lhop1 - Metamath Proof Explorer

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Theorem lhop1 24617

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 12431	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 / 2) ∈ ℝ⁺)
4		breq2 5034	. . . . . . . . . 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 344	. . . . . . . . 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 3260	. . . . . . . 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 3566	. . . . . . 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 3331	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ {𝑣 ∈ (𝐴(,)𝐵) ∣ (abs‘(𝑣 − 𝐴)) < 𝑑} ↔ (𝑣 ∈ (𝐴(,)𝐵) ∧ (abs‘(𝑣 − 𝐴)) < 𝑑))
10		eliooord 12784	. . . . . . . . . . . . . . . . . . . . . 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 536	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → (𝑣 < (𝑑 + 𝐴) ↔ (𝑣 < 𝐵 ∧ 𝑣 < (𝑑 + 𝐴))))
14		ioossre 12786	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴(,)𝐵) ⊆ ℝ
15		simpr 488	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → 𝑣 ∈ (𝐴(,)𝐵))
16	14, 15	sseldi 3913	. . . . . . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → 𝑑 ∈ ℝ⁺)
20	19	rpred 12419	. . . . . . . . . . . . . . . . . . . . 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 11225	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → ((𝑣 − 𝐴) < 𝑑 ↔ 𝑣 < (𝑑 + 𝐴)))
23	16	rexrd 10680	. . . . . . . . . . . . . . . . . . . 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 10659	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → (𝑑 + 𝐴) ∈ ℝ)
28	27	rexrd 10680	. . . . . . . . . . . . . . . . . . . . 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 12563	. . . . . . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . . . . . 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 315	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → (𝑣 < if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ↔ (𝑣 − 𝐴) < 𝑑))
33	18	rexrd 10680	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ^*)
34	25, 29	ifcld 4470	. . . . . . . . . . . . . . . . . . 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 12782	. . . . . . . . . . . . . . . . . . . 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 543	. . . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . . 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 10777	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → 𝐴 ≤ 𝑣)
40	18, 16, 39	abssubge0d 14783	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → (abs‘(𝑣 − 𝐴)) = (𝑣 − 𝐴))
41	40	breq1d 5040	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝐴(,)𝐵)) → ((abs‘(𝑣 − 𝐴)) < 𝑑 ↔ (𝑣 − 𝐴) < 𝑑))
42	32, 38, 41	3bitr4d 314	. . . . . . . . . . . . . . . . 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 4144	. . . . . . . . . . . . . . . 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 12558	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 ∈ ℝ^* ∧ (𝑑 + 𝐴) ∈ ℝ^*) → if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ≤ 𝐵)
46	44, 28, 45	syl2anc 587	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ≤ 𝐵)
47		iooss2 12762	. . . . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴))) ⊆ (𝐴(,)𝐵))
49		sseqin2 4142	. . . . . . . . . . . . . . . . 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 221	. . . . . . . . . . . . . . . 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 2835	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → {𝑣 ∈ (𝐴(,)𝐵) ∣ (abs‘(𝑣 − 𝐴)) < 𝑑} = (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴))))
52	51	eleq2d 2875	. . . . . . . . . . . . . 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 288	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ ℝ⁺) → ((𝑣 ∈ (𝐴(,)𝐵) ∧ (abs‘(𝑣 − 𝐴)) < 𝑑) ↔ 𝑣 ∈ (𝐴(,)if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)))))
54		lbioo 12757	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ¬ 𝐴 ∈ (𝐴(,)𝐵)
55		eleq1 2877	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ (𝐴(,)𝐵) ↔ 𝐴 ∈ (𝐴(,)𝐵)))
56	54, 55	mtbiri 330	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝐴 → ¬ 𝑦 ∈ (𝐴(,)𝐵))
57	56	necon2ai 3016	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (𝐴(,)𝐵) → 𝑦 ≠ 𝐴)
58	57	biantrurd 536	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (𝐴(,)𝐵) → ((abs‘(𝑦 − 𝐴)) < 𝑑 ↔ (𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑)))
59	58	bicomd 226	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (𝐴(,)𝐵) → ((𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑) ↔ (abs‘(𝑦 − 𝐴)) < 𝑑))
60		fveq2 6645	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = 𝑦 → ((ℝ D 𝐹)‘𝑧) = ((ℝ D 𝐹)‘𝑦))
61		fveq2 6645	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = 𝑦 → ((ℝ D 𝐺)‘𝑧) = ((ℝ D 𝐺)‘𝑦))
62	60, 61	oveq12d 7153	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = 𝑦 → (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)) = (((ℝ D 𝐹)‘𝑦) / ((ℝ D 𝐺)‘𝑦)))
63		eqid 2798	. . . . . . . . . . . . . . . . . . . . 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 7168	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)) ∈ V
65	62, 63, 64	fvmpt3i 6750	. . . . . . . . . . . . . . . . . . . 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 7157	. . . . . . . . . . . . . . . . . . 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 5040	. . . . . . . . . . . . . . . . . 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 348	. . . . . . . . . . . . . . . . 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 3132	. . . . . . . . . . . . . . . 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 7158	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝑦 → (abs‘(𝑣 − 𝐴)) = (abs‘(𝑦 − 𝐴)))
71	70	breq1d 5040	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝑦 → ((abs‘(𝑣 − 𝐴)) < 𝑑 ↔ (abs‘(𝑦 − 𝐴)) < 𝑑))
72	71	ralrab 3633	. . . . . . . . . . . . . . . 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 281	. . . . . . . . . . . . . . 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 3364	. . . . . . . . . . . . . . . 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 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 10680	. . . . . . . . . . . . . . . . . . . . 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 12419	. . . . . . . . . . . . . . . . . . . . . 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 10659	. . . . . . . . . . . . . . . . . . . . 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 12805	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝑑 + 𝐴) ∈ ℝ) → (𝐴(,](𝑑 + 𝐴)) ⊆ ℝ)
103	98, 101, 102	syl2anc 587	. . . . . . . . . . . . . . . . . . . 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 10659	. . . . . . . . . . . . . . . . . . . . . . 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 10680	. . . . . . . . . . . . . . . . . . . . . 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 4459	. . . . . . . . . . . . . . . . . . . . 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 12453	. . . . . . . . . . . . . . . . . . . . . 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 10680	. . . . . . . . . . . . . . . . . . . . . . 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 12563	. . . . . . . . . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . . . . . . . . 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 12559	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐵 ∈ ℝ^* ∧ (𝑑 + 𝐴) ∈ ℝ^*) → if(𝐵 ≤ (𝑑 + 𝐴), 𝐵, (𝑑 + 𝐴)) ≤ (𝑑 + 𝐴))
116	77, 111, 115	syl2anc 587	. . . . . . . . . . . . . . . . . . . . 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 12768	. . . . . . . . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . . . . . . . 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 1339	. . . . . . . . . . . . . . . . . . . 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 3916	. . . . . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . . . . . . 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 24616	. . . . . . . . . . . . . . . . . 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 12421	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
127		2cnd 11703	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 2 ∈ ℂ)
128		2ne0 11729	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ≠ 0
129	128	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 2 ≠ 0)
130	126, 127, 129	divcan2d 11407	. . . . . . . . . . . . . . . . . . 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 5056	. . . . . . . . . . . . . . . . 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 243	. . . . . . . . . . . . . . 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	syl5bi 245	. . . . . . . . . . . . . 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 243	. . . . . . . . . . . 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 6645	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑣 → (𝐹‘𝑧) = (𝐹‘𝑣))
140		fveq2 6645	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑣 → (𝐺‘𝑧) = (𝐺‘𝑣))
141	139, 140	oveq12d 7153	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑣 → ((𝐹‘𝑧) / (𝐺‘𝑧)) = ((𝐹‘𝑣) / (𝐺‘𝑣)))
142		eqid 2798	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧))) = (𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧)))
143		ovex 7168	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑧) / (𝐺‘𝑧)) ∈ V
144	141, 142, 143	fvmpt3i 6750	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ (𝐴(,)𝐵) → ((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧)))‘𝑣) = ((𝐹‘𝑣) / (𝐺‘𝑣)))
145	144	fvoveq1d 7157	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ (𝐴(,)𝐵) → (abs‘(((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧)))‘𝑣) − 𝐶)) = (abs‘(((𝐹‘𝑣) / (𝐺‘𝑣)) − 𝐶)))
146	145	breq1d 5040	. . . . . . . . . . . . 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 344	. . . . . . . . . . . 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 262	. . . . . . . . . 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 3154	. . . . . . 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 3233	. . . . . 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 3154	. . . 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 614	. . 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 24510	. . . . . . . 8 ⊢ (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ
158	84	feq2d 6473	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ ↔ (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ))
159	157, 158	mpbii 236	. . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ)
160	159	ffvelrnda 6828	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑧) ∈ ℂ)
161		dvf 24510	. . . . . . . 8 ⊢ (ℝ D 𝐺):dom (ℝ D 𝐺)⟶ℂ
162	86	feq2d 6473	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D 𝐺):dom (ℝ D 𝐺)⟶ℂ ↔ (ℝ D 𝐺):(𝐴(,)𝐵)⟶ℂ))
163	161, 162	mpbii 236	. . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐺):(𝐴(,)𝐵)⟶ℂ)
164	163	ffvelrnda 6828	. . . . . 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 6488	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D 𝐺) Fn (𝐴(,)𝐵))
167		fnfvelrn 6825	. . . . . . . . . 10 ⊢ (((ℝ D 𝐺) Fn (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐺)‘𝑧) ∈ ran (ℝ D 𝐺))
168	166, 167	sylan 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐺)‘𝑧) ∈ ran (ℝ D 𝐺))
169		eleq1 2877	. . . . . . . . 9 ⊢ (((ℝ D 𝐺)‘𝑧) = 0 → (((ℝ D 𝐺)‘𝑧) ∈ ran (ℝ D 𝐺) ↔ 0 ∈ ran (ℝ D 𝐺)))
170	168, 169	syl5ibcom 248	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐺)‘𝑧) = 0 → 0 ∈ ran (ℝ D 𝐺)))
171	170	necon3bd 3001	. . . . . . 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 11405	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)) ∈ ℂ)
174	173	fmpttd 6856	. . . 4 ⊢ (𝜑 → (𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧))):(𝐴(,)𝐵)⟶ℂ)
175		ax-resscn 10583	. . . . . 6 ⊢ ℝ ⊆ ℂ
176	14, 175	sstri 3924	. . . . 5 ⊢ (𝐴(,)𝐵) ⊆ ℂ
177	176	a1i 11	. . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ)
178	17	recnd 10658	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ)
179	174, 177, 178	ellimc3 24482	. . 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	ffvelrnda 6828	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑧) ∈ ℝ)
181	180	recnd 10658	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑧) ∈ ℂ)
182	82	ffvelrnda 6828	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑧) ∈ ℝ)
183	182	recnd 10658	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑧) ∈ ℂ)
184	92	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ¬ 0 ∈ ran 𝐺)
185	82	ffnd 6488	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 Fn (𝐴(,)𝐵))
186		fnfvelrn 6825	. . . . . . . . . 10 ⊢ ((𝐺 Fn (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑧) ∈ ran 𝐺)
187	185, 186	sylan 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑧) ∈ ran 𝐺)
188		eleq1 2877	. . . . . . . . 9 ⊢ ((𝐺‘𝑧) = 0 → ((𝐺‘𝑧) ∈ ran 𝐺 ↔ 0 ∈ ran 𝐺))
189	187, 188	syl5ibcom 248	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((𝐺‘𝑧) = 0 → 0 ∈ ran 𝐺))
190	189	necon3bd 3001	. . . . . . 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 11405	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((𝐹‘𝑧) / (𝐺‘𝑧)) ∈ ℂ)
193	192	fmpttd 6856	. . . 4 ⊢ (𝜑 → (𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧))):(𝐴(,)𝐵)⟶ℂ)
194	193, 177, 178	ellimc3 24482	. . 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 297	. 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 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 {crab 3110 ∩ cin 3880 ⊆ wss 3881 ifcif 4425 class class class wbr 5030 ↦ cmpt 5110 dom cdm 5519 ran crn 5520 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 + caddc 10529 · cmul 10531 ℝ^*cxr 10663 < clt 10664 ≤ cle 10665 − cmin 10859 / cdiv 11286 2c2 11680 ℝ⁺crp 12377 (,)cioo 12726 (,]cioc 12727 abscabs 14585 lim_ℂ climc 24465 D cdv 24466

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cn 21832 df-cnp 21833 df-haus 21920 df-cmp 21992 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483 df-limc 24469 df-dv 24470

This theorem is referenced by: lhop2 24618 lhop 24619 fourierdlem61 42809

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