lhop2 - Metamath Proof Explorer

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Theorem lhop2 23778

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

**Hypotheses**
Ref	Expression
lhop2.a	⊢ (𝜑 → 𝐴 ∈ ℝ^*)
lhop2.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
lhop2.l	⊢ (𝜑 → 𝐴 < 𝐵)
lhop2.f	⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)
lhop2.g	⊢ (𝜑 → 𝐺:(𝐴(,)𝐵)⟶ℝ)
lhop2.if	⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
lhop2.ig	⊢ (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵))
lhop2.f0	⊢ (𝜑 → 0 ∈ (𝐹 lim_ℂ 𝐵))
lhop2.g0	⊢ (𝜑 → 0 ∈ (𝐺 lim_ℂ 𝐵))
lhop2.gn0	⊢ (𝜑 → ¬ 0 ∈ ran 𝐺)
lhop2.gd0	⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐺))
lhop2.c	⊢ (𝜑 → 𝐶 ∈ ((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧))) lim_ℂ 𝐵))

**Assertion**
Ref	Expression
lhop2	⊢ (𝜑 → 𝐶 ∈ ((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧))) lim_ℂ 𝐵))

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

Proof of Theorem lhop2 Dummy variables 𝑥 𝑎 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		qssre 11798	. . 3 ⊢ ℚ ⊆ ℝ
2		lhop2.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
3		lhop2.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ)
4	3	rexrd 10089	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
5		lhop2.l	. . . 4 ⊢ (𝜑 → 𝐴 < 𝐵)
6		qbtwnxr 12031	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ∃𝑎 ∈ ℚ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))
7	2, 4, 5, 6	syl3anc 1326	. . 3 ⊢ (𝜑 → ∃𝑎 ∈ ℚ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))
8		ssrexv 3667	. . 3 ⊢ (ℚ ⊆ ℝ → (∃𝑎 ∈ ℚ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵) → ∃𝑎 ∈ ℝ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵)))
9	1, 7, 8	mpsyl 68	. 2 ⊢ (𝜑 → ∃𝑎 ∈ ℝ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))
10		simpr 477	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝑎(,)𝐵)) → 𝑧 ∈ (𝑎(,)𝐵))
11		simprl 794	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 𝑎 ∈ ℝ)
12	11	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝑎(,)𝐵)) → 𝑎 ∈ ℝ)
13	3	ad2antrr 762	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝑎(,)𝐵)) → 𝐵 ∈ ℝ)
14		elioore 12205	. . . . . . . 8 ⊢ (𝑧 ∈ (𝑎(,)𝐵) → 𝑧 ∈ ℝ)
15	14	adantl 482	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝑎(,)𝐵)) → 𝑧 ∈ ℝ)
16		iooneg 12292	. . . . . . 7 ⊢ ((𝑎 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑧 ∈ (𝑎(,)𝐵) ↔ -𝑧 ∈ (-𝐵(,)-𝑎)))
17	12, 13, 15, 16	syl3anc 1326	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝑎(,)𝐵)) → (𝑧 ∈ (𝑎(,)𝐵) ↔ -𝑧 ∈ (-𝐵(,)-𝑎)))
18	10, 17	mpbid 222	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝑎(,)𝐵)) → -𝑧 ∈ (-𝐵(,)-𝑎))
19	18	adantrr 753	. . . 4 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ (𝑧 ∈ (𝑎(,)𝐵) ∧ -𝑧 ≠ -𝐵)) → -𝑧 ∈ (-𝐵(,)-𝑎))
20		lhop2.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)
21	20	ad2antrr 762	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → 𝐹:(𝐴(,)𝐵)⟶ℝ)
22		elioore 12205	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (-𝐵(,)-𝑎) → 𝑥 ∈ ℝ)
23	22	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → 𝑥 ∈ ℝ)
24	23	recnd 10068	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → 𝑥 ∈ ℂ)
25	24	negnegd 10383	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → --𝑥 = 𝑥)
26		simpr 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → 𝑥 ∈ (-𝐵(,)-𝑎))
27	25, 26	eqeltrd 2701	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → --𝑥 ∈ (-𝐵(,)-𝑎))
28	11	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → 𝑎 ∈ ℝ)
29	3	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → 𝐵 ∈ ℝ)
30	23	renegcld 10457	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → -𝑥 ∈ ℝ)
31		iooneg 12292	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ -𝑥 ∈ ℝ) → (-𝑥 ∈ (𝑎(,)𝐵) ↔ --𝑥 ∈ (-𝐵(,)-𝑎)))
32	28, 29, 30, 31	syl3anc 1326	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → (-𝑥 ∈ (𝑎(,)𝐵) ↔ --𝑥 ∈ (-𝐵(,)-𝑎)))
33	27, 32	mpbird 247	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → -𝑥 ∈ (𝑎(,)𝐵))
34	2	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 𝐴 ∈ ℝ^*)
35		simprrl 804	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 𝐴 < 𝑎)
36	11	rexrd 10089	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 𝑎 ∈ ℝ^*)
37		xrltle 11982	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑎 ∈ ℝ^*) → (𝐴 < 𝑎 → 𝐴 ≤ 𝑎))
38	34, 36, 37	syl2anc 693	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (𝐴 < 𝑎 → 𝐴 ≤ 𝑎))
39	35, 38	mpd 15	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 𝐴 ≤ 𝑎)
40		iooss1 12210	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≤ 𝑎) → (𝑎(,)𝐵) ⊆ (𝐴(,)𝐵))
41	34, 39, 40	syl2anc 693	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (𝑎(,)𝐵) ⊆ (𝐴(,)𝐵))
42	41	sselda 3603	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ -𝑥 ∈ (𝑎(,)𝐵)) → -𝑥 ∈ (𝐴(,)𝐵))
43	33, 42	syldan 487	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → -𝑥 ∈ (𝐴(,)𝐵))
44	21, 43	ffvelrnd 6360	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → (𝐹‘-𝑥) ∈ ℝ)
45	44	recnd 10068	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → (𝐹‘-𝑥) ∈ ℂ)
46		lhop2.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺:(𝐴(,)𝐵)⟶ℝ)
47	46	ad2antrr 762	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → 𝐺:(𝐴(,)𝐵)⟶ℝ)
48	47, 43	ffvelrnd 6360	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → (𝐺‘-𝑥) ∈ ℝ)
49	48	recnd 10068	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → (𝐺‘-𝑥) ∈ ℂ)
50		lhop2.gn0	. . . . . . 7 ⊢ (𝜑 → ¬ 0 ∈ ran 𝐺)
51	50	ad2antrr 762	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → ¬ 0 ∈ ran 𝐺)
52	46	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 𝐺:(𝐴(,)𝐵)⟶ℝ)
53		ax-resscn 9993	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
54		fss 6056	. . . . . . . . . . . 12 ⊢ ((𝐺:(𝐴(,)𝐵)⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐺:(𝐴(,)𝐵)⟶ℂ)
55	52, 53, 54	sylancl 694	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 𝐺:(𝐴(,)𝐵)⟶ℂ)
56	55	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → 𝐺:(𝐴(,)𝐵)⟶ℂ)
57		ffn 6045	. . . . . . . . . 10 ⊢ (𝐺:(𝐴(,)𝐵)⟶ℂ → 𝐺 Fn (𝐴(,)𝐵))
58	56, 57	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → 𝐺 Fn (𝐴(,)𝐵))
59		fnfvelrn 6356	. . . . . . . . 9 ⊢ ((𝐺 Fn (𝐴(,)𝐵) ∧ -𝑥 ∈ (𝐴(,)𝐵)) → (𝐺‘-𝑥) ∈ ran 𝐺)
60	58, 43, 59	syl2anc 693	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → (𝐺‘-𝑥) ∈ ran 𝐺)
61		eleq1 2689	. . . . . . . 8 ⊢ ((𝐺‘-𝑥) = 0 → ((𝐺‘-𝑥) ∈ ran 𝐺 ↔ 0 ∈ ran 𝐺))
62	60, 61	syl5ibcom 235	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → ((𝐺‘-𝑥) = 0 → 0 ∈ ran 𝐺))
63	62	necon3bd 2808	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → (¬ 0 ∈ ran 𝐺 → (𝐺‘-𝑥) ≠ 0))
64	51, 63	mpd 15	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → (𝐺‘-𝑥) ≠ 0)
65	45, 49, 64	divcld 10801	. . . 4 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → ((𝐹‘-𝑥) / (𝐺‘-𝑥)) ∈ ℂ)
66		limcresi 23649	. . . . . 6 ⊢ ((𝑧 ∈ ℝ ↦ -𝑧) lim_ℂ 𝐵) ⊆ (((𝑧 ∈ ℝ ↦ -𝑧) ↾ (𝑎(,)𝐵)) lim_ℂ 𝐵)
67		ioossre 12235	. . . . . . . 8 ⊢ (𝑎(,)𝐵) ⊆ ℝ
68		resmpt 5449	. . . . . . . 8 ⊢ ((𝑎(,)𝐵) ⊆ ℝ → ((𝑧 ∈ ℝ ↦ -𝑧) ↾ (𝑎(,)𝐵)) = (𝑧 ∈ (𝑎(,)𝐵) ↦ -𝑧))
69	67, 68	ax-mp 5	. . . . . . 7 ⊢ ((𝑧 ∈ ℝ ↦ -𝑧) ↾ (𝑎(,)𝐵)) = (𝑧 ∈ (𝑎(,)𝐵) ↦ -𝑧)
70	69	oveq1i 6660	. . . . . 6 ⊢ (((𝑧 ∈ ℝ ↦ -𝑧) ↾ (𝑎(,)𝐵)) lim_ℂ 𝐵) = ((𝑧 ∈ (𝑎(,)𝐵) ↦ -𝑧) lim_ℂ 𝐵)
71	66, 70	sseqtri 3637	. . . . 5 ⊢ ((𝑧 ∈ ℝ ↦ -𝑧) lim_ℂ 𝐵) ⊆ ((𝑧 ∈ (𝑎(,)𝐵) ↦ -𝑧) lim_ℂ 𝐵)
72		eqid 2622	. . . . . . . 8 ⊢ (𝑧 ∈ ℝ ↦ -𝑧) = (𝑧 ∈ ℝ ↦ -𝑧)
73	72	negcncf 22721	. . . . . . 7 ⊢ (ℝ ⊆ ℂ → (𝑧 ∈ ℝ ↦ -𝑧) ∈ (ℝ–cn→ℂ))
74	53, 73	mp1i 13	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (𝑧 ∈ ℝ ↦ -𝑧) ∈ (ℝ–cn→ℂ))
75	3	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 𝐵 ∈ ℝ)
76		negeq 10273	. . . . . 6 ⊢ (𝑧 = 𝐵 → -𝑧 = -𝐵)
77	74, 75, 76	cnmptlimc 23654	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → -𝐵 ∈ ((𝑧 ∈ ℝ ↦ -𝑧) lim_ℂ 𝐵))
78	71, 77	sseldi 3601	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → -𝐵 ∈ ((𝑧 ∈ (𝑎(,)𝐵) ↦ -𝑧) lim_ℂ 𝐵))
79	75	renegcld 10457	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → -𝐵 ∈ ℝ)
80	11	renegcld 10457	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → -𝑎 ∈ ℝ)
81	80	rexrd 10089	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → -𝑎 ∈ ℝ^*)
82		simprrr 805	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 𝑎 < 𝐵)
83	11, 75	ltnegd 10605	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (𝑎 < 𝐵 ↔ -𝐵 < -𝑎))
84	82, 83	mpbid 222	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → -𝐵 < -𝑎)
85		eqid 2622	. . . . . . 7 ⊢ (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥)) = (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥))
86	44, 85	fmptd 6385	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥)):(-𝐵(,)-𝑎)⟶ℝ)
87		eqid 2622	. . . . . . 7 ⊢ (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥)) = (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥))
88	48, 87	fmptd 6385	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥)):(-𝐵(,)-𝑎)⟶ℝ)
89		reelprrecn 10028	. . . . . . . . . . 11 ⊢ ℝ ∈ {ℝ, ℂ}
90	89	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → ℝ ∈ {ℝ, ℂ})
91		neg1cn 11124	. . . . . . . . . . 11 ⊢ -1 ∈ ℂ
92	91	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → -1 ∈ ℂ)
93	20	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 𝐹:(𝐴(,)𝐵)⟶ℝ)
94	93	ffvelrnda 6359	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑦) ∈ ℝ)
95	94	recnd 10068	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑦) ∈ ℂ)
96		fvexd 6203	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑦) ∈ V)
97		1cnd 10056	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → 1 ∈ ℂ)
98		simpr 477	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
99	98	recnd 10068	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
100		1cnd 10056	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ ℝ) → 1 ∈ ℂ)
101	90	dvmptid 23720	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (ℝ D (𝑥 ∈ ℝ ↦ 𝑥)) = (𝑥 ∈ ℝ ↦ 1))
102		ioossre 12235	. . . . . . . . . . . . 13 ⊢ (-𝐵(,)-𝑎) ⊆ ℝ
103	102	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (-𝐵(,)-𝑎) ⊆ ℝ)
104		eqid 2622	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
105	104	tgioo2 22606	. . . . . . . . . . . 12 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
106		iooretop 22569	. . . . . . . . . . . . 13 ⊢ (-𝐵(,)-𝑎) ∈ (topGen‘ran (,))
107	106	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (-𝐵(,)-𝑎) ∈ (topGen‘ran (,)))
108	90, 99, 100, 101, 103, 105, 104, 107	dvmptres 23726	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ 𝑥)) = (𝑥 ∈ (-𝐵(,)-𝑎) ↦ 1))
109	90, 24, 97, 108	dvmptneg 23729	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ -𝑥)) = (𝑥 ∈ (-𝐵(,)-𝑎) ↦ -1))
110	93	feqmptd 6249	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 𝐹 = (𝑦 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑦)))
111	110	oveq2d 6666	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (ℝ D 𝐹) = (ℝ D (𝑦 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑦))))
112		dvf 23671	. . . . . . . . . . . . 13 ⊢ (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ
113		lhop2.if	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
114	113	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
115	114	feq2d 6031	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ ↔ (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ))
116	112, 115	mpbii 223	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ)
117	116	feqmptd 6249	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (ℝ D 𝐹) = (𝑦 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑦)))
118	111, 117	eqtr3d 2658	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (ℝ D (𝑦 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑦))) = (𝑦 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑦)))
119		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑦 = -𝑥 → (𝐹‘𝑦) = (𝐹‘-𝑥))
120		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑦 = -𝑥 → ((ℝ D 𝐹)‘𝑦) = ((ℝ D 𝐹)‘-𝑥))
121	90, 90, 43, 92, 95, 96, 109, 118, 119, 120	dvmptco 23735	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥))) = (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (((ℝ D 𝐹)‘-𝑥) · -1)))
122	116	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ)
123	122, 43	ffvelrnd 6360	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → ((ℝ D 𝐹)‘-𝑥) ∈ ℂ)
124	123, 92	mulcomd 10061	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → (((ℝ D 𝐹)‘-𝑥) · -1) = (-1 · ((ℝ D 𝐹)‘-𝑥)))
125	123	mulm1d 10482	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → (-1 · ((ℝ D 𝐹)‘-𝑥)) = -((ℝ D 𝐹)‘-𝑥))
126	124, 125	eqtrd 2656	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → (((ℝ D 𝐹)‘-𝑥) · -1) = -((ℝ D 𝐹)‘-𝑥))
127	126	mpteq2dva 4744	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (((ℝ D 𝐹)‘-𝑥) · -1)) = (𝑥 ∈ (-𝐵(,)-𝑎) ↦ -((ℝ D 𝐹)‘-𝑥)))
128	121, 127	eqtrd 2656	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥))) = (𝑥 ∈ (-𝐵(,)-𝑎) ↦ -((ℝ D 𝐹)‘-𝑥)))
129	128	dmeqd 5326	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → dom (ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥))) = dom (𝑥 ∈ (-𝐵(,)-𝑎) ↦ -((ℝ D 𝐹)‘-𝑥)))
130		negex 10279	. . . . . . . 8 ⊢ -((ℝ D 𝐹)‘-𝑥) ∈ V
131		eqid 2622	. . . . . . . 8 ⊢ (𝑥 ∈ (-𝐵(,)-𝑎) ↦ -((ℝ D 𝐹)‘-𝑥)) = (𝑥 ∈ (-𝐵(,)-𝑎) ↦ -((ℝ D 𝐹)‘-𝑥))
132	130, 131	dmmpti 6023	. . . . . . 7 ⊢ dom (𝑥 ∈ (-𝐵(,)-𝑎) ↦ -((ℝ D 𝐹)‘-𝑥)) = (-𝐵(,)-𝑎)
133	129, 132	syl6eq 2672	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → dom (ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥))) = (-𝐵(,)-𝑎))
134	52	ffvelrnda 6359	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑦) ∈ ℝ)
135	134	recnd 10068	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑦) ∈ ℂ)
136		fvexd 6203	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐺)‘𝑦) ∈ V)
137	52	feqmptd 6249	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 𝐺 = (𝑦 ∈ (𝐴(,)𝐵) ↦ (𝐺‘𝑦)))
138	137	oveq2d 6666	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (ℝ D 𝐺) = (ℝ D (𝑦 ∈ (𝐴(,)𝐵) ↦ (𝐺‘𝑦))))
139		dvf 23671	. . . . . . . . . . . . 13 ⊢ (ℝ D 𝐺):dom (ℝ D 𝐺)⟶ℂ
140		lhop2.ig	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵))
141	140	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → dom (ℝ D 𝐺) = (𝐴(,)𝐵))
142	141	feq2d 6031	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → ((ℝ D 𝐺):dom (ℝ D 𝐺)⟶ℂ ↔ (ℝ D 𝐺):(𝐴(,)𝐵)⟶ℂ))
143	139, 142	mpbii 223	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (ℝ D 𝐺):(𝐴(,)𝐵)⟶ℂ)
144	143	feqmptd 6249	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (ℝ D 𝐺) = (𝑦 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐺)‘𝑦)))
145	138, 144	eqtr3d 2658	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (ℝ D (𝑦 ∈ (𝐴(,)𝐵) ↦ (𝐺‘𝑦))) = (𝑦 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐺)‘𝑦)))
146		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑦 = -𝑥 → (𝐺‘𝑦) = (𝐺‘-𝑥))
147		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑦 = -𝑥 → ((ℝ D 𝐺)‘𝑦) = ((ℝ D 𝐺)‘-𝑥))
148	90, 90, 43, 92, 135, 136, 109, 145, 146, 147	dvmptco 23735	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥))) = (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (((ℝ D 𝐺)‘-𝑥) · -1)))
149	143	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → (ℝ D 𝐺):(𝐴(,)𝐵)⟶ℂ)
150	149, 43	ffvelrnd 6360	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → ((ℝ D 𝐺)‘-𝑥) ∈ ℂ)
151	150, 92	mulcomd 10061	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → (((ℝ D 𝐺)‘-𝑥) · -1) = (-1 · ((ℝ D 𝐺)‘-𝑥)))
152	150	mulm1d 10482	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → (-1 · ((ℝ D 𝐺)‘-𝑥)) = -((ℝ D 𝐺)‘-𝑥))
153	151, 152	eqtrd 2656	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → (((ℝ D 𝐺)‘-𝑥) · -1) = -((ℝ D 𝐺)‘-𝑥))
154	153	mpteq2dva 4744	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (((ℝ D 𝐺)‘-𝑥) · -1)) = (𝑥 ∈ (-𝐵(,)-𝑎) ↦ -((ℝ D 𝐺)‘-𝑥)))
155	148, 154	eqtrd 2656	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥))) = (𝑥 ∈ (-𝐵(,)-𝑎) ↦ -((ℝ D 𝐺)‘-𝑥)))
156	155	dmeqd 5326	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → dom (ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥))) = dom (𝑥 ∈ (-𝐵(,)-𝑎) ↦ -((ℝ D 𝐺)‘-𝑥)))
157		negex 10279	. . . . . . . 8 ⊢ -((ℝ D 𝐺)‘-𝑥) ∈ V
158		eqid 2622	. . . . . . . 8 ⊢ (𝑥 ∈ (-𝐵(,)-𝑎) ↦ -((ℝ D 𝐺)‘-𝑥)) = (𝑥 ∈ (-𝐵(,)-𝑎) ↦ -((ℝ D 𝐺)‘-𝑥))
159	157, 158	dmmpti 6023	. . . . . . 7 ⊢ dom (𝑥 ∈ (-𝐵(,)-𝑎) ↦ -((ℝ D 𝐺)‘-𝑥)) = (-𝐵(,)-𝑎)
160	156, 159	syl6eq 2672	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → dom (ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥))) = (-𝐵(,)-𝑎))
161	43	adantrr 753	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ (𝑥 ∈ (-𝐵(,)-𝑎) ∧ -𝑥 ≠ 𝐵)) → -𝑥 ∈ (𝐴(,)𝐵))
162		limcresi 23649	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ↦ -𝑥) lim_ℂ -𝐵) ⊆ (((𝑥 ∈ ℝ ↦ -𝑥) ↾ (-𝐵(,)-𝑎)) lim_ℂ -𝐵)
163		resmpt 5449	. . . . . . . . . . 11 ⊢ ((-𝐵(,)-𝑎) ⊆ ℝ → ((𝑥 ∈ ℝ ↦ -𝑥) ↾ (-𝐵(,)-𝑎)) = (𝑥 ∈ (-𝐵(,)-𝑎) ↦ -𝑥))
164	102, 163	ax-mp 5	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ↦ -𝑥) ↾ (-𝐵(,)-𝑎)) = (𝑥 ∈ (-𝐵(,)-𝑎) ↦ -𝑥)
165	164	oveq1i 6660	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ↦ -𝑥) ↾ (-𝐵(,)-𝑎)) lim_ℂ -𝐵) = ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ -𝑥) lim_ℂ -𝐵)
166	162, 165	sseqtri 3637	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ↦ -𝑥) lim_ℂ -𝐵) ⊆ ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ -𝑥) lim_ℂ -𝐵)
167	75	recnd 10068	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 𝐵 ∈ ℂ)
168	167	negnegd 10383	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → --𝐵 = 𝐵)
169		eqid 2622	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ ↦ -𝑥) = (𝑥 ∈ ℝ ↦ -𝑥)
170	169	negcncf 22721	. . . . . . . . . . 11 ⊢ (ℝ ⊆ ℂ → (𝑥 ∈ ℝ ↦ -𝑥) ∈ (ℝ–cn→ℂ))
171	53, 170	mp1i 13	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (𝑥 ∈ ℝ ↦ -𝑥) ∈ (ℝ–cn→ℂ))
172		negeq 10273	. . . . . . . . . 10 ⊢ (𝑥 = -𝐵 → -𝑥 = --𝐵)
173	171, 79, 172	cnmptlimc 23654	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → --𝐵 ∈ ((𝑥 ∈ ℝ ↦ -𝑥) lim_ℂ -𝐵))
174	168, 173	eqeltrrd 2702	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 𝐵 ∈ ((𝑥 ∈ ℝ ↦ -𝑥) lim_ℂ -𝐵))
175	166, 174	sseldi 3601	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 𝐵 ∈ ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ -𝑥) lim_ℂ -𝐵))
176		lhop2.f0	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ (𝐹 lim_ℂ 𝐵))
177	176	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 0 ∈ (𝐹 lim_ℂ 𝐵))
178	110	oveq1d 6665	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (𝐹 lim_ℂ 𝐵) = ((𝑦 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑦)) lim_ℂ 𝐵))
179	177, 178	eleqtrd 2703	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 0 ∈ ((𝑦 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑦)) lim_ℂ 𝐵))
180		eliooord 12233	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (-𝐵(,)-𝑎) → (-𝐵 < 𝑥 ∧ 𝑥 < -𝑎))
181	180	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → (-𝐵 < 𝑥 ∧ 𝑥 < -𝑎))
182	181	simpld 475	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → -𝐵 < 𝑥)
183	29, 23, 182	ltnegcon1d 10607	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → -𝑥 < 𝐵)
184	30, 183	ltned 10173	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → -𝑥 ≠ 𝐵)
185	184	neneqd 2799	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → ¬ -𝑥 = 𝐵)
186	185	pm2.21d 118	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → (-𝑥 = 𝐵 → (𝐹‘-𝑥) = 0))
187	186	impr 649	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ (𝑥 ∈ (-𝐵(,)-𝑎) ∧ -𝑥 = 𝐵)) → (𝐹‘-𝑥) = 0)
188	161, 95, 175, 179, 119, 187	limcco 23657	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 0 ∈ ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥)) lim_ℂ -𝐵))
189		lhop2.g0	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ (𝐺 lim_ℂ 𝐵))
190	189	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 0 ∈ (𝐺 lim_ℂ 𝐵))
191	137	oveq1d 6665	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (𝐺 lim_ℂ 𝐵) = ((𝑦 ∈ (𝐴(,)𝐵) ↦ (𝐺‘𝑦)) lim_ℂ 𝐵))
192	190, 191	eleqtrd 2703	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 0 ∈ ((𝑦 ∈ (𝐴(,)𝐵) ↦ (𝐺‘𝑦)) lim_ℂ 𝐵))
193	185	pm2.21d 118	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → (-𝑥 = 𝐵 → (𝐺‘-𝑥) = 0))
194	193	impr 649	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ (𝑥 ∈ (-𝐵(,)-𝑎) ∧ -𝑥 = 𝐵)) → (𝐺‘-𝑥) = 0)
195	161, 135, 175, 192, 146, 194	limcco 23657	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 0 ∈ ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥)) lim_ℂ -𝐵))
196	60, 87	fmptd 6385	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥)):(-𝐵(,)-𝑎)⟶ran 𝐺)
197		frn 6053	. . . . . . . 8 ⊢ ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥)):(-𝐵(,)-𝑎)⟶ran 𝐺 → ran (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥)) ⊆ ran 𝐺)
198	196, 197	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → ran (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥)) ⊆ ran 𝐺)
199	50	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → ¬ 0 ∈ ran 𝐺)
200	198, 199	ssneldd 3606	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → ¬ 0 ∈ ran (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥)))
201		lhop2.gd0	. . . . . . . 8 ⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐺))
202	201	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → ¬ 0 ∈ ran (ℝ D 𝐺))
203	155	rneqd 5353	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → ran (ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥))) = ran (𝑥 ∈ (-𝐵(,)-𝑎) ↦ -((ℝ D 𝐺)‘-𝑥)))
204	203	eleq2d 2687	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (0 ∈ ran (ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥))) ↔ 0 ∈ ran (𝑥 ∈ (-𝐵(,)-𝑎) ↦ -((ℝ D 𝐺)‘-𝑥))))
205	158, 157	elrnmpti 5376	. . . . . . . . 9 ⊢ (0 ∈ ran (𝑥 ∈ (-𝐵(,)-𝑎) ↦ -((ℝ D 𝐺)‘-𝑥)) ↔ ∃𝑥 ∈ (-𝐵(,)-𝑎)0 = -((ℝ D 𝐺)‘-𝑥))
206		eqcom 2629	. . . . . . . . . . 11 ⊢ (0 = -((ℝ D 𝐺)‘-𝑥) ↔ -((ℝ D 𝐺)‘-𝑥) = 0)
207	150	negeq0d 10384	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → (((ℝ D 𝐺)‘-𝑥) = 0 ↔ -((ℝ D 𝐺)‘-𝑥) = 0))
208		ffn 6045	. . . . . . . . . . . . . . 15 ⊢ ((ℝ D 𝐺):(𝐴(,)𝐵)⟶ℂ → (ℝ D 𝐺) Fn (𝐴(,)𝐵))
209	149, 208	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → (ℝ D 𝐺) Fn (𝐴(,)𝐵))
210		fnfvelrn 6356	. . . . . . . . . . . . . 14 ⊢ (((ℝ D 𝐺) Fn (𝐴(,)𝐵) ∧ -𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐺)‘-𝑥) ∈ ran (ℝ D 𝐺))
211	209, 43, 210	syl2anc 693	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → ((ℝ D 𝐺)‘-𝑥) ∈ ran (ℝ D 𝐺))
212		eleq1 2689	. . . . . . . . . . . . 13 ⊢ (((ℝ D 𝐺)‘-𝑥) = 0 → (((ℝ D 𝐺)‘-𝑥) ∈ ran (ℝ D 𝐺) ↔ 0 ∈ ran (ℝ D 𝐺)))
213	211, 212	syl5ibcom 235	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → (((ℝ D 𝐺)‘-𝑥) = 0 → 0 ∈ ran (ℝ D 𝐺)))
214	207, 213	sylbird 250	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → (-((ℝ D 𝐺)‘-𝑥) = 0 → 0 ∈ ran (ℝ D 𝐺)))
215	206, 214	syl5bi 232	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → (0 = -((ℝ D 𝐺)‘-𝑥) → 0 ∈ ran (ℝ D 𝐺)))
216	215	rexlimdva 3031	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (∃𝑥 ∈ (-𝐵(,)-𝑎)0 = -((ℝ D 𝐺)‘-𝑥) → 0 ∈ ran (ℝ D 𝐺)))
217	205, 216	syl5bi 232	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (0 ∈ ran (𝑥 ∈ (-𝐵(,)-𝑎) ↦ -((ℝ D 𝐺)‘-𝑥)) → 0 ∈ ran (ℝ D 𝐺)))
218	204, 217	sylbid 230	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (0 ∈ ran (ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥))) → 0 ∈ ran (ℝ D 𝐺)))
219	202, 218	mtod 189	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → ¬ 0 ∈ ran (ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥))))
220	116	ffvelrnda 6359	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑧) ∈ ℂ)
221	143	ffvelrnda 6359	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐺)‘𝑧) ∈ ℂ)
222	201	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ¬ 0 ∈ ran (ℝ D 𝐺))
223	143, 208	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (ℝ D 𝐺) Fn (𝐴(,)𝐵))
224		fnfvelrn 6356	. . . . . . . . . . . . 13 ⊢ (((ℝ D 𝐺) Fn (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐺)‘𝑧) ∈ ran (ℝ D 𝐺))
225	223, 224	sylan 488	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐺)‘𝑧) ∈ ran (ℝ D 𝐺))
226		eleq1 2689	. . . . . . . . . . . 12 ⊢ (((ℝ D 𝐺)‘𝑧) = 0 → (((ℝ D 𝐺)‘𝑧) ∈ ran (ℝ D 𝐺) ↔ 0 ∈ ran (ℝ D 𝐺)))
227	225, 226	syl5ibcom 235	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐺)‘𝑧) = 0 → 0 ∈ ran (ℝ D 𝐺)))
228	227	necon3bd 2808	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (¬ 0 ∈ ran (ℝ D 𝐺) → ((ℝ D 𝐺)‘𝑧) ≠ 0))
229	222, 228	mpd 15	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐺)‘𝑧) ≠ 0)
230	220, 221, 229	divcld 10801	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)) ∈ ℂ)
231		lhop2.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧))) lim_ℂ 𝐵))
232	231	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 𝐶 ∈ ((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧))) lim_ℂ 𝐵))
233		fveq2 6191	. . . . . . . . 9 ⊢ (𝑧 = -𝑥 → ((ℝ D 𝐹)‘𝑧) = ((ℝ D 𝐹)‘-𝑥))
234		fveq2 6191	. . . . . . . . 9 ⊢ (𝑧 = -𝑥 → ((ℝ D 𝐺)‘𝑧) = ((ℝ D 𝐺)‘-𝑥))
235	233, 234	oveq12d 6668	. . . . . . . 8 ⊢ (𝑧 = -𝑥 → (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧)) = (((ℝ D 𝐹)‘-𝑥) / ((ℝ D 𝐺)‘-𝑥)))
236	185	pm2.21d 118	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → (-𝑥 = 𝐵 → (((ℝ D 𝐹)‘-𝑥) / ((ℝ D 𝐺)‘-𝑥)) = 𝐶))
237	236	impr 649	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ (𝑥 ∈ (-𝐵(,)-𝑎) ∧ -𝑥 = 𝐵)) → (((ℝ D 𝐹)‘-𝑥) / ((ℝ D 𝐺)‘-𝑥)) = 𝐶)
238	161, 230, 175, 232, 235, 237	limcco 23657	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 𝐶 ∈ ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (((ℝ D 𝐹)‘-𝑥) / ((ℝ D 𝐺)‘-𝑥))) lim_ℂ -𝐵))
239		nfcv 2764	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥ℝ
240		nfcv 2764	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 D
241		nfmpt1 4747	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥))
242	239, 240, 241	nfov 6676	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥)))
243		nfcv 2764	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑦
244	242, 243	nffv 6198	. . . . . . . . . . 11 ⊢ Ⅎ𝑥((ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥)))‘𝑦)
245		nfcv 2764	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 /
246		nfmpt1 4747	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥))
247	239, 240, 246	nfov 6676	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥)))
248	247, 243	nffv 6198	. . . . . . . . . . 11 ⊢ Ⅎ𝑥((ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥)))‘𝑦)
249	244, 245, 248	nfov 6676	. . . . . . . . . 10 ⊢ Ⅎ𝑥(((ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥)))‘𝑦) / ((ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥)))‘𝑦))
250		nfcv 2764	. . . . . . . . . 10 ⊢ Ⅎ𝑦(((ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥)))‘𝑥) / ((ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥)))‘𝑥))
251		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → ((ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥)))‘𝑦) = ((ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥)))‘𝑥))
252		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → ((ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥)))‘𝑦) = ((ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥)))‘𝑥))
253	251, 252	oveq12d 6668	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (((ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥)))‘𝑦) / ((ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥)))‘𝑦)) = (((ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥)))‘𝑥) / ((ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥)))‘𝑥)))
254	249, 250, 253	cbvmpt 4749	. . . . . . . . 9 ⊢ (𝑦 ∈ (-𝐵(,)-𝑎) ↦ (((ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥)))‘𝑦) / ((ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥)))‘𝑦))) = (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (((ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥)))‘𝑥) / ((ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥)))‘𝑥)))
255	128	fveq1d 6193	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → ((ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥)))‘𝑥) = ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ -((ℝ D 𝐹)‘-𝑥))‘𝑥))
256	131	fvmpt2 6291	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (-𝐵(,)-𝑎) ∧ -((ℝ D 𝐹)‘-𝑥) ∈ V) → ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ -((ℝ D 𝐹)‘-𝑥))‘𝑥) = -((ℝ D 𝐹)‘-𝑥))
257	130, 256	mpan2 707	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (-𝐵(,)-𝑎) → ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ -((ℝ D 𝐹)‘-𝑥))‘𝑥) = -((ℝ D 𝐹)‘-𝑥))
258	255, 257	sylan9eq 2676	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → ((ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥)))‘𝑥) = -((ℝ D 𝐹)‘-𝑥))
259	155	fveq1d 6193	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → ((ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥)))‘𝑥) = ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ -((ℝ D 𝐺)‘-𝑥))‘𝑥))
260	158	fvmpt2 6291	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (-𝐵(,)-𝑎) ∧ -((ℝ D 𝐺)‘-𝑥) ∈ V) → ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ -((ℝ D 𝐺)‘-𝑥))‘𝑥) = -((ℝ D 𝐺)‘-𝑥))
261	157, 260	mpan2 707	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (-𝐵(,)-𝑎) → ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ -((ℝ D 𝐺)‘-𝑥))‘𝑥) = -((ℝ D 𝐺)‘-𝑥))
262	259, 261	sylan9eq 2676	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → ((ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥)))‘𝑥) = -((ℝ D 𝐺)‘-𝑥))
263	258, 262	oveq12d 6668	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → (((ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥)))‘𝑥) / ((ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥)))‘𝑥)) = (-((ℝ D 𝐹)‘-𝑥) / -((ℝ D 𝐺)‘-𝑥)))
264	201	ad2antrr 762	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → ¬ 0 ∈ ran (ℝ D 𝐺))
265	213	necon3bd 2808	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → (¬ 0 ∈ ran (ℝ D 𝐺) → ((ℝ D 𝐺)‘-𝑥) ≠ 0))
266	264, 265	mpd 15	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → ((ℝ D 𝐺)‘-𝑥) ≠ 0)
267	123, 150, 266	div2negd 10816	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → (-((ℝ D 𝐹)‘-𝑥) / -((ℝ D 𝐺)‘-𝑥)) = (((ℝ D 𝐹)‘-𝑥) / ((ℝ D 𝐺)‘-𝑥)))
268	263, 267	eqtrd 2656	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → (((ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥)))‘𝑥) / ((ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥)))‘𝑥)) = (((ℝ D 𝐹)‘-𝑥) / ((ℝ D 𝐺)‘-𝑥)))
269	268	mpteq2dva 4744	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (((ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥)))‘𝑥) / ((ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥)))‘𝑥))) = (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (((ℝ D 𝐹)‘-𝑥) / ((ℝ D 𝐺)‘-𝑥))))
270	254, 269	syl5eq 2668	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (𝑦 ∈ (-𝐵(,)-𝑎) ↦ (((ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥)))‘𝑦) / ((ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥)))‘𝑦))) = (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (((ℝ D 𝐹)‘-𝑥) / ((ℝ D 𝐺)‘-𝑥))))
271	270	oveq1d 6665	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → ((𝑦 ∈ (-𝐵(,)-𝑎) ↦ (((ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥)))‘𝑦) / ((ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥)))‘𝑦))) lim_ℂ -𝐵) = ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (((ℝ D 𝐹)‘-𝑥) / ((ℝ D 𝐺)‘-𝑥))) lim_ℂ -𝐵))
272	238, 271	eleqtrrd 2704	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 𝐶 ∈ ((𝑦 ∈ (-𝐵(,)-𝑎) ↦ (((ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥)))‘𝑦) / ((ℝ D (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥)))‘𝑦))) lim_ℂ -𝐵))
273	79, 81, 84, 86, 88, 133, 160, 188, 195, 200, 219, 272	lhop1 23777	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 𝐶 ∈ ((𝑦 ∈ (-𝐵(,)-𝑎) ↦ (((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥))‘𝑦) / ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥))‘𝑦))) lim_ℂ -𝐵))
274		nffvmpt1 6199	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥))‘𝑦)
275		nffvmpt1 6199	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥))‘𝑦)
276	274, 245, 275	nfov 6676	. . . . . . . 8 ⊢ Ⅎ𝑥(((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥))‘𝑦) / ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥))‘𝑦))
277		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑦(((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥))‘𝑥) / ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥))‘𝑥))
278		fveq2 6191	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥))‘𝑦) = ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥))‘𝑥))
279		fveq2 6191	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥))‘𝑦) = ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥))‘𝑥))
280	278, 279	oveq12d 6668	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥))‘𝑦) / ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥))‘𝑦)) = (((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥))‘𝑥) / ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥))‘𝑥)))
281	276, 277, 280	cbvmpt 4749	. . . . . . 7 ⊢ (𝑦 ∈ (-𝐵(,)-𝑎) ↦ (((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥))‘𝑦) / ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥))‘𝑦))) = (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥))‘𝑥) / ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥))‘𝑥)))
282		fvex 6201	. . . . . . . . . 10 ⊢ (𝐹‘-𝑥) ∈ V
283	85	fvmpt2 6291	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (-𝐵(,)-𝑎) ∧ (𝐹‘-𝑥) ∈ V) → ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥))‘𝑥) = (𝐹‘-𝑥))
284	26, 282, 283	sylancl 694	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥))‘𝑥) = (𝐹‘-𝑥))
285		fvex 6201	. . . . . . . . . 10 ⊢ (𝐺‘-𝑥) ∈ V
286	87	fvmpt2 6291	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (-𝐵(,)-𝑎) ∧ (𝐺‘-𝑥) ∈ V) → ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥))‘𝑥) = (𝐺‘-𝑥))
287	26, 285, 286	sylancl 694	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥))‘𝑥) = (𝐺‘-𝑥))
288	284, 287	oveq12d 6668	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑥 ∈ (-𝐵(,)-𝑎)) → (((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥))‘𝑥) / ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥))‘𝑥)) = ((𝐹‘-𝑥) / (𝐺‘-𝑥)))
289	288	mpteq2dva 4744	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (𝑥 ∈ (-𝐵(,)-𝑎) ↦ (((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥))‘𝑥) / ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥))‘𝑥))) = (𝑥 ∈ (-𝐵(,)-𝑎) ↦ ((𝐹‘-𝑥) / (𝐺‘-𝑥))))
290	281, 289	syl5eq 2668	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (𝑦 ∈ (-𝐵(,)-𝑎) ↦ (((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥))‘𝑦) / ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥))‘𝑦))) = (𝑥 ∈ (-𝐵(,)-𝑎) ↦ ((𝐹‘-𝑥) / (𝐺‘-𝑥))))
291	290	oveq1d 6665	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → ((𝑦 ∈ (-𝐵(,)-𝑎) ↦ (((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐹‘-𝑥))‘𝑦) / ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ (𝐺‘-𝑥))‘𝑦))) lim_ℂ -𝐵) = ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ ((𝐹‘-𝑥) / (𝐺‘-𝑥))) lim_ℂ -𝐵))
292	273, 291	eleqtrd 2703	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 𝐶 ∈ ((𝑥 ∈ (-𝐵(,)-𝑎) ↦ ((𝐹‘-𝑥) / (𝐺‘-𝑥))) lim_ℂ -𝐵))
293		negeq 10273	. . . . . 6 ⊢ (𝑥 = -𝑧 → -𝑥 = --𝑧)
294	293	fveq2d 6195	. . . . 5 ⊢ (𝑥 = -𝑧 → (𝐹‘-𝑥) = (𝐹‘--𝑧))
295	293	fveq2d 6195	. . . . 5 ⊢ (𝑥 = -𝑧 → (𝐺‘-𝑥) = (𝐺‘--𝑧))
296	294, 295	oveq12d 6668	. . . 4 ⊢ (𝑥 = -𝑧 → ((𝐹‘-𝑥) / (𝐺‘-𝑥)) = ((𝐹‘--𝑧) / (𝐺‘--𝑧)))
297	79	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝑎(,)𝐵)) → -𝐵 ∈ ℝ)
298		eliooord 12233	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝑎(,)𝐵) → (𝑎 < 𝑧 ∧ 𝑧 < 𝐵))
299	298	adantl 482	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝑎(,)𝐵)) → (𝑎 < 𝑧 ∧ 𝑧 < 𝐵))
300	299	simprd 479	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝑎(,)𝐵)) → 𝑧 < 𝐵)
301	15, 13	ltnegd 10605	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝑎(,)𝐵)) → (𝑧 < 𝐵 ↔ -𝐵 < -𝑧))
302	300, 301	mpbid 222	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝑎(,)𝐵)) → -𝐵 < -𝑧)
303	297, 302	gtned 10172	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝑎(,)𝐵)) → -𝑧 ≠ -𝐵)
304	303	neneqd 2799	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝑎(,)𝐵)) → ¬ -𝑧 = -𝐵)
305	304	pm2.21d 118	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝑎(,)𝐵)) → (-𝑧 = -𝐵 → ((𝐹‘--𝑧) / (𝐺‘--𝑧)) = 𝐶))
306	305	impr 649	. . . 4 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ (𝑧 ∈ (𝑎(,)𝐵) ∧ -𝑧 = -𝐵)) → ((𝐹‘--𝑧) / (𝐺‘--𝑧)) = 𝐶)
307	19, 65, 78, 292, 296, 306	limcco 23657	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 𝐶 ∈ ((𝑧 ∈ (𝑎(,)𝐵) ↦ ((𝐹‘--𝑧) / (𝐺‘--𝑧))) lim_ℂ 𝐵))
308	15	recnd 10068	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝑎(,)𝐵)) → 𝑧 ∈ ℂ)
309	308	negnegd 10383	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝑎(,)𝐵)) → --𝑧 = 𝑧)
310	309	fveq2d 6195	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝑎(,)𝐵)) → (𝐹‘--𝑧) = (𝐹‘𝑧))
311	309	fveq2d 6195	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝑎(,)𝐵)) → (𝐺‘--𝑧) = (𝐺‘𝑧))
312	310, 311	oveq12d 6668	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝑎(,)𝐵)) → ((𝐹‘--𝑧) / (𝐺‘--𝑧)) = ((𝐹‘𝑧) / (𝐺‘𝑧)))
313	312	mpteq2dva 4744	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (𝑧 ∈ (𝑎(,)𝐵) ↦ ((𝐹‘--𝑧) / (𝐺‘--𝑧))) = (𝑧 ∈ (𝑎(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧))))
314	313	oveq1d 6665	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → ((𝑧 ∈ (𝑎(,)𝐵) ↦ ((𝐹‘--𝑧) / (𝐺‘--𝑧))) lim_ℂ 𝐵) = ((𝑧 ∈ (𝑎(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧))) lim_ℂ 𝐵))
315	41	resmptd 5452	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → ((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧))) ↾ (𝑎(,)𝐵)) = (𝑧 ∈ (𝑎(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧))))
316	315	oveq1d 6665	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧))) ↾ (𝑎(,)𝐵)) lim_ℂ 𝐵) = ((𝑧 ∈ (𝑎(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧))) lim_ℂ 𝐵))
317		fss 6056	. . . . . . . . 9 ⊢ ((𝐹:(𝐴(,)𝐵)⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:(𝐴(,)𝐵)⟶ℂ)
318	93, 53, 317	sylancl 694	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 𝐹:(𝐴(,)𝐵)⟶ℂ)
319	318	ffvelrnda 6359	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑧) ∈ ℂ)
320	55	ffvelrnda 6359	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑧) ∈ ℂ)
321	50	ad2antrr 762	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ¬ 0 ∈ ran 𝐺)
322	55, 57	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 𝐺 Fn (𝐴(,)𝐵))
323		fnfvelrn 6356	. . . . . . . . . . 11 ⊢ ((𝐺 Fn (𝐴(,)𝐵) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑧) ∈ ran 𝐺)
324	322, 323	sylan 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑧) ∈ ran 𝐺)
325		eleq1 2689	. . . . . . . . . 10 ⊢ ((𝐺‘𝑧) = 0 → ((𝐺‘𝑧) ∈ ran 𝐺 ↔ 0 ∈ ran 𝐺))
326	324, 325	syl5ibcom 235	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((𝐺‘𝑧) = 0 → 0 ∈ ran 𝐺))
327	326	necon3bd 2808	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (¬ 0 ∈ ran 𝐺 → (𝐺‘𝑧) ≠ 0))
328	321, 327	mpd 15	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑧) ≠ 0)
329	319, 320, 328	divcld 10801	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((𝐹‘𝑧) / (𝐺‘𝑧)) ∈ ℂ)
330		eqid 2622	. . . . . 6 ⊢ (𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧))) = (𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧)))
331	329, 330	fmptd 6385	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧))):(𝐴(,)𝐵)⟶ℂ)
332		ioossre 12235	. . . . . . 7 ⊢ (𝐴(,)𝐵) ⊆ ℝ
333	332, 53	sstri 3612	. . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ ℂ
334	333	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (𝐴(,)𝐵) ⊆ ℂ)
335		eqid 2622	. . . . 5 ⊢ ((TopOpen‘ℂ_fld) ↾_t ((𝐴(,)𝐵) ∪ {𝐵})) = ((TopOpen‘ℂ_fld) ↾_t ((𝐴(,)𝐵) ∪ {𝐵}))
336		ssun2 3777	. . . . . . 7 ⊢ {𝐵} ⊆ ((𝑎(,)𝐵) ∪ {𝐵})
337		snssg 4327	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → (𝐵 ∈ ((𝑎(,)𝐵) ∪ {𝐵}) ↔ {𝐵} ⊆ ((𝑎(,)𝐵) ∪ {𝐵})))
338	75, 337	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (𝐵 ∈ ((𝑎(,)𝐵) ∪ {𝐵}) ↔ {𝐵} ⊆ ((𝑎(,)𝐵) ∪ {𝐵})))
339	336, 338	mpbiri 248	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 𝐵 ∈ ((𝑎(,)𝐵) ∪ {𝐵}))
340	104	cnfldtopon 22586	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
341	332	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (𝐴(,)𝐵) ⊆ ℝ)
342	75	snssd 4340	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → {𝐵} ⊆ ℝ)
343	341, 342	unssd 3789	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → ((𝐴(,)𝐵) ∪ {𝐵}) ⊆ ℝ)
344	343, 53	syl6ss 3615	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → ((𝐴(,)𝐵) ∪ {𝐵}) ⊆ ℂ)
345		resttopon 20965	. . . . . . . . 9 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ ((𝐴(,)𝐵) ∪ {𝐵}) ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t ((𝐴(,)𝐵) ∪ {𝐵})) ∈ (TopOn‘((𝐴(,)𝐵) ∪ {𝐵})))
346	340, 344, 345	sylancr 695	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → ((TopOpen‘ℂ_fld) ↾_t ((𝐴(,)𝐵) ∪ {𝐵})) ∈ (TopOn‘((𝐴(,)𝐵) ∪ {𝐵})))
347		topontop 20718	. . . . . . . 8 ⊢ (((TopOpen‘ℂ_fld) ↾_t ((𝐴(,)𝐵) ∪ {𝐵})) ∈ (TopOn‘((𝐴(,)𝐵) ∪ {𝐵})) → ((TopOpen‘ℂ_fld) ↾_t ((𝐴(,)𝐵) ∪ {𝐵})) ∈ Top)
348	346, 347	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → ((TopOpen‘ℂ_fld) ↾_t ((𝐴(,)𝐵) ∪ {𝐵})) ∈ Top)
349		indi 3873	. . . . . . . . . 10 ⊢ ((𝑎(,)+∞) ∩ ((𝐴(,)𝐵) ∪ {𝐵})) = (((𝑎(,)+∞) ∩ (𝐴(,)𝐵)) ∪ ((𝑎(,)+∞) ∩ {𝐵}))
350		pnfxr 10092	. . . . . . . . . . . . . 14 ⊢ +∞ ∈ ℝ^*
351	350	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → +∞ ∈ ℝ^*)
352	4	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 𝐵 ∈ ℝ^*)
353		iooin 12209	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℝ^* ∧ +∞ ∈ ℝ^) ∧ (𝐴 ∈ ℝ^ ∧ 𝐵 ∈ ℝ^*)) → ((𝑎(,)+∞) ∩ (𝐴(,)𝐵)) = (if(𝑎 ≤ 𝐴, 𝐴, 𝑎)(,)if(+∞ ≤ 𝐵, +∞, 𝐵)))
354	36, 351, 34, 352, 353	syl22anc 1327	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → ((𝑎(,)+∞) ∩ (𝐴(,)𝐵)) = (if(𝑎 ≤ 𝐴, 𝐴, 𝑎)(,)if(+∞ ≤ 𝐵, +∞, 𝐵)))
355		xrltnle 10105	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑎 ∈ ℝ^*) → (𝐴 < 𝑎 ↔ ¬ 𝑎 ≤ 𝐴))
356	34, 36, 355	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (𝐴 < 𝑎 ↔ ¬ 𝑎 ≤ 𝐴))
357	35, 356	mpbid 222	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → ¬ 𝑎 ≤ 𝐴)
358	357	iffalsed 4097	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → if(𝑎 ≤ 𝐴, 𝐴, 𝑎) = 𝑎)
359		ltpnf 11954	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞)
360	75, 359	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 𝐵 < +∞)
361		xrltnle 10105	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝐵 < +∞ ↔ ¬ +∞ ≤ 𝐵))
362	352, 350, 361	sylancl 694	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (𝐵 < +∞ ↔ ¬ +∞ ≤ 𝐵))
363	360, 362	mpbid 222	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → ¬ +∞ ≤ 𝐵)
364	363	iffalsed 4097	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → if(+∞ ≤ 𝐵, +∞, 𝐵) = 𝐵)
365	358, 364	oveq12d 6668	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (if(𝑎 ≤ 𝐴, 𝐴, 𝑎)(,)if(+∞ ≤ 𝐵, +∞, 𝐵)) = (𝑎(,)𝐵))
366	354, 365	eqtrd 2656	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → ((𝑎(,)+∞) ∩ (𝐴(,)𝐵)) = (𝑎(,)𝐵))
367		elioopnf 12267	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ ℝ^* → (𝐵 ∈ (𝑎(,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝑎 < 𝐵)))
368	36, 367	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (𝐵 ∈ (𝑎(,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝑎 < 𝐵)))
369	75, 82, 368	mpbir2and 957	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 𝐵 ∈ (𝑎(,)+∞))
370	369	snssd 4340	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → {𝐵} ⊆ (𝑎(,)+∞))
371		sseqin2 3817	. . . . . . . . . . . 12 ⊢ ({𝐵} ⊆ (𝑎(,)+∞) ↔ ((𝑎(,)+∞) ∩ {𝐵}) = {𝐵})
372	370, 371	sylib 208	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → ((𝑎(,)+∞) ∩ {𝐵}) = {𝐵})
373	366, 372	uneq12d 3768	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (((𝑎(,)+∞) ∩ (𝐴(,)𝐵)) ∪ ((𝑎(,)+∞) ∩ {𝐵})) = ((𝑎(,)𝐵) ∪ {𝐵}))
374	349, 373	syl5eq 2668	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → ((𝑎(,)+∞) ∩ ((𝐴(,)𝐵) ∪ {𝐵})) = ((𝑎(,)𝐵) ∪ {𝐵}))
375		retop 22565	. . . . . . . . . . 11 ⊢ (topGen‘ran (,)) ∈ Top
376	375	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (topGen‘ran (,)) ∈ Top)
377		reex 10027	. . . . . . . . . . . 12 ⊢ ℝ ∈ V
378	377	ssex 4802	. . . . . . . . . . 11 ⊢ (((𝐴(,)𝐵) ∪ {𝐵}) ⊆ ℝ → ((𝐴(,)𝐵) ∪ {𝐵}) ∈ V)
379	343, 378	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → ((𝐴(,)𝐵) ∪ {𝐵}) ∈ V)
380		iooretop 22569	. . . . . . . . . . 11 ⊢ (𝑎(,)+∞) ∈ (topGen‘ran (,))
381	380	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (𝑎(,)+∞) ∈ (topGen‘ran (,)))
382		elrestr 16089	. . . . . . . . . 10 ⊢ (((topGen‘ran (,)) ∈ Top ∧ ((𝐴(,)𝐵) ∪ {𝐵}) ∈ V ∧ (𝑎(,)+∞) ∈ (topGen‘ran (,))) → ((𝑎(,)+∞) ∩ ((𝐴(,)𝐵) ∪ {𝐵})) ∈ ((topGen‘ran (,)) ↾_t ((𝐴(,)𝐵) ∪ {𝐵})))
383	376, 379, 381, 382	syl3anc 1326	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → ((𝑎(,)+∞) ∩ ((𝐴(,)𝐵) ∪ {𝐵})) ∈ ((topGen‘ran (,)) ↾_t ((𝐴(,)𝐵) ∪ {𝐵})))
384	374, 383	eqeltrrd 2702	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → ((𝑎(,)𝐵) ∪ {𝐵}) ∈ ((topGen‘ran (,)) ↾_t ((𝐴(,)𝐵) ∪ {𝐵})))
385		eqid 2622	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
386	104, 385	rerest 22607	. . . . . . . . 9 ⊢ (((𝐴(,)𝐵) ∪ {𝐵}) ⊆ ℝ → ((TopOpen‘ℂ_fld) ↾_t ((𝐴(,)𝐵) ∪ {𝐵})) = ((topGen‘ran (,)) ↾_t ((𝐴(,)𝐵) ∪ {𝐵})))
387	343, 386	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → ((TopOpen‘ℂ_fld) ↾_t ((𝐴(,)𝐵) ∪ {𝐵})) = ((topGen‘ran (,)) ↾_t ((𝐴(,)𝐵) ∪ {𝐵})))
388	384, 387	eleqtrrd 2704	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → ((𝑎(,)𝐵) ∪ {𝐵}) ∈ ((TopOpen‘ℂ_fld) ↾_t ((𝐴(,)𝐵) ∪ {𝐵})))
389		isopn3i 20886	. . . . . . 7 ⊢ ((((TopOpen‘ℂ_fld) ↾_t ((𝐴(,)𝐵) ∪ {𝐵})) ∈ Top ∧ ((𝑎(,)𝐵) ∪ {𝐵}) ∈ ((TopOpen‘ℂ_fld) ↾_t ((𝐴(,)𝐵) ∪ {𝐵}))) → ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴(,)𝐵) ∪ {𝐵})))‘((𝑎(,)𝐵) ∪ {𝐵})) = ((𝑎(,)𝐵) ∪ {𝐵}))
390	348, 388, 389	syl2anc 693	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴(,)𝐵) ∪ {𝐵})))‘((𝑎(,)𝐵) ∪ {𝐵})) = ((𝑎(,)𝐵) ∪ {𝐵}))
391	339, 390	eleqtrrd 2704	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 𝐵 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t ((𝐴(,)𝐵) ∪ {𝐵})))‘((𝑎(,)𝐵) ∪ {𝐵})))
392	331, 41, 334, 104, 335, 391	limcres 23650	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → (((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧))) ↾ (𝑎(,)𝐵)) lim_ℂ 𝐵) = ((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧))) lim_ℂ 𝐵))
393	314, 316, 392	3eqtr2d 2662	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → ((𝑧 ∈ (𝑎(,)𝐵) ↦ ((𝐹‘--𝑧) / (𝐺‘--𝑧))) lim_ℂ 𝐵) = ((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧))) lim_ℂ 𝐵))
394	307, 393	eleqtrd 2703	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ (𝐴 < 𝑎 ∧ 𝑎 < 𝐵))) → 𝐶 ∈ ((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧))) lim_ℂ 𝐵))
395	9, 394	rexlimddv 3035	1 ⊢ (𝜑 → 𝐶 ∈ ((𝑧 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑧) / (𝐺‘𝑧))) lim_ℂ 𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∃wrex 2913 Vcvv 3200 ∪ cun 3572 ∩ cin 3573 ⊆ wss 3574 ifcif 4086 {csn 4177 {cpr 4179 class class class wbr 4653 ↦ cmpt 4729 dom cdm 5114 ran crn 5115 ↾ cres 5116 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 · cmul 9941 +∞cpnf 10071 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 -cneg 10267 / cdiv 10684 ℚcq 11788 (,)cioo 12175 ↾_t crest 16081 TopOpenctopn 16082 topGenctg 16098 ℂ_fldccnfld 19746 Topctop 20698 TopOnctopon 20715 intcnt 20821 –cn→ccncf 22679 lim_ℂ climc 23626 D cdv 23627

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631

This theorem is referenced by: lhop 23779 fourierdlem60 40383

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