251, 253	ltnled 11292	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑦 ≤ 𝑥 ∧ 𝑥 ∈ ℝ⁺)) → (𝑡 < 𝑠 ↔ ¬ 𝑠 ≤ 𝑡))
255	246, 254	mpbird 257	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑦 ≤ 𝑥 ∧ 𝑥 ∈ ℝ⁺)) → 𝑡 < 𝑠)
256	220, 232	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℝ)
257		rerpdivcl 12949	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((ψ‘𝑥) ∈ ℝ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
258	256, 257	mpancom 689	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
259	258	ad2antll 730	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑦 ≤ 𝑥 ∧ 𝑥 ∈ ℝ⁺)) → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
260		resubcl 11457	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((ψ‘𝑥) / 𝑥) ∈ ℝ ∧ 1 ∈ ℝ) → (((ψ‘𝑥) / 𝑥) − 1) ∈ ℝ)
261	259, 43, 260	sylancl 587	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑦 ≤ 𝑥 ∧ 𝑥 ∈ ℝ⁺)) → (((ψ‘𝑥) / 𝑥) − 1) ∈ ℝ)
262	261	recnd 11172	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑦 ≤ 𝑥 ∧ 𝑥 ∈ ℝ⁺)) → (((ψ‘𝑥) / 𝑥) − 1) ∈ ℂ)
263	262	abscld 15374	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑦 ≤ 𝑥 ∧ 𝑥 ∈ ℝ⁺)) → (abs‘(((ψ‘𝑥) / 𝑥) − 1)) ∈ ℝ)
264		lelttr 11235	. . . . . . . . . . . . . . . . . 18 ⊢ (((abs‘(((ψ‘𝑥) / 𝑥) − 1)) ∈ ℝ ∧ 𝑡 ∈ ℝ ∧ 𝑠 ∈ ℝ) → (((abs‘(((ψ‘𝑥) / 𝑥) − 1)) ≤ 𝑡 ∧ 𝑡 < 𝑠) → (abs‘(((ψ‘𝑥) / 𝑥) − 1)) < 𝑠))
265	263, 251, 253, 264	syl3anc 1374	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑦 ≤ 𝑥 ∧ 𝑥 ∈ ℝ⁺)) → (((abs‘(((ψ‘𝑥) / 𝑥) − 1)) ≤ 𝑡 ∧ 𝑡 < 𝑠) → (abs‘(((ψ‘𝑥) / 𝑥) − 1)) < 𝑠))
266	255, 265	mpan2d 695	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑦 ≤ 𝑥 ∧ 𝑥 ∈ ℝ⁺)) → ((abs‘(((ψ‘𝑥) / 𝑥) − 1)) ≤ 𝑡 → (abs‘(((ψ‘𝑥) / 𝑥) − 1)) < 𝑠))
267	244, 266	sylbird 260	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑦 ≤ 𝑥 ∧ 𝑥 ∈ ℝ⁺)) → ((abs‘((𝑅‘𝑥) / 𝑥)) ≤ 𝑡 → (abs‘(((ψ‘𝑥) / 𝑥) − 1)) < 𝑠))
268	227, 267	embantd 59	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) ∧ (𝑦 ≤ 𝑥 ∧ 𝑥 ∈ ℝ⁺)) → ((𝑥 ∈ (𝑦[,)+∞) → (abs‘((𝑅‘𝑥) / 𝑥)) ≤ 𝑡) → (abs‘(((ψ‘𝑥) / 𝑥) − 1)) < 𝑠))
269	268	exp32 420	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) → (𝑦 ≤ 𝑥 → (𝑥 ∈ ℝ⁺ → ((𝑥 ∈ (𝑦[,)+∞) → (abs‘((𝑅‘𝑥) / 𝑥)) ≤ 𝑡) → (abs‘(((ψ‘𝑥) / 𝑥) − 1)) < 𝑠))))
270	269	com24 95	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) → ((𝑥 ∈ (𝑦[,)+∞) → (abs‘((𝑅‘𝑥) / 𝑥)) ≤ 𝑡) → (𝑥 ∈ ℝ⁺ → (𝑦 ≤ 𝑥 → (abs‘(((ψ‘𝑥) / 𝑥) − 1)) < 𝑠))))
271	270	ralimdv2 3147	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) → (∀𝑥 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑥) / 𝑥)) ≤ 𝑡 → ∀𝑥 ∈ ℝ⁺ (𝑦 ≤ 𝑥 → (abs‘(((ψ‘𝑥) / 𝑥) − 1)) < 𝑠)))
272	219, 271	biimtrid 242	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) ∧ 𝑦 ∈ ℝ⁺) → (∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 → ∀𝑥 ∈ ℝ⁺ (𝑦 ≤ 𝑥 → (abs‘(((ψ‘𝑥) / 𝑥) − 1)) < 𝑠)))
273	272	reximdva 3151	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ (𝑡 ∈ (0[,]𝐴) ∧ ¬ 𝑠 ≤ 𝑡)) → (∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 → ∃𝑦 ∈ ℝ⁺ ∀𝑥 ∈ ℝ⁺ (𝑦 ≤ 𝑥 → (abs‘(((ψ‘𝑥) / 𝑥) − 1)) < 𝑠)))
274	273	anassrs 467	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑡 ∈ (0[,]𝐴)) ∧ ¬ 𝑠 ≤ 𝑡) → (∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 → ∃𝑦 ∈ ℝ⁺ ∀𝑥 ∈ ℝ⁺ (𝑦 ≤ 𝑥 → (abs‘(((ψ‘𝑥) / 𝑥) − 1)) < 𝑠)))
275	274	impancom 451	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑡 ∈ (0[,]𝐴)) ∧ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡) → (¬ 𝑠 ≤ 𝑡 → ∃𝑦 ∈ ℝ⁺ ∀𝑥 ∈ ℝ⁺ (𝑦 ≤ 𝑥 → (abs‘(((ψ‘𝑥) / 𝑥) − 1)) < 𝑠)))
276	275	expimpd 453	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ⁺) ∧ 𝑡 ∈ (0[,]𝐴)) → ((∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 ∧ ¬ 𝑠 ≤ 𝑡) → ∃𝑦 ∈ ℝ⁺ ∀𝑥 ∈ ℝ⁺ (𝑦 ≤ 𝑥 → (abs‘(((ψ‘𝑥) / 𝑥) − 1)) < 𝑠)))
277	276	rexlimdva 3139	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ⁺) → (∃𝑡 ∈ (0[,]𝐴)(∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ (𝑦[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑡 ∧ ¬ 𝑠 ≤ 𝑡) → ∃𝑦 ∈ ℝ⁺ ∀𝑥 ∈ ℝ⁺ (𝑦 ≤ 𝑥 → (abs‘(((ψ‘𝑥) / 𝑥) − 1)) < 𝑠)))
278	213, 277	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ⁺) → ∃𝑦 ∈ ℝ⁺ ∀𝑥 ∈ ℝ⁺ (𝑦 ≤ 𝑥 → (abs‘(((ψ‘𝑥) / 𝑥) − 1)) < 𝑠))
279		ssrexv 4005	. . . 4 ⊢ (ℝ⁺ ⊆ ℝ → (∃𝑦 ∈ ℝ⁺ ∀𝑥 ∈ ℝ⁺ (𝑦 ≤ 𝑥 → (abs‘(((ψ‘𝑥) / 𝑥) − 1)) < 𝑠) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ℝ⁺ (𝑦 ≤ 𝑥 → (abs‘(((ψ‘𝑥) / 𝑥) − 1)) < 𝑠)))
280	1, 278, 279	mpsyl 68	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ⁺) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ℝ⁺ (𝑦 ≤ 𝑥 → (abs‘(((ψ‘𝑥) / 𝑥) − 1)) < 𝑠))
281	280	ralrimiva 3130	. 2 ⊢ (𝜑 → ∀𝑠 ∈ ℝ⁺ ∃𝑦 ∈ ℝ ∀𝑥 ∈ ℝ⁺ (𝑦 ≤ 𝑥 → (abs‘(((ψ‘𝑥) / 𝑥) − 1)) < 𝑠))
282	258	recnd 11172	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) / 𝑥) ∈ ℂ)
283	282	rgen 3054	. . . 4 ⊢ ∀𝑥 ∈ ℝ⁺ ((ψ‘𝑥) / 𝑥) ∈ ℂ
284	283	a1i 11	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ((ψ‘𝑥) / 𝑥) ∈ ℂ)
285	1	a1i 11	. . 3 ⊢ (𝜑 → ℝ⁺ ⊆ ℝ)
286		1cnd 11139	. . 3 ⊢ (𝜑 → 1 ∈ ℂ)
287	284, 285, 286	rlim2 15431	. 2 ⊢ (𝜑 → ((𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ⇝_𝑟 1 ↔ ∀𝑠 ∈ ℝ⁺ ∃𝑦 ∈ ℝ ∀𝑥 ∈ ℝ⁺ (𝑦 ≤ 𝑥 → (abs‘(((ψ‘𝑥) / 𝑥) − 1)) < 𝑠)))
288	281, 287	mpbird 257	1 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ⇝_𝑟 1)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∃wrex 3062 {crab 3401 ⊆ wss 3903 ∅c0 4287 class class class wbr 5100 ↦ cmpt 5181 ‘cfv 6500 (class class class)co 7368 infcinf 9356 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 +∞cpnf 11175 ℝ^*cxr 11177 < clt 11178 ≤ cle 11179 − cmin 11376 / cdiv 11806 2c2 12212 3c3 12213 ℕ₀cn0 12413 ℤcz 12500 ℝ⁺crp 12917 [,)cico 13275 [,]cicc 13276 ↑cexp 13996 abscabs 15169 ⇝_𝑟 crli 15420 TopOpenctopn 17353 ℂ_fldccnfld 21321 Cn ccn 23180 ×_t ctx 23516 –cn→ccncf 24837 ψcchp 27071

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-oadd 8411 df-er 8645 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-fi 9326 df-sup 9357 df-inf 9358 df-oi 9427 df-dju 9825 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13277 df-ioc 13278 df-ico 13279 df-icc 13280 df-fz 13436 df-fzo 13583 df-fl 13724 df-mod 13802 df-seq 13937 df-exp 13997 df-fac 14209 df-bc 14238 df-hash 14266 df-shft 15002 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-limsup 15406 df-clim 15423 df-rlim 15424 df-sum 15622 df-ef 16002 df-sin 16004 df-cos 16005 df-pi 16007 df-dvds 16192 df-gcd 16434 df-prm 16611 df-pc 16777 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-rest 17354 df-topn 17355 df-0g 17373 df-gsum 17374 df-topgen 17375 df-pt 17376 df-prds 17379 df-xrs 17435 df-qtop 17440 df-imas 17441 df-xps 17443 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-mulg 19010 df-cntz 19258 df-cmn 19723 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-fbas 21318 df-fg 21319 df-cnfld 21322 df-top 22850 df-topon 22867 df-topsp 22889 df-bases 22902 df-cld 22975 df-ntr 22976 df-cls 22977 df-nei 23054 df-lp 23092 df-perf 23093 df-cn 23183 df-cnp 23184 df-haus 23271 df-tx 23518 df-hmeo 23711 df-fil 23802 df-fm 23894 df-flim 23895 df-flf 23896 df-xms 24276 df-ms 24277 df-tms 24278 df-cncf 24839 df-limc 25835 df-dv 25836 df-log 26533 df-vma 27076 df-chp 27077

This theorem is referenced by: pntleml 27590

W3C validator