lhop1 - Metamath Proof Explorer

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Theorem lhop1 26073

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 13111	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 / 2) ∈ ℝ⁺)
4		breq2 5170	. . . . . . . . . 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 3229	. . . . . . . 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 3631	. . . . . . 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 3465	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ {𝑣 ∈ (𝐴(,)𝐵) ∣ (abs‘(𝑣 − 𝐴)) < 𝑑} ↔ (𝑣 ∈ (𝐴(,)𝐵) ∧ (abs‘(𝑣 − 𝐴)) < 𝑑))
10		eliooord 13466	. . . . . . . . . . . . . . . . . . . . . 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 13468	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴(,)𝐵) ⊆ ℝ
15		simpr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → 𝑣 ∈ (𝐴(,)𝐵))
16	14, 15	sselid 4006	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → 𝑣 ∈ ℝ)
17		lhop1.a	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐴 ∈ ℝ)
18	17	ad3antrrr 729	. . . . . . . . . . . . . . . . . . . 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 13099	. . . . . . . . . . . . . . . . . . . . 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 11886	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → ((𝑣 − 𝐴) < 𝑑 ↔ 𝑣 < (𝑑 + 𝐴)))
23	16	rexrd 11340	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → 𝑣 ∈ ℝ^*)
24		lhop1.b	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
25	24	ad3antrrr 729	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ^*)
26	17	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → 𝐴 ∈ ℝ)
27	20, 26	readdcld 11319	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → (𝑑 + 𝐴) ∈ ℝ)
28	27	rexrd 11340	. . . . . . . . . . . . . . . . . . . . 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 13244	. . . . . . . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . . . . . . 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 11340	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ^*)
34	25, 29	ifcld 4594	. . . . . . . . . . . . . . . . . . 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 13464	. . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . 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 11438	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → 𝐴 ≤ 𝑣)
40	18, 16, 39	abssubge0d 15480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → (abs‘(𝑣 − 𝐴)) = (𝑣 − 𝐴))
41	40	breq1d 5176	. . . . . . . . . . . . . . . . . 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 4247	. . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → 𝐵 ∈ ℝ^*)
45		xrmin1 13239	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 ∈ ℝ^* ∧ (𝑑 + 𝐴) ∈ ℝ^*) → if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ≤ 𝐵)
46	44, 28, 45	syl2anc 583	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ≤ 𝐵)
47		iooss2 13443	. . . . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴))) ⊆ (𝐴(,)𝐵))
49		sseqin2 4244	. . . . . . . . . . . . . . . . 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 2782	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → {𝑣 ∈ (𝐴(,)𝐵) ∣ (abs‘(𝑣 − 𝐴)) < 𝑑} = (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴))))
52	51	eleq2d 2830	. . . . . . . . . . . . . 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 13438	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ¬ 𝐴 ∈ (𝐴(,)𝐵)
55		eleq1 2832	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ (𝐴(,)𝐵) ↔ 𝐴 ∈ (𝐴(,)𝐵)))
56	54, 55	mtbiri 327	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝐴 → ¬ 𝑦 ∈ (𝐴(,)𝐵))
57	56	necon2ai 2976	. . . . . . . . . . . . . . . . . . . 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 6920	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = 𝑦 → ((ℝ D 𝐹)‘𝑧) = ((ℝ D 𝐹)‘𝑦))
61		fveq2 6920	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = 𝑦 → ((ℝ D 𝐺)‘𝑧) = ((ℝ D 𝐺)‘𝑦))
62	60, 61	oveq12d 7466	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = 𝑦 → (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)) = (((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)))
63		eqid 2740	. . . . . . . . . . . . . . . . . . . . 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 7481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)) ∈ V
65	62, 63, 64	fvmpt3i 7034	. . . . . . . . . . . . . . . . . . . 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 7470	. . . . . . . . . . . . . . . . . . 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 5176	. . . . . . . . . . . . . . . . . 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 3097	. . . . . . . . . . . . . . . 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 7471	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝑦 → (abs‘(𝑣 − 𝐴)) = (abs‘(𝑦 − 𝐴)))
71	70	breq1d 5176	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝑦 → ((abs‘(𝑣 − 𝐴)) < 𝑑 ↔ (abs‘(𝑦 − 𝐴)) < 𝑑))
72	71	ralrab 3715	. . . . . . . . . . . . . . . 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 716	. . . . . . . . . . . . . . . . 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 3334	. . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . . . . . 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 11340	. . . . . . . . . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . . . . . . . . . . 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 13099	. . . . . . . . . . . . . . . . . . . . . 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 11319	. . . . . . . . . . . . . . . . . . . . 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 13487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝑑 + 𝐴) ∈ ℝ) → (𝐴(,](𝑑 + 𝐴)) ⊆ ℝ)
103	98, 101, 102	syl2anc 583	. . . . . . . . . . . . . . . . . . . 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 11319	. . . . . . . . . . . . . . . . . . . . . . 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 11340	. . . . . . . . . . . . . . . . . . . . . 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 4583	. . . . . . . . . . . . . . . . . . . . 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 13133	. . . . . . . . . . . . . . . . . . . . . 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 11340	. . . . . . . . . . . . . . . . . . . . . . 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 13244	. . . . . . . . . . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . . . . . . . . . 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 712	. . . . . . . . . . . . . . . . . . . . 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 13240	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐵 ∈ ℝ^* ∧ (𝑑 + 𝐴) ∈ ℝ^*) → if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ≤ (𝑑 + 𝐴))
116	77, 111, 115	syl2anc 583	. . . . . . . . . . . . . . . . . . . . 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 13449	. . . . . . . . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . . . . . . . . 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 1342	. . . . . . . . . . . . . . . . . . . 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 4009	. . . . . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . . . . . . 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 779	. . . . . . . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . . . . . 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 26072	. . . . . . . . . . . . . . . . . 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 13101	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
127		2cnd 12371	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 2 ∈ ℂ)
128		2ne0 12397	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ≠ 0
129	128	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 2 ≠ 0)
130	126, 127, 129	divcan2d 12072	. . . . . . . . . . . . . . . . . . 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 5192	. . . . . . . . . . . . . . . . 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 6920	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑣 → (𝐹‘𝑧) = (𝐹‘𝑣))
140		fveq2 6920	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑣 → (𝐺‘𝑧) = (𝐺‘𝑣))
141	139, 140	oveq12d 7466	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑣 → ((𝐹‘𝑧) / (𝐺‘𝑧)) = ((𝐹‘𝑣) / (𝐺‘𝑣)))
142		eqid 2740	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧))) = (𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧)))
143		ovex 7481	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑧) / (𝐺‘𝑧)) ∈ V
144	141, 142, 143	fvmpt3i 7034	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ (𝐴(,)𝐵) → ((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧)))‘𝑣) = ((𝐹‘𝑣) / (𝐺‘𝑣)))
145	144	fvoveq1d 7470	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ (𝐴(,)𝐵) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧)))‘𝑣) − 𝐶)) = (abs‘(((𝐹‘𝑣) / (𝐺‘𝑣)) − 𝐶)))
146	145	breq1d 5176	. . . . . . . . . . . . 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 3160	. . . . . . 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 3174	. . . . . 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 3160	. . . 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 611	. . 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 25962	. . . . . . . 8 ⊢ (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ
158	84	feq2d 6733	. . . . . . . 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 7118	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑧) ∈ ℂ)
161		dvf 25962	. . . . . . . 8 ⊢ (ℝ D 𝐺):dom (ℝ D 𝐺)⟶ℂ
162	86	feq2d 6733	. . . . . . . 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 7118	. . . . . 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 6748	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D 𝐺) Fn (𝐴(,)𝐵))
167		fnfvelrn 7114	. . . . . . . . . 10 ⊢ (((ℝ D 𝐺) Fn (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐺)‘𝑧) ∈ ran (ℝ D 𝐺))
168	166, 167	sylan 579	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐺)‘𝑧) ∈ ran (ℝ D 𝐺))
169		eleq1 2832	. . . . . . . . 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 2960	. . . . . . 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 12070	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)) ∈ ℂ)
174	173	fmpttd 7149	. . . 4 ⊢ (𝜑 → (𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧))):(𝐴(,)𝐵)⟶ℂ)
175		ax-resscn 11241	. . . . . 6 ⊢ ℝ ⊆ ℂ
176	14, 175	sstri 4018	. . . . 5 ⊢ (𝐴(,)𝐵) ⊆ ℂ
177	176	a1i 11	. . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ)
178	17	recnd 11318	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ)
179	174, 177, 178	ellimc3 25934	. . 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 7118	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑧) ∈ ℝ)
181	180	recnd 11318	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑧) ∈ ℂ)
182	82	ffvelcdmda 7118	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑧) ∈ ℝ)
183	182	recnd 11318	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑧) ∈ ℂ)
184	92	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ¬ 0 ∈ ran 𝐺)
185	82	ffnd 6748	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 Fn (𝐴(,)𝐵))
186		fnfvelrn 7114	. . . . . . . . . 10 ⊢ ((𝐺 Fn (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑧) ∈ ran 𝐺)
187	185, 186	sylan 579	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑧) ∈ ran 𝐺)
188		eleq1 2832	. . . . . . . . 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 2960	. . . . . . 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 12070	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((𝐹‘𝑧) / (𝐺‘𝑧)) ∈ ℂ)
193	192	fmpttd 7149	. . . 4 ⊢ (𝜑 → (𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧))):(𝐴(,)𝐵)⟶ℂ)
194	193, 177, 178	ellimc3 25934	. . 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 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ∃wrex 3076 {crab 3443 ∩ cin 3975 ⊆ wss 3976 ifcif 4548 class class class wbr 5166 ↦ cmpt 5249 dom cdm 5700 ran crn 5701 Fn wfn 6568 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 ℝcr 11183 0cc0 11184 + caddc 11187 · cmul 11189 ℝ^*cxr 11323 < clt 11324 ≤ cle 11325 − cmin 11520 / cdiv 11947 2c2 12348 ℝ⁺crp 13057 (,)cioo 13407 (,]cioc 13408 abscabs 15283 lim_ℂ climc 25917 D cdv 25918

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-cmp 23416 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-limc 25921 df-dv 25922

This theorem is referenced by: lhop2 26074 lhop 26075 fourierdlem61 46088

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