lhop1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > lhop1		Structured version Visualization version GIF version

Theorem lhop1 25992

Description: L'Hôpital's Rule for limits from the right. If 𝐹 and 𝐺 are differentiable real functions on (𝐴, 𝐵), and 𝐹 and 𝐺 both approach 0 at 𝐴, and 𝐺(𝑥) and 𝐺' (𝑥) are not zero on (𝐴, 𝐵), and the limit of 𝐹' (𝑥) / 𝐺' (𝑥) at 𝐴 is 𝐶, then the limit 𝐹(𝑥) / 𝐺(𝑥) at 𝐴 also exists and equals 𝐶. (Contributed by Mario Carneiro, 29-Dec-2016.)

**Hypotheses**
Ref	Expression
lhop1.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
lhop1.b	⊢ (𝜑 → 𝐵 ∈ ℝ^*)
lhop1.l	⊢ (𝜑 → 𝐴 < 𝐵)
lhop1.f	⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)
lhop1.g	⊢ (𝜑 → 𝐺:(𝐴(,)𝐵)⟶ℝ)
lhop1.if	⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
lhop1.ig	⊢ (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵))
lhop1.f0	⊢ (𝜑 → 0 ∈ (𝐹 lim_ℂ 𝐴))
lhop1.g0	⊢ (𝜑 → 0 ∈ (𝐺 lim_ℂ 𝐴))
lhop1.gn0	⊢ (𝜑 → ¬ 0 ∈ ran 𝐺)
lhop1.gd0	⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐺))
lhop1.c	⊢ (𝜑 → 𝐶 ∈ ((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧))) lim_ℂ 𝐴))

**Assertion**
Ref	Expression
lhop1	⊢ (𝜑 → 𝐶 ∈ ((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧))) lim_ℂ 𝐴))

Distinct variable groups: 𝑧,𝐵 𝜑,𝑧 𝑧,𝐴 𝑧,𝐶 𝑧,𝐹 𝑧,𝐺

Proof of Theorem lhop1 Dummy variables 𝑒 𝑑 𝑟 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lhop1.c	. 2 ⊢ (𝜑 → 𝐶 ∈ ((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧))) lim_ℂ 𝐴))
2		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
3	2	rphalfcld 12975	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 / 2) ∈ ℝ⁺)
4		breq2 5104	. . . . . . . . . 10 ⊢ (𝑒 = (𝑥 / 2) → ((abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)))‘𝑦) − 𝐶)) < 𝑒 ↔ (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)))‘𝑦) − 𝐶)) < (𝑥 / 2)))
5	4	imbi2d 340	. . . . . . . . 9 ⊢ (𝑒 = (𝑥 / 2) → (((𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)))‘𝑦) − 𝐶)) < 𝑒) ↔ ((𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)))‘𝑦) − 𝐶)) < (𝑥 / 2))))
6	5	rexralbidv 3204	. . . . . . . 8 ⊢ (𝑒 = (𝑥 / 2) → (∃𝑑 ∈ ℝ⁺ ∀𝑦 ∈ (𝐴(,)𝐵)((𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)))‘𝑦) − 𝐶)) < 𝑒) ↔ ∃𝑑 ∈ ℝ⁺ ∀𝑦 ∈ (𝐴(,)𝐵)((𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)))‘𝑦) − 𝐶)) < (𝑥 / 2))))
7	6	rspcv 3574	. . . . . . 7 ⊢ ((𝑥 / 2) ∈ ℝ⁺ → (∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑦 ∈ (𝐴(,)𝐵)((𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)))‘𝑦) − 𝐶)) < 𝑒) → ∃𝑑 ∈ ℝ⁺ ∀𝑦 ∈ (𝐴(,)𝐵)((𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)))‘𝑦) − 𝐶)) < (𝑥 / 2))))
8	3, 7	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑦 ∈ (𝐴(,)𝐵)((𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)))‘𝑦) − 𝐶)) < 𝑒) → ∃𝑑 ∈ ℝ⁺ ∀𝑦 ∈ (𝐴(,)𝐵)((𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)))‘𝑦) − 𝐶)) < (𝑥 / 2))))
9		rabid 3422	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ {𝑣 ∈ (𝐴(,)𝐵) ∣ (abs‘(𝑣 − 𝐴)) < 𝑑} ↔ (𝑣 ∈ (𝐴(,)𝐵) ∧ (abs‘(𝑣 − 𝐴)) < 𝑑))
10		eliooord 13335	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 ∈ (𝐴(,)𝐵) → (𝐴 < 𝑣 ∧ 𝑣 < 𝐵))
11	10	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → (𝐴 < 𝑣 ∧ 𝑣 < 𝐵))
12	11	simprd 495	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → 𝑣 < 𝐵)
13	12	biantrurd 532	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → (𝑣 < (𝑑 + 𝐴) ↔ (𝑣 < 𝐵 ∧ 𝑣 < (𝑑 + 𝐴))))
14		ioossre 13337	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴(,)𝐵) ⊆ ℝ
15		simpr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → 𝑣 ∈ (𝐴(,)𝐵))
16	14, 15	sselid 3933	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → 𝑣 ∈ ℝ)
17		lhop1.a	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐴 ∈ ℝ)
18	17	ad3antrrr 731	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ)
19		simpr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → 𝑑 ∈ ℝ⁺)
20	19	rpred 12963	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → 𝑑 ∈ ℝ)
21	20	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → 𝑑 ∈ ℝ)
22	16, 18, 21	ltsubaddd 11747	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → ((𝑣 − 𝐴) < 𝑑 ↔ 𝑣 < (𝑑 + 𝐴)))
23	16	rexrd 11196	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → 𝑣 ∈ ℝ^*)
24		lhop1.b	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
25	24	ad3antrrr 731	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ^*)
26	17	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → 𝐴 ∈ ℝ)
27	20, 26	readdcld 11175	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → (𝑑 + 𝐴) ∈ ℝ)
28	27	rexrd 11196	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → (𝑑 + 𝐴) ∈ ℝ^*)
29	28	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → (𝑑 + 𝐴) ∈ ℝ^*)
30		xrltmin 13111	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝑑 + 𝐴) ∈ ℝ^*) → (𝑣 < if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ↔ (𝑣 < 𝐵 ∧ 𝑣 < (𝑑 + 𝐴))))
31	23, 25, 29, 30	syl3anc 1374	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → (𝑣 < if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ↔ (𝑣 < 𝐵 ∧ 𝑣 < (𝑑 + 𝐴))))
32	13, 22, 31	3bitr4rd 312	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → (𝑣 < if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ↔ (𝑣 − 𝐴) < 𝑑))
33	18	rexrd 11196	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ^*)
34	25, 29	ifcld 4528	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ∈ ℝ^*)
35	11	simpld 494	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝑣)
36		elioo5 13333	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℝ^* ∧ if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ∈ ℝ^* ∧ 𝑣 ∈ ℝ^*) → (𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴))) ↔ (𝐴 < 𝑣 ∧ 𝑣 < if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))))
37	36	baibd 539	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℝ^* ∧ if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ∈ ℝ^* ∧ 𝑣 ∈ ℝ^*) ∧ 𝐴 < 𝑣) → (𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴))) ↔ 𝑣 < if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴))))
38	33, 34, 23, 35, 37	syl31anc 1376	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → (𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴))) ↔ 𝑣 < if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴))))
39	18, 16, 35	ltled 11295	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → 𝐴 ≤ 𝑣)
40	18, 16, 39	abssubge0d 15371	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → (abs‘(𝑣 − 𝐴)) = (𝑣 − 𝐴))
41	40	breq1d 5110	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → ((abs‘(𝑣 − 𝐴)) < 𝑑 ↔ (𝑣 − 𝐴) < 𝑑))
42	32, 38, 41	3bitr4d 311	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → (𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴))) ↔ (abs‘(𝑣 − 𝐴)) < 𝑑))
43	42	rabbi2dva 4180	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → ((𝐴(,)𝐵) ∩ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) = {𝑣 ∈ (𝐴(,)𝐵) ∣ (abs‘(𝑣 − 𝐴)) < 𝑑})
44	24	ad2antrr 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → 𝐵 ∈ ℝ^*)
45		xrmin1 13106	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 ∈ ℝ^* ∧ (𝑑 + 𝐴) ∈ ℝ^*) → if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ≤ 𝐵)
46	44, 28, 45	syl2anc 585	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ≤ 𝐵)
47		iooss2 13311	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∈ ℝ^* ∧ if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ≤ 𝐵) → (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴))) ⊆ (𝐴(,)𝐵))
48	44, 46, 47	syl2anc 585	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴))) ⊆ (𝐴(,)𝐵))
49		sseqin2 4177	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴))) ⊆ (𝐴(,)𝐵) ↔ ((𝐴(,)𝐵) ∩ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) = (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴))))
50	48, 49	sylib 218	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → ((𝐴(,)𝐵) ∩ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) = (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴))))
51	43, 50	eqtr3d 2774	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → {𝑣 ∈ (𝐴(,)𝐵) ∣ (abs‘(𝑣 − 𝐴)) < 𝑑} = (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴))))
52	51	eleq2d 2823	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → (𝑣 ∈ {𝑣 ∈ (𝐴(,)𝐵) ∣ (abs‘(𝑣 − 𝐴)) < 𝑑} ↔ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))))
53	9, 52	bitr3id 285	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → ((𝑣 ∈ (𝐴(,)𝐵) ∧ (abs‘(𝑣 − 𝐴)) < 𝑑) ↔ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))))
54		lbioo 13306	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ¬ 𝐴 ∈ (𝐴(,)𝐵)
55		eleq1 2825	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ (𝐴(,)𝐵) ↔ 𝐴 ∈ (𝐴(,)𝐵)))
56	54, 55	mtbiri 327	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝐴 → ¬ 𝑦 ∈ (𝐴(,)𝐵))
57	56	necon2ai 2962	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (𝐴(,)𝐵) → 𝑦 ≠ 𝐴)
58	57	biantrurd 532	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (𝐴(,)𝐵) → ((abs‘(𝑦 − 𝐴)) < 𝑑 ↔ (𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑)))
59	58	bicomd 223	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (𝐴(,)𝐵) → ((𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑) ↔ (abs‘(𝑦 − 𝐴)) < 𝑑))
60		fveq2 6844	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = 𝑦 → ((ℝ D 𝐹)‘𝑧) = ((ℝ D 𝐹)‘𝑦))
61		fveq2 6844	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = 𝑦 → ((ℝ D 𝐺)‘𝑧) = ((ℝ D 𝐺)‘𝑦))
62	60, 61	oveq12d 7388	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = 𝑦 → (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)) = (((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)))
63		eqid 2737	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧))) = (𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)))
64		ovex 7403	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)) ∈ V
65	62, 63, 64	fvmpt3i 6957	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (𝐴(,)𝐵) → ((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)))‘𝑦) = (((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)))
66	65	fvoveq1d 7392	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (𝐴(,)𝐵) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)))‘𝑦) − 𝐶)) = (abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)))
67	66	breq1d 5110	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (𝐴(,)𝐵) → ((abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)))‘𝑦) − 𝐶)) < (𝑥 / 2) ↔ (abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2)))
68	59, 67	imbi12d 344	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (𝐴(,)𝐵) → (((𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)))‘𝑦) − 𝐶)) < (𝑥 / 2)) ↔ ((abs‘(𝑦 − 𝐴)) < 𝑑 → (abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))))
69	68	ralbiia 3082	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑦 ∈ (𝐴(,)𝐵)((𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)))‘𝑦) − 𝐶)) < (𝑥 / 2)) ↔ ∀𝑦 ∈ (𝐴(,)𝐵)((abs‘(𝑦 − 𝐴)) < 𝑑 → (abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2)))
70		fvoveq1 7393	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝑦 → (abs‘(𝑣 − 𝐴)) = (abs‘(𝑦 − 𝐴)))
71	70	breq1d 5110	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝑦 → ((abs‘(𝑣 − 𝐴)) < 𝑑 ↔ (abs‘(𝑦 − 𝐴)) < 𝑑))
72	71	ralrab 3654	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑦 ∈ {𝑣 ∈ (𝐴(,)𝐵) ∣ (abs‘(𝑣 − 𝐴)) < 𝑑} (abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2) ↔ ∀𝑦 ∈ (𝐴(,)𝐵)((abs‘(𝑦 − 𝐴)) < 𝑑 → (abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2)))
73	69, 72	bitr4i 278	. . . . . . . . . . . . . . 15 ⊢ (∀𝑦 ∈ (𝐴(,)𝐵)((𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)))‘𝑦) − 𝐶)) < (𝑥 / 2)) ↔ ∀𝑦 ∈ {𝑣 ∈ (𝐴(,)𝐵) ∣ (abs‘(𝑣 − 𝐴)) < 𝑑} (abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))
74	51	adantrr 718	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴))))) → {𝑣 ∈ (𝐴(,)𝐵) ∣ (abs‘(𝑣 − 𝐴)) < 𝑑} = (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴))))
75	74	raleqdv 3298	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴))))) → (∀𝑦 ∈ {𝑣 ∈ (𝐴(,)𝐵) ∣ (abs‘(𝑣 − 𝐴)) < 𝑑} (abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2) ↔ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2)))
76	17	ad2antrr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) → 𝐴 ∈ ℝ)
77	24	ad2antrr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) → 𝐵 ∈ ℝ^*)
78		lhop1.l	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐴 < 𝐵)
79	78	ad2antrr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) → 𝐴 < 𝐵)
80		lhop1.f	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)
81	80	ad2antrr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) → 𝐹:(𝐴(,)𝐵)⟶ℝ)
82		lhop1.g	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐺:(𝐴(,)𝐵)⟶ℝ)
83	82	ad2antrr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) → 𝐺:(𝐴(,)𝐵)⟶ℝ)
84		lhop1.if	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
85	84	ad2antrr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
86		lhop1.ig	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵))
87	86	ad2antrr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) → dom (ℝ D 𝐺) = (𝐴(,)𝐵))
88		lhop1.f0	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 0 ∈ (𝐹 lim_ℂ 𝐴))
89	88	ad2antrr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) → 0 ∈ (𝐹 lim_ℂ 𝐴))
90		lhop1.g0	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 0 ∈ (𝐺 lim_ℂ 𝐴))
91	90	ad2antrr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) → 0 ∈ (𝐺 lim_ℂ 𝐴))
92		lhop1.gn0	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ¬ 0 ∈ ran 𝐺)
93	92	ad2antrr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) → ¬ 0 ∈ ran 𝐺)
94		lhop1.gd0	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐺))
95	94	ad2antrr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) → ¬ 0 ∈ ran (ℝ D 𝐺))
96	1	ad2antrr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) → 𝐶 ∈ ((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧))) lim_ℂ 𝐴))
97	3	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) → (𝑥 / 2) ∈ ℝ⁺)
98	76	rexrd 11196	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) → 𝐴 ∈ ℝ^*)
99		simprll 779	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) → 𝑑 ∈ ℝ⁺)
100	99	rpred 12963	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) → 𝑑 ∈ ℝ)
101	100, 76	readdcld 11175	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) → (𝑑 + 𝐴) ∈ ℝ)
102		iocssre 13357	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝑑 + 𝐴) ∈ ℝ) → (𝐴(,](𝑑 + 𝐴)) ⊆ ℝ)
103	98, 101, 102	syl2anc 585	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) → (𝐴(,](𝑑 + 𝐴)) ⊆ ℝ)
104	77	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) ∧ 𝐵 ≤ (𝑑 + 𝐴)) → 𝐵 ∈ ℝ^*)
105	100	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) ∧ ¬ 𝐵 ≤ (𝑑 + 𝐴)) → 𝑑 ∈ ℝ)
106	76	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) ∧ ¬ 𝐵 ≤ (𝑑 + 𝐴)) → 𝐴 ∈ ℝ)
107	105, 106	readdcld 11175	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) ∧ ¬ 𝐵 ≤ (𝑑 + 𝐴)) → (𝑑 + 𝐴) ∈ ℝ)
108	107	rexrd 11196	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) ∧ ¬ 𝐵 ≤ (𝑑 + 𝐴)) → (𝑑 + 𝐴) ∈ ℝ^*)
109	104, 108	ifclda 4517	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) → if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ∈ ℝ^*)
110	76, 99	ltaddrp2d 12997	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) → 𝐴 < (𝑑 + 𝐴))
111	101	rexrd 11196	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) → (𝑑 + 𝐴) ∈ ℝ^*)
112		xrltmin 13111	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝑑 + 𝐴) ∈ ℝ^*) → (𝐴 < if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ↔ (𝐴 < 𝐵 ∧ 𝐴 < (𝑑 + 𝐴))))
113	98, 77, 111, 112	syl3anc 1374	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) → (𝐴 < if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ↔ (𝐴 < 𝐵 ∧ 𝐴 < (𝑑 + 𝐴))))
114	79, 110, 113	mpbir2and 714	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) → 𝐴 < if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))
115		xrmin2 13107	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐵 ∈ ℝ^* ∧ (𝑑 + 𝐴) ∈ ℝ^*) → if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ≤ (𝑑 + 𝐴))
116	77, 111, 115	syl2anc 585	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) → if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ≤ (𝑑 + 𝐴))
117		elioc1 13317	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝑑 + 𝐴) ∈ ℝ^) → (if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ∈ (𝐴(,](𝑑 + 𝐴)) ↔ (if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ∈ ℝ^ ∧ 𝐴 < if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ∧ if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ≤ (𝑑 + 𝐴))))
118	98, 111, 117	syl2anc 585	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) → (if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ∈ (𝐴(,](𝑑 + 𝐴)) ↔ (if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ∈ ℝ^* ∧ 𝐴 < if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ∧ if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ≤ (𝑑 + 𝐴))))
119	109, 114, 116, 118	mpbir3and 1344	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) → if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ∈ (𝐴(,](𝑑 + 𝐴)))
120	103, 119	sseldd 3936	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) → if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ∈ ℝ)
121	77, 111, 45	syl2anc 585	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) → if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ≤ 𝐵)
122		simprlr 780	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) → 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴))))
123		simprr 773	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) → ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))
124		eqid 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 + (𝑟 / 2)) = (𝐴 + (𝑟 / 2))
125	76, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 96, 97, 120, 121, 122, 123, 124	lhop1lem 25991	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) → (abs‘(((𝐹‘𝑣) / (𝐺‘𝑣)) − 𝐶)) < (2 · (𝑥 / 2)))
126	2	rpcnd 12965	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
127		2cnd 12237	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 2 ∈ ℂ)
128		2ne0 12263	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ≠ 0
129	128	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 2 ≠ 0)
130	126, 127, 129	divcan2d 11933	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (2 · (𝑥 / 2)) = 𝑥)
131	130	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) → (2 · (𝑥 / 2)) = 𝑥)
132	125, 131	breqtrd 5126	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ ((𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))) ∧ ∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2))) → (abs‘(((𝐹‘𝑣) / (𝐺‘𝑣)) − 𝐶)) < 𝑥)
133	132	expr 456	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴))))) → (∀𝑦 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))(abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2) → (abs‘(((𝐹‘𝑣) / (𝐺‘𝑣)) − 𝐶)) < 𝑥))
134	75, 133	sylbid 240	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴))))) → (∀𝑦 ∈ {𝑣 ∈ (𝐴(,)𝐵) ∣ (abs‘(𝑣 − 𝐴)) < 𝑑} (abs‘((((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)) − 𝐶)) < (𝑥 / 2) → (abs‘(((𝐹‘𝑣) / (𝐺‘𝑣)) − 𝐶)) < 𝑥))
135	73, 134	biimtrid 242	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑑 ∈ ℝ⁺ ∧ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴))))) → (∀𝑦 ∈ (𝐴(,)𝐵)((𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)))‘𝑦) − 𝐶)) < (𝑥 / 2)) → (abs‘(((𝐹‘𝑣) / (𝐺‘𝑣)) − 𝐶)) < 𝑥))
136	135	expr 456	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → (𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴))) → (∀𝑦 ∈ (𝐴(,)𝐵)((𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)))‘𝑦) − 𝐶)) < (𝑥 / 2)) → (abs‘(((𝐹‘𝑣) / (𝐺‘𝑣)) − 𝐶)) < 𝑥)))
137	53, 136	sylbid 240	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → ((𝑣 ∈ (𝐴(,)𝐵) ∧ (abs‘(𝑣 − 𝐴)) < 𝑑) → (∀𝑦 ∈ (𝐴(,)𝐵)((𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)))‘𝑦) − 𝐶)) < (𝑥 / 2)) → (abs‘(((𝐹‘𝑣) / (𝐺‘𝑣)) − 𝐶)) < 𝑥)))
138	137	expdimp 452	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → ((abs‘(𝑣 − 𝐴)) < 𝑑 → (∀𝑦 ∈ (𝐴(,)𝐵)((𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)))‘𝑦) − 𝐶)) < (𝑥 / 2)) → (abs‘(((𝐹‘𝑣) / (𝐺‘𝑣)) − 𝐶)) < 𝑥)))
139		fveq2 6844	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑣 → (𝐹‘𝑧) = (𝐹‘𝑣))
140		fveq2 6844	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑣 → (𝐺‘𝑧) = (𝐺‘𝑣))
141	139, 140	oveq12d 7388	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑣 → ((𝐹‘𝑧) / (𝐺‘𝑧)) = ((𝐹‘𝑣) / (𝐺‘𝑣)))
142		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧))) = (𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧)))
143		ovex 7403	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑧) / (𝐺‘𝑧)) ∈ V
144	141, 142, 143	fvmpt3i 6957	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ (𝐴(,)𝐵) → ((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧)))‘𝑣) = ((𝐹‘𝑣) / (𝐺‘𝑣)))
145	144	fvoveq1d 7392	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ (𝐴(,)𝐵) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧)))‘𝑣) − 𝐶)) = (abs‘(((𝐹‘𝑣) / (𝐺‘𝑣)) − 𝐶)))
146	145	breq1d 5110	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ (𝐴(,)𝐵) → ((abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧)))‘𝑣) − 𝐶)) < 𝑥 ↔ (abs‘(((𝐹‘𝑣) / (𝐺‘𝑣)) − 𝐶)) < 𝑥))
147	146	imbi2d 340	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ (𝐴(,)𝐵) → ((∀𝑦 ∈ (𝐴(,)𝐵)((𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)))‘𝑦) − 𝐶)) < (𝑥 / 2)) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧)))‘𝑣) − 𝐶)) < 𝑥) ↔ (∀𝑦 ∈ (𝐴(,)𝐵)((𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)))‘𝑦) − 𝐶)) < (𝑥 / 2)) → (abs‘(((𝐹‘𝑣) / (𝐺‘𝑣)) − 𝐶)) < 𝑥)))
148	147	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → ((∀𝑦 ∈ (𝐴(,)𝐵)((𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)))‘𝑦) − 𝐶)) < (𝑥 / 2)) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧)))‘𝑣) − 𝐶)) < 𝑥) ↔ (∀𝑦 ∈ (𝐴(,)𝐵)((𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)))‘𝑦) − 𝐶)) < (𝑥 / 2)) → (abs‘(((𝐹‘𝑣) / (𝐺‘𝑣)) − 𝐶)) < 𝑥)))
149	138, 148	sylibrd 259	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → ((abs‘(𝑣 − 𝐴)) < 𝑑 → (∀𝑦 ∈ (𝐴(,)𝐵)((𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)))‘𝑦) − 𝐶)) < (𝑥 / 2)) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧)))‘𝑣) − 𝐶)) < 𝑥)))
150	149	adantld 490	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → ((𝑣 ≠ 𝐴 ∧ (abs‘(𝑣 − 𝐴)) < 𝑑) → (∀𝑦 ∈ (𝐴(,)𝐵)((𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)))‘𝑦) − 𝐶)) < (𝑥 / 2)) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧)))‘𝑣) − 𝐶)) < 𝑥)))
151	150	com23 86	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → (∀𝑦 ∈ (𝐴(,)𝐵)((𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)))‘𝑦) − 𝐶)) < (𝑥 / 2)) → ((𝑣 ≠ 𝐴 ∧ (abs‘(𝑣 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧)))‘𝑣) − 𝐶)) < 𝑥)))
152	151	ralrimdva 3138	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → (∀𝑦 ∈ (𝐴(,)𝐵)((𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)))‘𝑦) − 𝐶)) < (𝑥 / 2)) → ∀𝑣 ∈ (𝐴(,)𝐵)((𝑣 ≠ 𝐴 ∧ (abs‘(𝑣 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧)))‘𝑣) − 𝐶)) < 𝑥)))
153	152	reximdva 3151	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (∃𝑑 ∈ ℝ⁺ ∀𝑦 ∈ (𝐴(,)𝐵)((𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)))‘𝑦) − 𝐶)) < (𝑥 / 2)) → ∃𝑑 ∈ ℝ⁺ ∀𝑣 ∈ (𝐴(,)𝐵)((𝑣 ≠ 𝐴 ∧ (abs‘(𝑣 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧)))‘𝑣) − 𝐶)) < 𝑥)))
154	8, 153	syld 47	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑦 ∈ (𝐴(,)𝐵)((𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)))‘𝑦) − 𝐶)) < 𝑒) → ∃𝑑 ∈ ℝ⁺ ∀𝑣 ∈ (𝐴(,)𝐵)((𝑣 ≠ 𝐴 ∧ (abs‘(𝑣 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧)))‘𝑣) − 𝐶)) < 𝑥)))
155	154	ralrimdva 3138	. . . 4 ⊢ (𝜑 → (∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑦 ∈ (𝐴(,)𝐵)((𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)))‘𝑦) − 𝐶)) < 𝑒) → ∀𝑥 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑣 ∈ (𝐴(,)𝐵)((𝑣 ≠ 𝐴 ∧ (abs‘(𝑣 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧)))‘𝑣) − 𝐶)) < 𝑥)))
156	155	anim2d 613	. . 3 ⊢ (𝜑 → ((𝐶 ∈ ℂ ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑦 ∈ (𝐴(,)𝐵)((𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)))‘𝑦) − 𝐶)) < 𝑒)) → (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑣 ∈ (𝐴(,)𝐵)((𝑣 ≠ 𝐴 ∧ (abs‘(𝑣 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧)))‘𝑣) − 𝐶)) < 𝑥))))
157		dvf 25881	. . . . . . . 8 ⊢ (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ
158	84	feq2d 6656	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ ↔ (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ))
159	157, 158	mpbii 233	. . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ)
160	159	ffvelcdmda 7040	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑧) ∈ ℂ)
161		dvf 25881	. . . . . . . 8 ⊢ (ℝ D 𝐺):dom (ℝ D 𝐺)⟶ℂ
162	86	feq2d 6656	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D 𝐺):dom (ℝ D 𝐺)⟶ℂ ↔ (ℝ D 𝐺):(𝐴(,)𝐵)⟶ℂ))
163	161, 162	mpbii 233	. . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐺):(𝐴(,)𝐵)⟶ℂ)
164	163	ffvelcdmda 7040	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐺)‘𝑧) ∈ ℂ)
165	94	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ¬ 0 ∈ ran (ℝ D 𝐺))
166	163	ffnd 6673	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D 𝐺) Fn (𝐴(,)𝐵))
167		fnfvelrn 7036	. . . . . . . . . 10 ⊢ (((ℝ D 𝐺) Fn (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐺)‘𝑧) ∈ ran (ℝ D 𝐺))
168	166, 167	sylan 581	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐺)‘𝑧) ∈ ran (ℝ D 𝐺))
169		eleq1 2825	. . . . . . . . 9 ⊢ (((ℝ D 𝐺)‘𝑧) = 0 → (((ℝ D 𝐺)‘𝑧) ∈ ran (ℝ D 𝐺) ↔ 0 ∈ ran (ℝ D 𝐺)))
170	168, 169	syl5ibcom 245	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐺)‘𝑧) = 0 → 0 ∈ ran (ℝ D 𝐺)))
171	170	necon3bd 2947	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (¬ 0 ∈ ran (ℝ D 𝐺) → ((ℝ D 𝐺)‘𝑧) ≠ 0))
172	165, 171	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐺)‘𝑧) ≠ 0)
173	160, 164, 172	divcld 11931	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)) ∈ ℂ)
174	173	fmpttd 7071	. . . 4 ⊢ (𝜑 → (𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧))):(𝐴(,)𝐵)⟶ℂ)
175		ax-resscn 11097	. . . . . 6 ⊢ ℝ ⊆ ℂ
176	14, 175	sstri 3945	. . . . 5 ⊢ (𝐴(,)𝐵) ⊆ ℂ
177	176	a1i 11	. . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ)
178	17	recnd 11174	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ)
179	174, 177, 178	ellimc3 25853	. . 3 ⊢ (𝜑 → (𝐶 ∈ ((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧))) lim_ℂ 𝐴) ↔ (𝐶 ∈ ℂ ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑦 ∈ (𝐴(,)𝐵)((𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)))‘𝑦) − 𝐶)) < 𝑒))))
180	80	ffvelcdmda 7040	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑧) ∈ ℝ)
181	180	recnd 11174	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑧) ∈ ℂ)
182	82	ffvelcdmda 7040	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑧) ∈ ℝ)
183	182	recnd 11174	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑧) ∈ ℂ)
184	92	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ¬ 0 ∈ ran 𝐺)
185	82	ffnd 6673	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 Fn (𝐴(,)𝐵))
186		fnfvelrn 7036	. . . . . . . . . 10 ⊢ ((𝐺 Fn (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑧) ∈ ran 𝐺)
187	185, 186	sylan 581	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑧) ∈ ran 𝐺)
188		eleq1 2825	. . . . . . . . 9 ⊢ ((𝐺‘𝑧) = 0 → ((𝐺‘𝑧) ∈ ran 𝐺 ↔ 0 ∈ ran 𝐺))
189	187, 188	syl5ibcom 245	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((𝐺‘𝑧) = 0 → 0 ∈ ran 𝐺))
190	189	necon3bd 2947	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (¬ 0 ∈ ran 𝐺 → (𝐺‘𝑧) ≠ 0))
191	184, 190	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑧) ≠ 0)
192	181, 183, 191	divcld 11931	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((𝐹‘𝑧) / (𝐺‘𝑧)) ∈ ℂ)
193	192	fmpttd 7071	. . . 4 ⊢ (𝜑 → (𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧))):(𝐴(,)𝐵)⟶ℂ)
194	193, 177, 178	ellimc3 25853	. . 3 ⊢ (𝜑 → (𝐶 ∈ ((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧))) lim_ℂ 𝐴) ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑣 ∈ (𝐴(,)𝐵)((𝑣 ≠ 𝐴 ∧ (abs‘(𝑣 − 𝐴)) < 𝑑) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧)))‘𝑣) − 𝐶)) < 𝑥))))
195	156, 179, 194	3imtr4d 294	. 2 ⊢ (𝜑 → (𝐶 ∈ ((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧))) lim_ℂ 𝐴) → 𝐶 ∈ ((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧))) lim_ℂ 𝐴)))
196	1, 195	mpd 15	1 ⊢ (𝜑 → 𝐶 ∈ ((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧))) lim_ℂ 𝐴))

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 ∩ cin 3902 ⊆ wss 3903 ifcif 4481 class class class wbr 5100 ↦ cmpt 5181 dom cdm 5634 ran crn 5635 Fn wfn 6497 ⟶wf 6498 ‘cfv 6502 (class class class)co 7370 ℂcc 11038 ℝcr 11039 0cc0 11040 + caddc 11043 · cmul 11045 ℝ^*cxr 11179 < clt 11180 ≤ cle 11181 − cmin 11378 / cdiv 11808 2c2 12214 ℝ⁺crp 12919 (,)cioo 13275 (,]cioc 13276 abscabs 15171 lim_ℂ climc 25836 D cdv 25837

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 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692 ax-cnex 11096 ax-resscn 11097 ax-1cn 11098 ax-icn 11099 ax-addcl 11100 ax-addrcl 11101 ax-mulcl 11102 ax-mulrcl 11103 ax-mulcom 11104 ax-addass 11105 ax-mulass 11106 ax-distr 11107 ax-i2m1 11108 ax-1ne0 11109 ax-1rid 11110 ax-rnegex 11111 ax-rrecex 11112 ax-cnre 11113 ax-pre-lttri 11114 ax-pre-lttrn 11115 ax-pre-ltadd 11116 ax-pre-mulgt0 11117 ax-pre-sup 11118 ax-addf 11119

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 5529 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5587 df-se 5588 df-we 5589 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6269 df-ord 6330 df-on 6331 df-lim 6332 df-suc 6333 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-isom 6511 df-riota 7327 df-ov 7373 df-oprab 7374 df-mpo 7375 df-of 7634 df-om 7821 df-1st 7945 df-2nd 7946 df-supp 8115 df-frecs 8235 df-wrecs 8266 df-recs 8315 df-rdg 8353 df-1o 8409 df-2o 8410 df-er 8647 df-map 8779 df-pm 8780 df-ixp 8850 df-en 8898 df-dom 8899 df-sdom 8900 df-fin 8901 df-fsupp 9279 df-fi 9328 df-sup 9359 df-inf 9360 df-oi 9429 df-card 9865 df-pnf 11182 df-mnf 11183 df-xr 11184 df-ltxr 11185 df-le 11186 df-sub 11380 df-neg 11381 df-div 11809 df-nn 12160 df-2 12222 df-3 12223 df-4 12224 df-5 12225 df-6 12226 df-7 12227 df-8 12228 df-9 12229 df-n0 12416 df-z 12503 df-dec 12622 df-uz 12766 df-q 12876 df-rp 12920 df-xneg 13040 df-xadd 13041 df-xmul 13042 df-ioo 13279 df-ioc 13280 df-ico 13281 df-icc 13282 df-fz 13438 df-fzo 13585 df-seq 13939 df-exp 13999 df-hash 14268 df-cj 15036 df-re 15037 df-im 15038 df-sqrt 15172 df-abs 15173 df-struct 17088 df-sets 17105 df-slot 17123 df-ndx 17135 df-base 17151 df-ress 17172 df-plusg 17204 df-mulr 17205 df-starv 17206 df-sca 17207 df-vsca 17208 df-ip 17209 df-tset 17210 df-ple 17211 df-ds 17213 df-unif 17214 df-hom 17215 df-cco 17216 df-rest 17356 df-topn 17357 df-0g 17375 df-gsum 17376 df-topgen 17377 df-pt 17378 df-prds 17381 df-xrs 17437 df-qtop 17442 df-imas 17443 df-xps 17445 df-mre 17519 df-mrc 17520 df-acs 17522 df-mgm 18579 df-sgrp 18658 df-mnd 18674 df-submnd 18723 df-mulg 19015 df-cntz 19263 df-cmn 19728 df-psmet 21318 df-xmet 21319 df-met 21320 df-bl 21321 df-mopn 21322 df-fbas 21323 df-fg 21324 df-cnfld 21327 df-top 22855 df-topon 22872 df-topsp 22894 df-bases 22907 df-cld 22980 df-ntr 22981 df-cls 22982 df-nei 23059 df-lp 23097 df-perf 23098 df-cn 23188 df-cnp 23189 df-haus 23276 df-cmp 23348 df-tx 23523 df-hmeo 23716 df-fil 23807 df-fm 23899 df-flim 23900 df-flf 23901 df-xms 24281 df-ms 24282 df-tms 24283 df-cncf 24844 df-limc 25840 df-dv 25841

This theorem is referenced by: lhop2 25993 lhop 25994 fourierdlem61 46554

W3C validator