lhop1 - Metamath Proof Explorer

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Theorem lhop1 26078

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 488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
3	2	rphalfcld 13051	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 / 2) ∈ ℝ⁺)
4		breq2 5106	. . . . . . . . . 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 342	. . . . . . . . 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 3230	. . . . . . . 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 3579	. . . . . . 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 3437	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ {𝑣 ∈ (𝐴(,)𝐵) ∣ (abs‘(𝑣 − 𝐴)) < 𝑑} ↔ (𝑣 ∈ (𝐴(,)𝐵) ∧ (abs‘(𝑣 − 𝐴)) < 𝑑))
10		eliooord 13411	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 ∈ (𝐴(,)𝐵) → (𝐴 < 𝑣 ∧ 𝑣 < 𝐵))
11	10	adantl 485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → (𝐴 < 𝑣 ∧ 𝑣 < 𝐵))
12	11	simprd 499	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → 𝑣 < 𝐵)
13	12	biantrurd 540	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → (𝑣 < (𝑑 + 𝐴) ↔ (𝑣 < 𝐵 ∧ 𝑣 < (𝑑 + 𝐴))))
14		ioossre 13413	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴(,)𝐵) ⊆ ℝ
15		simpr 488	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → 𝑣 ∈ (𝐴(,)𝐵))
16	14, 15	sselid 3936	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → 𝑣 ∈ ℝ)
17		lhop1.a	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐴 ∈ ℝ)
18	17	ad3antrrr 740	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ)
19		simpr 488	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → 𝑑 ∈ ℝ⁺)
20	19	rpred 13039	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → 𝑑 ∈ ℝ)
21	20	adantr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → 𝑑 ∈ ℝ)
22	16, 18, 21	ltsubaddd 11785	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → ((𝑣 − 𝐴) < 𝑑 ↔ 𝑣 < (𝑑 + 𝐴)))
23	16	rexrd 11234	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → 𝑣 ∈ ℝ^*)
24		lhop1.b	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
25	24	ad3antrrr 740	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ^*)
26	17	ad2antrr 736	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → 𝐴 ∈ ℝ)
27	20, 26	readdcld 11213	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → (𝑑 + 𝐴) ∈ ℝ)
28	27	rexrd 11234	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → (𝑑 + 𝐴) ∈ ℝ^*)
29	28	adantr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → (𝑑 + 𝐴) ∈ ℝ^*)
30		xrltmin 13187	. . . . . . . . . . . . . . . . . . . 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 1392	. . . . . . . . . . . . . . . . . . 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 314	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → (𝑣 < if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ↔ (𝑣 − 𝐴) < 𝑑))
33	18	rexrd 11234	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ^*)
34	25, 29	ifcld 4529	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ∈ ℝ^*)
35	11	simpld 498	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝑣)
36		elioo5 13409	. . . . . . . . . . . . . . . . . . . 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 547	. . . . . . . . . . . . . . . . . . 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 1394	. . . . . . . . . . . . . . . . . 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 11333	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → 𝐴 ≤ 𝑣)
40	18, 16, 39	abssubge0d 15463	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → (abs‘(𝑣 − 𝐴)) = (𝑣 − 𝐴))
41	40	breq1d 5112	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → ((abs‘(𝑣 − 𝐴)) < 𝑑 ↔ (𝑣 − 𝐴) < 𝑑))
42	32, 38, 41	3bitr4d 313	. . . . . . . . . . . . . . . . 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 4179	. . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → 𝐵 ∈ ℝ^*)
45		xrmin1 13182	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 ∈ ℝ^* ∧ (𝑑 + 𝐴) ∈ ℝ^*) → if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ≤ 𝐵)
46	44, 28, 45	syl2anc 593	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ≤ 𝐵)
47		iooss2 13387	. . . . . . . . . . . . . . . . . 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 593	. . . . . . . . . . . . . . . . 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 220	. . . . . . . . . . . . . . . 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 2801	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → {𝑣 ∈ (𝐴(,)𝐵) ∣ (abs‘(𝑣 − 𝐴)) < 𝑑} = (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴))))
52	51	eleq2d 2850	. . . . . . . . . . . . . 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 287	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → ((𝑣 ∈ (𝐴(,)𝐵) ∧ (abs‘(𝑣 − 𝐴)) < 𝑑) ↔ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))))
54		lbioo 13382	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ¬ 𝐴 ∈ (𝐴(,)𝐵)
55		eleq1 2852	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ (𝐴(,)𝐵) ↔ 𝐴 ∈ (𝐴(,)𝐵)))
56	54, 55	mtbiri 329	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝐴 → ¬ 𝑦 ∈ (𝐴(,)𝐵))
57	56	necon2ai 2988	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (𝐴(,)𝐵) → 𝑦 ≠ 𝐴)
58	57	biantrurd 540	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (𝐴(,)𝐵) → ((abs‘(𝑦 − 𝐴)) < 𝑑 ↔ (𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑)))
59	58	bicomd 225	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (𝐴(,)𝐵) → ((𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑) ↔ (abs‘(𝑦 − 𝐴)) < 𝑑))
60		fveq2 6869	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = 𝑦 → ((ℝ D 𝐹)‘𝑧) = ((ℝ D 𝐹)‘𝑦))
61		fveq2 6869	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = 𝑦 → ((ℝ D 𝐺)‘𝑧) = ((ℝ D 𝐺)‘𝑦))
62	60, 61	oveq12d 7416	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = 𝑦 → (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)) = (((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)))
63		eqid 2764	. . . . . . . . . . . . . . . . . . . . 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 7431	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)) ∈ V
65	62, 63, 64	fvmpt3i 6983	. . . . . . . . . . . . . . . . . . . 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 7420	. . . . . . . . . . . . . . . . . . 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 5112	. . . . . . . . . . . . . . . . . 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 346	. . . . . . . . . . . . . . . . 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 3108	. . . . . . . . . . . . . . . 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 7421	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝑦 → (abs‘(𝑣 − 𝐴)) = (abs‘(𝑦 − 𝐴)))
71	70	breq1d 5112	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝑦 → ((abs‘(𝑣 − 𝐴)) < 𝑑 ↔ (abs‘(𝑦 − 𝐴)) < 𝑑))
72	71	ralrab 3659	. . . . . . . . . . . . . . . 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 280	. . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . . 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 3322	. . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . 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 11234	. . . . . . . . . . . . . . . . . . . . 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 788	. . . . . . . . . . . . . . . . . . . . . . 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 13039	. . . . . . . . . . . . . . . . . . . . . 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 11213	. . . . . . . . . . . . . . . . . . . . 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 13433	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝑑 + 𝐴) ∈ ℝ) → (𝐴(,](𝑑 + 𝐴)) ⊆ ℝ)
103	98, 101, 102	syl2anc 593	. . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . . 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 11213	. . . . . . . . . . . . . . . . . . . . . . 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 11234	. . . . . . . . . . . . . . . . . . . . . 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 4518	. . . . . . . . . . . . . . . . . . . . 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 13073	. . . . . . . . . . . . . . . . . . . . . 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 11234	. . . . . . . . . . . . . . . . . . . . . . 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 13187	. . . . . . . . . . . . . . . . . . . . . . 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 1392	. . . . . . . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . . . . . 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 13183	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐵 ∈ ℝ^* ∧ (𝑑 + 𝐴) ∈ ℝ^*) → if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ≤ (𝑑 + 𝐴))
116	77, 111, 115	syl2anc 593	. . . . . . . . . . . . . . . . . . . . 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 13393	. . . . . . . . . . . . . . . . . . . . . 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 593	. . . . . . . . . . . . . . . . . . . . 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 1357	. . . . . . . . . . . . . . . . . . . 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 3939	. . . . . . . . . . . . . . . . . . 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 593	. . . . . . . . . . . . . . . . . . 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 789	. . . . . . . . . . . . . . . . . . 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 782	. . . . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . . . . 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 26077	. . . . . . . . . . . . . . . . . 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 13041	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
127		2cnd 12298	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 2 ∈ ℂ)
128		2ne0 12326	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ≠ 0
129	128	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 2 ≠ 0)
130	126, 127, 129	divcan2d 11971	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (2 · (𝑥 / 2)) = 𝑥)
131	130	adantr 484	. . . . . . . . . . . . . . . . . 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 5128	. . . . . . . . . . . . . . . . 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 460	. . . . . . . . . . . . . . . 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 242	. . . . . . . . . . . . . . 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 244	. . . . . . . . . . . . . 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 460	. . . . . . . . . . . . 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 242	. . . . . . . . . . . 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 456	. . . . . . . . . . 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 6869	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑣 → (𝐹‘𝑧) = (𝐹‘𝑣))
140		fveq2 6869	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑣 → (𝐺‘𝑧) = (𝐺‘𝑣))
141	139, 140	oveq12d 7416	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑣 → ((𝐹‘𝑧) / (𝐺‘𝑧)) = ((𝐹‘𝑣) / (𝐺‘𝑣)))
142		eqid 2764	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧))) = (𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧)))
143		ovex 7431	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑧) / (𝐺‘𝑧)) ∈ V
144	141, 142, 143	fvmpt3i 6983	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ (𝐴(,)𝐵) → ((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧)))‘𝑣) = ((𝐹‘𝑣) / (𝐺‘𝑣)))
145	144	fvoveq1d 7420	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ (𝐴(,)𝐵) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧)))‘𝑣) − 𝐶)) = (abs‘(((𝐹‘𝑣) / (𝐺‘𝑣)) − 𝐶)))
146	145	breq1d 5112	. . . . . . . . . . . . 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 342	. . . . . . . . . . . 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 485	. . . . . . . . . . 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 261	. . . . . . . . . 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 494	. . . . . . . . 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 3164	. . . . . . 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 3177	. . . . . 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 3164	. . . 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 621	. . 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 25971	. . . . . . . 8 ⊢ (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ
158	84	feq2d 6677	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ ↔ (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ))
159	157, 158	mpbii 235	. . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ)
160	159	ffvelcdmda 7067	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑧) ∈ ℂ)
161		dvf 25971	. . . . . . . 8 ⊢ (ℝ D 𝐺):dom (ℝ D 𝐺)⟶ℂ
162	86	feq2d 6677	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D 𝐺):dom (ℝ D 𝐺)⟶ℂ ↔ (ℝ D 𝐺):(𝐴(,)𝐵)⟶ℂ))
163	161, 162	mpbii 235	. . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐺):(𝐴(,)𝐵)⟶ℂ)
164	163	ffvelcdmda 7067	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐺)‘𝑧) ∈ ℂ)
165	94	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ¬ 0 ∈ ran (ℝ D 𝐺))
166	163	ffnd 6694	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D 𝐺) Fn (𝐴(,)𝐵))
167		fnfvelrn 7063	. . . . . . . . . 10 ⊢ (((ℝ D 𝐺) Fn (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐺)‘𝑧) ∈ ran (ℝ D 𝐺))
168	166, 167	sylan 589	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐺)‘𝑧) ∈ ran (ℝ D 𝐺))
169		eleq1 2852	. . . . . . . . 9 ⊢ (((ℝ D 𝐺)‘𝑧) = 0 → (((ℝ D 𝐺)‘𝑧) ∈ ran (ℝ D 𝐺) ↔ 0 ∈ ran (ℝ D 𝐺)))
170	168, 169	syl5ibcom 247	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐺)‘𝑧) = 0 → 0 ∈ ran (ℝ D 𝐺)))
171	170	necon3bd 2973	. . . . . . 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 11969	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)) ∈ ℂ)
174	173	fmpttd 7098	. . . 4 ⊢ (𝜑 → (𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧))):(𝐴(,)𝐵)⟶ℂ)
175		ax-resscn 11132	. . . . . 6 ⊢ ℝ ⊆ ℂ
176	14, 175	sstri 3947	. . . . 5 ⊢ (𝐴(,)𝐵) ⊆ ℂ
177	176	a1i 11	. . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ)
178	17	recnd 11212	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ)
179	174, 177, 178	ellimc3 25943	. . 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 7067	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑧) ∈ ℝ)
181	180	recnd 11212	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑧) ∈ ℂ)
182	82	ffvelcdmda 7067	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑧) ∈ ℝ)
183	182	recnd 11212	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑧) ∈ ℂ)
184	92	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ¬ 0 ∈ ran 𝐺)
185	82	ffnd 6694	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 Fn (𝐴(,)𝐵))
186		fnfvelrn 7063	. . . . . . . . . 10 ⊢ ((𝐺 Fn (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑧) ∈ ran 𝐺)
187	185, 186	sylan 589	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑧) ∈ ran 𝐺)
188		eleq1 2852	. . . . . . . . 9 ⊢ ((𝐺‘𝑧) = 0 → ((𝐺‘𝑧) ∈ ran 𝐺 ↔ 0 ∈ ran 𝐺))
189	187, 188	syl5ibcom 247	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((𝐺‘𝑧) = 0 → 0 ∈ ran 𝐺))
190	189	necon3bd 2973	. . . . . . 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 11969	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((𝐹‘𝑧) / (𝐺‘𝑧)) ∈ ℂ)
193	192	fmpttd 7098	. . . 4 ⊢ (𝜑 → (𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧))):(𝐴(,)𝐵)⟶ℂ)
194	193, 177, 178	ellimc3 25943	. . 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 296	. 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 208 ∧ wa 399 ∧ w3a 1099 = wceq 1562 ∈ wcel 2144 ≠ wne 2959 ∀wral 3078 ∃wrex 3088 {crab 3416 ∩ cin 3905 ⊆ wss 3906 ifcif 4482 class class class wbr 5102 ↦ cmpt 5183 dom cdm 5649 ran crn 5650 Fn wfn 6518 ⟶wf 6519 ‘cfv 6523 (class class class)co 7398 ℂcc 11073 ℝcr 11074 0cc0 11075 + caddc 11078 · cmul 11080 ℝ^*cxr 11217 < clt 11218 ≤ cle 11219 − cmin 11416 / cdiv 11846 2c2 12274 ℝ⁺crp 12995 (,)cioo 13351 (,]cioc 13352 abscabs 15263 lim_ℂ climc 25926 D cdv 25927

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-rep 5229 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5544 df-eprel 5549 df-po 5557 df-so 5558 df-fr 5602 df-se 5603 df-we 5604 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-pred 6290 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-isom 6532 df-riota 7355 df-ov 7401 df-oprab 7402 df-mpo 7403 df-of 7662 df-om 7849 df-1st 7972 df-2nd 7973 df-supp 8143 df-frecs 8264 df-wrecs 8295 df-recs 8344 df-rdg 8383 df-1o 8439 df-2o 8440 df-er 8680 df-map 8812 df-pm 8813 df-ixp 8882 df-en 8930 df-dom 8931 df-sdom 8932 df-fin 8933 df-fsupp 9310 df-fi 9359 df-sup 9390 df-inf 9391 df-oi 9460 df-card 9899 df-pnf 11220 df-mnf 11221 df-xr 11222 df-ltxr 11223 df-le 11224 df-sub 11418 df-neg 11419 df-div 11847 df-nn 12213 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12484 df-z 12571 df-dec 12691 df-uz 12842 df-q 12952 df-rp 12996 df-xneg 13116 df-xadd 13117 df-xmul 13118 df-ioo 13355 df-ioc 13356 df-ico 13357 df-icc 13358 df-fz 13515 df-fzo 13662 df-seq 14017 df-exp 14077 df-hash 14346 df-cj 15128 df-re 15129 df-im 15130 df-sqrt 15264 df-abs 15265 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17248 df-ress 17269 df-plusg 17301 df-mulr 17302 df-starv 17303 df-sca 17304 df-vsca 17305 df-ip 17306 df-tset 17307 df-ple 17308 df-ds 17310 df-unif 17311 df-hom 17312 df-cco 17313 df-rest 17453 df-topn 17454 df-0g 17472 df-gsum 17473 df-topgen 17474 df-pt 17475 df-prds 17478 df-xrs 17534 df-qtop 17539 df-imas 17540 df-xps 17542 df-mre 17616 df-mrc 17617 df-acs 17619 df-mgm 18676 df-sgrp 18755 df-mnd 18771 df-submnd 18820 df-mulg 19112 df-cntz 19359 df-cmn 19824 df-psmet 21418 df-xmet 21419 df-met 21420 df-bl 21421 df-mopn 21422 df-fbas 21423 df-fg 21424 df-cnfld 21427 df-top 22956 df-topon 22973 df-topsp 22995 df-bases 23008 df-cld 23081 df-ntr 23082 df-cls 23083 df-nei 23160 df-lp 23198 df-perf 23199 df-cn 23289 df-cnp 23290 df-haus 23377 df-cmp 23449 df-tx 23624 df-hmeo 23817 df-fil 23908 df-fm 24000 df-flim 24001 df-flf 24002 df-xms 24382 df-ms 24383 df-tms 24384 df-cncf 24942 df-limc 25930 df-dv 25931

This theorem is referenced by: lhop2 26079 lhop 26080 fourierdlem61 46746

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