ftc1cnnc - Mathbox for Brendan Leahy

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Theorem ftc1cnnc 38442

Description: Choice-free proof of ftc1cn 26271. (Contributed by Brendan Leahy, 20-Nov-2017.)

**Hypotheses**
Ref	Expression
ftc1cnnc.g	⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)
ftc1cnnc.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
ftc1cnnc.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
ftc1cnnc.le	⊢ (𝜑 → 𝐴 ≤ 𝐵)
ftc1cnnc.f	⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))
ftc1cnnc.i	⊢ (𝜑 → 𝐹 ∈ 𝐿¹)

**Assertion**
Ref	Expression
ftc1cnnc	⊢ (𝜑 → (ℝ D 𝐺) = 𝐹)

Distinct variable groups: 𝑥,𝑡,𝐴 𝑥,𝐵,𝑡 𝑥,𝐹,𝑡 𝜑,𝑥,𝑡 Allowed substitution hints: 𝐺(𝑥, 𝑡)

Proof of Theorem ftc1cnnc Dummy variables 𝑦 𝑧 𝑠 𝑢 𝑣 𝑤 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvf 26135	. . . . 5 ⊢ (ℝ D 𝐺):dom (ℝ D 𝐺)⟶ℂ
2	1	a1i 11	. . . 4 ⊢ (𝜑 → (ℝ D 𝐺):dom (ℝ D 𝐺)⟶ℂ)
3	2	ffund 6708	. . 3 ⊢ (𝜑 → Fun (ℝ D 𝐺))
4		ax-resscn 11182	. . . . . . 7 ⊢ ℝ ⊆ ℂ
5	4	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℂ)
6		ftc1cnnc.g	. . . . . . 7 ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)
7		ftc1cnnc.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
8		ftc1cnnc.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
9		ftc1cnnc.le	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
10		ssidd 3954	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐵))
11		ioossre 13461	. . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ ℝ
12	11	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
13		ftc1cnnc.i	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝐿¹)
14		ftc1cnnc.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))
15		cncff 25122	. . . . . . . 8 ⊢ (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ) → 𝐹:(𝐴(,)𝐵)⟶ℂ)
16	14, 15	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℂ)
17	6, 7, 8, 9, 10, 12, 13, 16	ftc1lem2 26264	. . . . . 6 ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ)
18		iccssre 13483	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
19	7, 8, 18	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
20		tgioo4 25032	. . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
21		eqid 2760	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
22	5, 17, 19, 20, 21	dvbssntr 26128	. . . . 5 ⊢ (𝜑 → dom (ℝ D 𝐺) ⊆ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)))
23		iccntr 25049	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
24	7, 8, 23	syl2anc 596	. . . . 5 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
25	22, 24	sseqtrd 3967	. . . 4 ⊢ (𝜑 → dom (ℝ D 𝐺) ⊆ (𝐴(,)𝐵))
26		retop 24988	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) ∈ Top
27	20, 26	eqeltrri 2857	. . . . . . . . 9 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℝ) ∈ Top
28	27	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → ((TopOpen‘ℂ_fld) ↾_t ℝ) ∈ Top)
29	19	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝐴[,]𝐵) ⊆ ℝ)
30		iooretop 24992	. . . . . . . . . 10 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
31	30, 20	eleqtri 2858	. . . . . . . . 9 ⊢ (𝐴(,)𝐵) ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ)
32	31	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝐴(,)𝐵) ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ))
33		ioossicc 13487	. . . . . . . . 9 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
34	33	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
35		uniretop 24989	. . . . . . . . . 10 ⊢ ℝ = ∪ (topGen‘ran (,))
36	20	unieqi 4879	. . . . . . . . . 10 ⊢ ∪ (topGen‘ran (,)) = ∪ ((TopOpen‘ℂ_fld) ↾_t ℝ)
37	35, 36	eqtri 2783	. . . . . . . . 9 ⊢ ℝ = ∪ ((TopOpen‘ℂ_fld) ↾_t ℝ)
38	37	ssntr 23284	. . . . . . . 8 ⊢ (((((TopOpen‘ℂ_fld) ↾_t ℝ) ∈ Top ∧ (𝐴[,]𝐵) ⊆ ℝ) ∧ ((𝐴(,)𝐵) ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ) ∧ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))) → (𝐴(,)𝐵) ⊆ ((int‘((TopOpen‘ℂ_fld) ↾_t ℝ))‘(𝐴[,]𝐵)))
39	28, 29, 32, 34, 38	syl22anc 852	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝐴(,)𝐵) ⊆ ((int‘((TopOpen‘ℂ_fld) ↾_t ℝ))‘(𝐴[,]𝐵)))
40		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → 𝑐 ∈ (𝐴(,)𝐵))
41	39, 40	sseldd 3932	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → 𝑐 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t ℝ))‘(𝐴[,]𝐵)))
42	16	ffvelcdmda 7078	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑐) ∈ ℂ)
43		cnxmet 24999	. . . . . . . . . . . 12 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
44	11, 4	sstri 3940	. . . . . . . . . . . 12 ⊢ (𝐴(,)𝐵) ⊆ ℂ
45		xmetres2 24588	. . . . . . . . . . . 12 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ (𝐴(,)𝐵) ⊆ ℂ) → ((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵))) ∈ (∞Met‘(𝐴(,)𝐵)))
46	43, 44, 45	mp2an 705	. . . . . . . . . . 11 ⊢ ((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵))) ∈ (∞Met‘(𝐴(,)𝐵))
47	46	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ 𝑤 ∈ ℝ⁺) → ((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵))) ∈ (∞Met‘(𝐴(,)𝐵)))
48	43	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ 𝑤 ∈ ℝ⁺) → (abs ∘ − ) ∈ (∞Met‘ℂ))
49		ssid 3953	. . . . . . . . . . . . . . 15 ⊢ ℂ ⊆ ℂ
50		eqid 2760	. . . . . . . . . . . . . . . 16 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) = ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵))
51	21	cnfldtopon 25009	. . . . . . . . . . . . . . . . 17 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
52	51	toponrestid 23147	. . . . . . . . . . . . . . . 16 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
53	21, 50, 52	cncfcn 25139	. . . . . . . . . . . . . . 15 ⊢ (((𝐴(,)𝐵) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴(,)𝐵)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) Cn (TopOpen‘ℂ_fld)))
54	44, 49, 53	mp2an 705	. . . . . . . . . . . . . 14 ⊢ ((𝐴(,)𝐵)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) Cn (TopOpen‘ℂ_fld))
55	14, 54	eleqtrdi 2870	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) Cn (TopOpen‘ℂ_fld)))
56		resttopon 23387	. . . . . . . . . . . . . . . . 17 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ (𝐴(,)𝐵) ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) ∈ (TopOn‘(𝐴(,)𝐵)))
57	51, 44, 56	mp2an 705	. . . . . . . . . . . . . . . 16 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) ∈ (TopOn‘(𝐴(,)𝐵))
58	57	toponunii 23142	. . . . . . . . . . . . . . 15 ⊢ (𝐴(,)𝐵) = ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵))
59	58	eleq2i 2852	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ (𝐴(,)𝐵) ↔ 𝑐 ∈ ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)))
60	59	biimpi 219	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (𝐴(,)𝐵) → 𝑐 ∈ ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)))
61		eqid 2760	. . . . . . . . . . . . . 14 ⊢ ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) = ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵))
62	61	cncnpi 23504	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) Cn (TopOpen‘ℂ_fld)) ∧ 𝑐 ∈ ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵))) → 𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑐))
63	55, 60, 62	syl2an 608	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → 𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑐))
64		eqid 2760	. . . . . . . . . . . . . . . 16 ⊢ ((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵))) = ((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))
65	21	cnfldtopn 25008	. . . . . . . . . . . . . . . 16 ⊢ (TopOpen‘ℂ_fld) = (MetOpen‘(abs ∘ − ))
66		eqid 2760	. . . . . . . . . . . . . . . 16 ⊢ (MetOpen‘((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))) = (MetOpen‘((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵))))
67	64, 65, 66	metrest 24751	. . . . . . . . . . . . . . 15 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ (𝐴(,)𝐵) ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) = (MetOpen‘((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))))
68	43, 44, 67	mp2an 705	. . . . . . . . . . . . . 14 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) = (MetOpen‘((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵))))
69	68	oveq1i 7424	. . . . . . . . . . . . 13 ⊢ (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld)) = ((MetOpen‘((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))) CnP (TopOpen‘ℂ_fld))
70	69	fveq1i 6880	. . . . . . . . . . . 12 ⊢ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑐) = (((MetOpen‘((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))) CnP (TopOpen‘ℂ_fld))‘𝑐)
71	63, 70	eleqtrdi 2870	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → 𝐹 ∈ (((MetOpen‘((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))) CnP (TopOpen‘ℂ_fld))‘𝑐))
72	71	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ 𝑤 ∈ ℝ⁺) → 𝐹 ∈ (((MetOpen‘((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))) CnP (TopOpen‘ℂ_fld))‘𝑐))
73		simpr 490	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ 𝑤 ∈ ℝ⁺) → 𝑤 ∈ ℝ⁺)
74	66, 65	metcnpi2 24772	. . . . . . . . . 10 ⊢ (((((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵))) ∈ (∞Met‘(𝐴(,)𝐵)) ∧ (abs ∘ − ) ∈ (∞Met‘ℂ)) ∧ (𝐹 ∈ (((MetOpen‘((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))) CnP (TopOpen‘ℂ_fld))‘𝑐) ∧ 𝑤 ∈ ℝ⁺)) → ∃𝑣 ∈ ℝ⁺ ∀𝑢 ∈ (𝐴(,)𝐵)((𝑢((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))𝑐) < 𝑣 → ((𝐹‘𝑢)(abs ∘ − )(𝐹‘𝑐)) < 𝑤))
75	47, 48, 72, 73, 74	syl22anc 852	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ 𝑤 ∈ ℝ⁺) → ∃𝑣 ∈ ℝ⁺ ∀𝑢 ∈ (𝐴(,)𝐵)((𝑢((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))𝑐) < 𝑣 → ((𝐹‘𝑢)(abs ∘ − )(𝐹‘𝑐)) < 𝑤))
76		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑢 ∈ (𝐴(,)𝐵)) → 𝑢 ∈ (𝐴(,)𝐵))
77		simpllr 788	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑢 ∈ (𝐴(,)𝐵)) → 𝑐 ∈ (𝐴(,)𝐵))
78	76, 77	ovresd 7581	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑢 ∈ (𝐴(,)𝐵)) → (𝑢((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))𝑐) = (𝑢(abs ∘ − )𝑐))
79		elioore 13429	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ (𝐴(,)𝐵) → 𝑢 ∈ ℝ)
80	79	recnd 11262	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ (𝐴(,)𝐵) → 𝑢 ∈ ℂ)
81	44	sseli 3927	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 ∈ (𝐴(,)𝐵) → 𝑐 ∈ ℂ)
82	81	ad3antlr 744	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑢 ∈ (𝐴(,)𝐵)) → 𝑐 ∈ ℂ)
83		eqid 2760	. . . . . . . . . . . . . . . . . 18 ⊢ (abs ∘ − ) = (abs ∘ − )
84	83	cnmetdval 24997	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (𝑢(abs ∘ − )𝑐) = (abs‘(𝑢 − 𝑐)))
85	80, 82, 84	syl2an2 699	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑢 ∈ (𝐴(,)𝐵)) → (𝑢(abs ∘ − )𝑐) = (abs‘(𝑢 − 𝑐)))
86	78, 85	eqtrd 2795	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑢 ∈ (𝐴(,)𝐵)) → (𝑢((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))𝑐) = (abs‘(𝑢 − 𝑐)))
87	86	breq1d 5113	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑢 ∈ (𝐴(,)𝐵)) → ((𝑢((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))𝑐) < 𝑣 ↔ (abs‘(𝑢 − 𝑐)) < 𝑣))
88	16	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) → 𝐹:(𝐴(,)𝐵)⟶ℂ)
89	88	ffvelcdmda 7078	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑢 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑢) ∈ ℂ)
90	42	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑢 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑐) ∈ ℂ)
91	83	cnmetdval 24997	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑢) ∈ ℂ ∧ (𝐹‘𝑐) ∈ ℂ) → ((𝐹‘𝑢)(abs ∘ − )(𝐹‘𝑐)) = (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))))
92	89, 90, 91	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑢 ∈ (𝐴(,)𝐵)) → ((𝐹‘𝑢)(abs ∘ − )(𝐹‘𝑐)) = (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))))
93	92	breq1d 5113	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑢 ∈ (𝐴(,)𝐵)) → (((𝐹‘𝑢)(abs ∘ − )(𝐹‘𝑐)) < 𝑤 ↔ (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤))
94	87, 93	imbi12d 347	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑢 ∈ (𝐴(,)𝐵)) → (((𝑢((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))𝑐) < 𝑣 → ((𝐹‘𝑢)(abs ∘ − )(𝐹‘𝑐)) < 𝑤) ↔ ((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)))
95	94	ralbidva 3183	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) → (∀𝑢 ∈ (𝐴(,)𝐵)((𝑢((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))𝑐) < 𝑣 → ((𝐹‘𝑢)(abs ∘ − )(𝐹‘𝑐)) < 𝑤) ↔ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)))
96		simprll 791	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → 𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}))
97		eldifsni 4753	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) → 𝑧 ≠ 𝑐)
98	96, 97	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → 𝑧 ≠ 𝑐)
99	19	ssdifssd 4094	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((𝐴[,]𝐵) ∖ {𝑐}) ⊆ ℝ)
100	99	sselda 3931	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐})) → 𝑧 ∈ ℝ)
101	100	ad2ant2r 760	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤))) → 𝑧 ∈ ℝ)
102	101	ad2ant2r 760	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → 𝑧 ∈ ℝ)
103		elioore 13429	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 ∈ (𝐴(,)𝐵) → 𝑐 ∈ ℝ)
104	103	ad3antlr 744	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → 𝑐 ∈ ℝ)
105	102, 104	lttri2d 11374	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → (𝑧 ≠ 𝑐 ↔ (𝑧 < 𝑐 ∨ 𝑐 < 𝑧)))
106	105	biimpa 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑧 ≠ 𝑐) → (𝑧 < 𝑐 ∨ 𝑐 < 𝑧))
107		fveq2 6879	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑠 = 𝑧 → (𝐺‘𝑠) = (𝐺‘𝑧))
108	107	oveq1d 7429	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑠 = 𝑧 → ((𝐺‘𝑠) − (𝐺‘𝑐)) = ((𝐺‘𝑧) − (𝐺‘𝑐)))
109		oveq1 7421	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑠 = 𝑧 → (𝑠 − 𝑐) = (𝑧 − 𝑐))
110	108, 109	oveq12d 7432	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑠 = 𝑧 → (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)) = (((𝐺‘𝑧) − (𝐺‘𝑐)) / (𝑧 − 𝑐)))
111		eqid 2760	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐))) = (𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))
112		ovex 7447	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐺‘𝑧) − (𝐺‘𝑐)) / (𝑧 − 𝑐)) ∈ V
113	110, 111, 112	fvmpt 6987	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) → ((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) = (((𝐺‘𝑧) − (𝐺‘𝑐)) / (𝑧 − 𝑐)))
114	113	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣) → ((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) = (((𝐺‘𝑧) − (𝐺‘𝑐)) / (𝑧 − 𝑐)))
115	114	ad2antlr 740	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑧 < 𝑐) → ((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) = (((𝐺‘𝑧) − (𝐺‘𝑐)) / (𝑧 − 𝑐)))
116	17	ad4antr 745	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑧 < 𝑐) → 𝐺:(𝐴[,]𝐵)⟶ℂ)
117		eldifi 4078	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) → 𝑧 ∈ (𝐴[,]𝐵))
118	117	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣) → 𝑧 ∈ (𝐴[,]𝐵))
119	118	ad2antlr 740	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑧 < 𝑐) → 𝑧 ∈ (𝐴[,]𝐵))
120	116, 119	ffvelcdmd 7079	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑧 < 𝑐) → (𝐺‘𝑧) ∈ ℂ)
121	33	sseli 3927	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑐 ∈ (𝐴(,)𝐵) → 𝑐 ∈ (𝐴[,]𝐵))
122	17	ffvelcdmda 7078	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴[,]𝐵)) → (𝐺‘𝑐) ∈ ℂ)
123	121, 122	sylan2 605	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑐) ∈ ℂ)
124	123	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑧 < 𝑐) → (𝐺‘𝑐) ∈ ℂ)
125	102	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑧 < 𝑐) → 𝑧 ∈ ℝ)
126	125	recnd 11262	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑧 < 𝑐) → 𝑧 ∈ ℂ)
127	81	ad4antlr 746	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑧 < 𝑐) → 𝑐 ∈ ℂ)
128		ltne 11332	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑧 ∈ ℝ ∧ 𝑧 < 𝑐) → 𝑐 ≠ 𝑧)
129	128	necomd 3010	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑧 ∈ ℝ ∧ 𝑧 < 𝑐) → 𝑧 ≠ 𝑐)
130	102, 129	sylan 592	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑧 < 𝑐) → 𝑧 ≠ 𝑐)
131	120, 124, 126, 127, 130	div2subd 12066	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑧 < 𝑐) → (((𝐺‘𝑧) − (𝐺‘𝑐)) / (𝑧 − 𝑐)) = (((𝐺‘𝑐) − (𝐺‘𝑧)) / (𝑐 − 𝑧)))
132	115, 131	eqtrd 2795	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑧 < 𝑐) → ((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) = (((𝐺‘𝑐) − (𝐺‘𝑧)) / (𝑐 − 𝑧)))
133	132	fvoveq1d 7436	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑧 < 𝑐) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) = (abs‘((((𝐺‘𝑐) − (𝐺‘𝑧)) / (𝑐 − 𝑧)) − (𝐹‘𝑐))))
134	7	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → 𝐴 ∈ ℝ)
135	8	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → 𝐵 ∈ ℝ)
136	9	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → 𝐴 ≤ 𝐵)
137	14	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))
138	13	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → 𝐹 ∈ 𝐿¹)
139		simpllr 788	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → 𝑐 ∈ (𝐴(,)𝐵))
140		simplrl 789	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → 𝑤 ∈ ℝ⁺)
141		simplrr 790	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → 𝑣 ∈ ℝ⁺)
142		simprlr 792	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤))
143		fvoveq1 7437	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑢 = 𝑦 → (abs‘(𝑢 − 𝑐)) = (abs‘(𝑦 − 𝑐)))
144	143	breq1d 5113	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑢 = 𝑦 → ((abs‘(𝑢 − 𝑐)) < 𝑣 ↔ (abs‘(𝑦 − 𝑐)) < 𝑣))
145	144	imbrov2fvoveq 7439	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑢 = 𝑦 → (((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤) ↔ ((abs‘(𝑦 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑐))) < 𝑤)))
146	145	rspccva 3575	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤) ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((abs‘(𝑦 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑐))) < 𝑤))
147	142, 146	sylan 592	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((abs‘(𝑦 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑐))) < 𝑤))
148	96, 117	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → 𝑧 ∈ (𝐴[,]𝐵))
149		simprr 785	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → (abs‘(𝑧 − 𝑐)) < 𝑣)
150	121	ad3antlr 744	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → 𝑐 ∈ (𝐴[,]𝐵))
151	103	recnd 11262	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑐 ∈ (𝐴(,)𝐵) → 𝑐 ∈ ℂ)
152	151	subidd 11582	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 ∈ (𝐴(,)𝐵) → (𝑐 − 𝑐) = 0)
153	152	abs00bd 15379	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑐 ∈ (𝐴(,)𝐵) → (abs‘(𝑐 − 𝑐)) = 0)
154	153	ad3antlr 744	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → (abs‘(𝑐 − 𝑐)) = 0)
155	141	rpgt0d 13090	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → 0 < 𝑣)
156	154, 155	eqbrtrd 5127	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → (abs‘(𝑐 − 𝑐)) < 𝑣)
157	6, 134, 135, 136, 137, 138, 139, 111, 140, 141, 147, 148, 149, 150, 156	ftc1cnnclem 38441	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑧 < 𝑐) → (abs‘((((𝐺‘𝑐) − (𝐺‘𝑧)) / (𝑐 − 𝑧)) − (𝐹‘𝑐))) < 𝑤)
158	133, 157	eqbrtrd 5127	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑧 < 𝑐) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) < 𝑤)
159	113	fvoveq1d 7436	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) = (abs‘((((𝐺‘𝑧) − (𝐺‘𝑐)) / (𝑧 − 𝑐)) − (𝐹‘𝑐))))
160	159	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) = (abs‘((((𝐺‘𝑧) − (𝐺‘𝑐)) / (𝑧 − 𝑐)) − (𝐹‘𝑐))))
161	160	ad2antlr 740	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑐 < 𝑧) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) = (abs‘((((𝐺‘𝑧) − (𝐺‘𝑐)) / (𝑧 − 𝑐)) − (𝐹‘𝑐))))
162	6, 134, 135, 136, 137, 138, 139, 111, 140, 141, 147, 150, 156, 148, 149	ftc1cnnclem 38441	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑐 < 𝑧) → (abs‘((((𝐺‘𝑧) − (𝐺‘𝑐)) / (𝑧 − 𝑐)) − (𝐹‘𝑐))) < 𝑤)
163	161, 162	eqbrtrd 5127	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑐 < 𝑧) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) < 𝑤)
164	158, 163	jaodan 972	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ (𝑧 < 𝑐 ∨ 𝑐 < 𝑧)) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) < 𝑤)
165	106, 164	syldan 603	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑧 ≠ 𝑐) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) < 𝑤)
166	98, 165	mpdan 700	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) < 𝑤)
167	166	expr 462	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤))) → ((abs‘(𝑧 − 𝑐)) < 𝑣 → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) < 𝑤))
168	167	adantld 496	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤))) → ((𝑧 ≠ 𝑐 ∧ (abs‘(𝑧 − 𝑐)) < 𝑣) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) < 𝑤))
169	168	expr 462	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐})) → (∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤) → ((𝑧 ≠ 𝑐 ∧ (abs‘(𝑧 − 𝑐)) < 𝑣) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) < 𝑤)))
170	169	ralrimdva 3162	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) → (∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤) → ∀𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐})((𝑧 ≠ 𝑐 ∧ (abs‘(𝑧 − 𝑐)) < 𝑣) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) < 𝑤)))
171	95, 170	sylbid 243	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) → (∀𝑢 ∈ (𝐴(,)𝐵)((𝑢((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))𝑐) < 𝑣 → ((𝐹‘𝑢)(abs ∘ − )(𝐹‘𝑐)) < 𝑤) → ∀𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐})((𝑧 ≠ 𝑐 ∧ (abs‘(𝑧 − 𝑐)) < 𝑣) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) < 𝑤)))
172	171	anassrs 473	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑣 ∈ ℝ⁺) → (∀𝑢 ∈ (𝐴(,)𝐵)((𝑢((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))𝑐) < 𝑣 → ((𝐹‘𝑢)(abs ∘ − )(𝐹‘𝑐)) < 𝑤) → ∀𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐})((𝑧 ≠ 𝑐 ∧ (abs‘(𝑧 − 𝑐)) < 𝑣) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) < 𝑤)))
173	172	reximdva 3175	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ 𝑤 ∈ ℝ⁺) → (∃𝑣 ∈ ℝ⁺ ∀𝑢 ∈ (𝐴(,)𝐵)((𝑢((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))𝑐) < 𝑣 → ((𝐹‘𝑢)(abs ∘ − )(𝐹‘𝑐)) < 𝑤) → ∃𝑣 ∈ ℝ⁺ ∀𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐})((𝑧 ≠ 𝑐 ∧ (abs‘(𝑧 − 𝑐)) < 𝑣) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) < 𝑤)))
174	75, 173	mpd 16	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ 𝑤 ∈ ℝ⁺) → ∃𝑣 ∈ ℝ⁺ ∀𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐})((𝑧 ≠ 𝑐 ∧ (abs‘(𝑧 − 𝑐)) < 𝑣) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) < 𝑤))
175	174	ralrimiva 3154	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → ∀𝑤 ∈ ℝ⁺ ∃𝑣 ∈ ℝ⁺ ∀𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐})((𝑧 ≠ 𝑐 ∧ (abs‘(𝑧 − 𝑐)) < 𝑣) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) < 𝑤))
176	17	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → 𝐺:(𝐴[,]𝐵)⟶ℂ)
177	19, 4	sstrdi 3943	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
178	177	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝐴[,]𝐵) ⊆ ℂ)
179	121	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → 𝑐 ∈ (𝐴[,]𝐵))
180	176, 178, 179	dvlem 26124	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ 𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐})) → (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)) ∈ ℂ)
181	180	fmpttd 7109	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐))):((𝐴[,]𝐵) ∖ {𝑐})⟶ℂ)
182	177	ssdifssd 4094	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴[,]𝐵) ∖ {𝑐}) ⊆ ℂ)
183	182	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → ((𝐴[,]𝐵) ∖ {𝑐}) ⊆ ℂ)
184	81	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → 𝑐 ∈ ℂ)
185	181, 183, 184	ellimc3 26107	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → ((𝐹‘𝑐) ∈ ((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐))) lim_ℂ 𝑐) ↔ ((𝐹‘𝑐) ∈ ℂ ∧ ∀𝑤 ∈ ℝ⁺ ∃𝑣 ∈ ℝ⁺ ∀𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐})((𝑧 ≠ 𝑐 ∧ (abs‘(𝑧 − 𝑐)) < 𝑣) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) < 𝑤))))
186	42, 175, 185	mpbir2and 726	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑐) ∈ ((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐))) lim_ℂ 𝑐))
187		eqid 2760	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℝ) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
188	187, 21, 111, 5, 17, 19	eldv 26126	. . . . . . 7 ⊢ (𝜑 → (𝑐(ℝ D 𝐺)(𝐹‘𝑐) ↔ (𝑐 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t ℝ))‘(𝐴[,]𝐵)) ∧ (𝐹‘𝑐) ∈ ((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐))) lim_ℂ 𝑐))))
189	188	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝑐(ℝ D 𝐺)(𝐹‘𝑐) ↔ (𝑐 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t ℝ))‘(𝐴[,]𝐵)) ∧ (𝐹‘𝑐) ∈ ((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐))) lim_ℂ 𝑐))))
190	41, 186, 189	mpbir2and 726	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → 𝑐(ℝ D 𝐺)(𝐹‘𝑐))
191		vex 3454	. . . . . 6 ⊢ 𝑐 ∈ V
192		fvex 6892	. . . . . 6 ⊢ (𝐹‘𝑐) ∈ V
193	191, 192	breldm 5892	. . . . 5 ⊢ (𝑐(ℝ D 𝐺)(𝐹‘𝑐) → 𝑐 ∈ dom (ℝ D 𝐺))
194	190, 193	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → 𝑐 ∈ dom (ℝ D 𝐺))
195	25, 194	eqelssd 3952	. . 3 ⊢ (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵))
196		df-fn 6536	. . 3 ⊢ ((ℝ D 𝐺) Fn (𝐴(,)𝐵) ↔ (Fun (ℝ D 𝐺) ∧ dom (ℝ D 𝐺) = (𝐴(,)𝐵)))
197	3, 195, 196	sylanbrc 595	. 2 ⊢ (𝜑 → (ℝ D 𝐺) Fn (𝐴(,)𝐵))
198	16	ffnd 6704	. 2 ⊢ (𝜑 → 𝐹 Fn (𝐴(,)𝐵))
199	3	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → Fun (ℝ D 𝐺))
200		funbrfv 6927	. . 3 ⊢ (Fun (ℝ D 𝐺) → (𝑐(ℝ D 𝐺)(𝐹‘𝑐) → ((ℝ D 𝐺)‘𝑐) = (𝐹‘𝑐)))
201	199, 190, 200	sylc 66	. 2 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐺)‘𝑐) = (𝐹‘𝑐))
202	197, 198, 201	eqfnfvd 7026	1 ⊢ (𝜑 → (ℝ D 𝐺) = 𝐹)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 ∖ cdif 3896 ⊆ wss 3899 {csn 4584 ∪ cuni 4867 class class class wbr 5103 ↦ cmpt 5186 × cxp 5653 dom cdm 5655 ran crn 5656 ↾ cres 5657 ∘ ccom 5659 Fun wfun 6527 Fn wfn 6528 ⟶wf 6529 ‘cfv 6533 (class class class)co 7414 ℂcc 11123 ℝcr 11124 0cc0 11125 < clt 11268 ≤ cle 11269 − cmin 11466 / cdiv 11896 ℝ⁺crp 13043 (,)cioo 13399 [,]cicc 13402 abscabs 15322 ↾_t crest 17506 TopOpenctopn 17507 topGenctg 17523 ∞Metcxmet 21571 MetOpencmopn 21576 ℂ_fldccnfld 21586 Topctop 23119 TopOnctopon 23136 intcnt 23243 Cn ccn 23450 CnP ccnp 23451 –cn→ccncf 25105 𝐿¹cibl 25846 ∫citg 25847 lim_ℂ climc 26090 D cdv 26091

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-inf2 9621 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202 ax-pre-sup 11203 ax-addf 11204

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-symdif 4199 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-disj 5071 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-ofr 7680 df-om 7864 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8456 df-2o 8457 df-oadd 8460 df-omul 8461 df-er 8697 df-map 8829 df-pm 8830 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9333 df-fi 9382 df-sup 9413 df-inf 9414 df-oi 9483 df-dju 9907 df-card 9945 df-acn 9948 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-div 11897 df-nn 12259 df-2 12328 df-3 12329 df-4 12330 df-5 12331 df-6 12332 df-7 12333 df-8 12334 df-9 12335 df-n0 12530 df-z 12617 df-dec 12738 df-uz 12889 df-q 12999 df-rp 13044 df-xneg 13164 df-xadd 13165 df-xmul 13166 df-ioo 13403 df-ico 13405 df-icc 13406 df-fz 13563 df-fzo 13711 df-fl 13854 df-mod 13932 df-seq 14067 df-exp 14127 df-hash 14396 df-cj 15187 df-re 15188 df-im 15189 df-sqrt 15323 df-abs 15324 df-clim 15576 df-rlim 15577 df-sum 15775 df-struct 17240 df-sets 17257 df-slot 17275 df-ndx 17287 df-base 17303 df-ress 17324 df-plusg 17356 df-mulr 17357 df-starv 17358 df-sca 17359 df-vsca 17360 df-ip 17361 df-tset 17362 df-ple 17363 df-ds 17365 df-unif 17366 df-hom 17367 df-cco 17368 df-rest 17508 df-topn 17509 df-0g 17527 df-gsum 17528 df-topgen 17529 df-pt 17530 df-prds 17533 df-xrs 17589 df-qtop 17594 df-imas 17595 df-xps 17597 df-mre 17671 df-mrc 17672 df-acs 17674 df-mgm 18731 df-sgrp 18822 df-mnd 18838 df-submnd 18893 df-mulg 19192 df-cntz 19445 df-cmn 19910 df-psmet 21578 df-xmet 21579 df-met 21580 df-bl 21581 df-mopn 21582 df-fbas 21583 df-fg 21584 df-cnfld 21587 df-top 23120 df-topon 23137 df-topsp 23159 df-bases 23172 df-cld 23245 df-ntr 23246 df-cls 23247 df-nei 23324 df-lp 23362 df-perf 23363 df-cn 23453 df-cnp 23454 df-haus 23541 df-cmp 23613 df-tx 23789 df-hmeo 23982 df-fil 24073 df-fm 24165 df-flim 24166 df-flf 24167 df-xms 24547 df-ms 24548 df-tms 24549 df-cncf 25107 df-ovol 25693 df-vol 25694 df-mbf 25848 df-itg1 25849 df-itg2 25850 df-ibl 25851 df-itg 25852 df-0p 25899 df-limc 26094 df-dv 26095

This theorem is used by: ftc2nc 38452

W3C validator