ftc1cnnc - Mathbox for Brendan Leahy

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Theorem ftc1cnnc 37678

Description: Choice-free proof of ftc1cn 26098. (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 25956	. . . . 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 6740	. . 3 ⊢ (𝜑 → Fun (ℝ D 𝐺))
4		ax-resscn 11209	. . . . . . 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 4018	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐵))
11		ioossre 13444	. . . . . . . 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 24932	. . . . . . . 8 ⊢ (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ) → 𝐹:(𝐴(,)𝐵)⟶ℂ)
16	14, 15	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℂ)
17	6, 7, 8, 9, 10, 12, 13, 16	ftc1lem2 26091	. . . . . 6 ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ)
18		iccssre 13465	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
19	7, 8, 18	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
20		eqid 2734	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
21	20	tgioo2 24838	. . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
22	5, 17, 19, 21, 20	dvbssntr 25949	. . . . 5 ⊢ (𝜑 → dom (ℝ D 𝐺) ⊆ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)))
23		iccntr 24856	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
24	7, 8, 23	syl2anc 584	. . . . 5 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
25	22, 24	sseqtrd 4035	. . . 4 ⊢ (𝜑 → dom (ℝ D 𝐺) ⊆ (𝐴(,)𝐵))
26		retop 24797	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) ∈ Top
27	21, 26	eqeltrri 2835	. . . . . . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝐴[,]𝐵) ⊆ ℝ)
30		iooretop 24801	. . . . . . . . . 10 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
31	30, 21	eleqtri 2836	. . . . . . . . 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 13469	. . . . . . . . 9 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
34	33	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
35		uniretop 24798	. . . . . . . . . 10 ⊢ ℝ = ∪ (topGen‘ran (,))
36	21	unieqi 4923	. . . . . . . . . 10 ⊢ ∪ (topGen‘ran (,)) = ∪ ((TopOpen‘ℂ_fld) ↾_t ℝ)
37	35, 36	eqtri 2762	. . . . . . . . 9 ⊢ ℝ = ∪ ((TopOpen‘ℂ_fld) ↾_t ℝ)
38	37	ssntr 23081	. . . . . . . 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 839	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝐴(,)𝐵) ⊆ ((int‘((TopOpen‘ℂ_fld) ↾_t ℝ))‘(𝐴[,]𝐵)))
40		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → 𝑐 ∈ (𝐴(,)𝐵))
41	39, 40	sseldd 3995	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → 𝑐 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t ℝ))‘(𝐴[,]𝐵)))
42	16	ffvelcdmda 7103	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑐) ∈ ℂ)
43		cnxmet 24808	. . . . . . . . . . . 12 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
44	11, 4	sstri 4004	. . . . . . . . . . . 12 ⊢ (𝐴(,)𝐵) ⊆ ℂ
45		xmetres2 24386	. . . . . . . . . . . 12 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ (𝐴(,)𝐵) ⊆ ℂ) → ((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵))) ∈ (∞Met‘(𝐴(,)𝐵)))
46	43, 44, 45	mp2an 692	. . . . . . . . . . 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 4017	. . . . . . . . . . . . . . 15 ⊢ ℂ ⊆ ℂ
50		eqid 2734	. . . . . . . . . . . . . . . 16 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) = ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵))
51	20	cnfldtopon 24818	. . . . . . . . . . . . . . . . 17 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
52	51	toponrestid 22942	. . . . . . . . . . . . . . . 16 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
53	20, 50, 52	cncfcn 24949	. . . . . . . . . . . . . . 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 692	. . . . . . . . . . . . . 14 ⊢ ((𝐴(,)𝐵)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) Cn (TopOpen‘ℂ_fld))
55	14, 54	eleqtrdi 2848	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) Cn (TopOpen‘ℂ_fld)))
56		resttopon 23184	. . . . . . . . . . . . . . . . 17 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ (𝐴(,)𝐵) ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) ∈ (TopOn‘(𝐴(,)𝐵)))
57	51, 44, 56	mp2an 692	. . . . . . . . . . . . . . . 16 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) ∈ (TopOn‘(𝐴(,)𝐵))
58	57	toponunii 22937	. . . . . . . . . . . . . . 15 ⊢ (𝐴(,)𝐵) = ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵))
59	58	eleq2i 2830	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ (𝐴(,)𝐵) ↔ 𝑐 ∈ ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)))
60	59	biimpi 216	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (𝐴(,)𝐵) → 𝑐 ∈ ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)))
61		eqid 2734	. . . . . . . . . . . . . 14 ⊢ ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) = ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵))
62	61	cncnpi 23301	. . . . . . . . . . . . 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 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → 𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑐))
64		eqid 2734	. . . . . . . . . . . . . . . 16 ⊢ ((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵))) = ((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))
65	20	cnfldtopn 24817	. . . . . . . . . . . . . . . 16 ⊢ (TopOpen‘ℂ_fld) = (MetOpen‘(abs ∘ − ))
66		eqid 2734	. . . . . . . . . . . . . . . 16 ⊢ (MetOpen‘((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))) = (MetOpen‘((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵))))
67	64, 65, 66	metrest 24552	. . . . . . . . . . . . . . 15 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ (𝐴(,)𝐵) ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) = (MetOpen‘((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))))
68	43, 44, 67	mp2an 692	. . . . . . . . . . . . . 14 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) = (MetOpen‘((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵))))
69	68	oveq1i 7440	. . . . . . . . . . . . 13 ⊢ (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld)) = ((MetOpen‘((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))) CnP (TopOpen‘ℂ_fld))
70	69	fveq1i 6907	. . . . . . . . . . . 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 2848	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → 𝐹 ∈ (((MetOpen‘((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))) CnP (TopOpen‘ℂ_fld))‘𝑐))
72	71	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ 𝑤 ∈ ℝ⁺) → 𝐹 ∈ (((MetOpen‘((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))) CnP (TopOpen‘ℂ_fld))‘𝑐))
73		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ 𝑤 ∈ ℝ⁺) → 𝑤 ∈ ℝ⁺)
74	66, 65	metcnpi2 24573	. . . . . . . . . 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 839	. . . . . . . . 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 484	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑢 ∈ (𝐴(,)𝐵)) → 𝑢 ∈ (𝐴(,)𝐵))
77		simpllr 776	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑢 ∈ (𝐴(,)𝐵)) → 𝑐 ∈ (𝐴(,)𝐵))
78	76, 77	ovresd 7599	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑢 ∈ (𝐴(,)𝐵)) → (𝑢((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))𝑐) = (𝑢(abs ∘ − )𝑐))
79		elioore 13413	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ (𝐴(,)𝐵) → 𝑢 ∈ ℝ)
80	79	recnd 11286	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ (𝐴(,)𝐵) → 𝑢 ∈ ℂ)
81	44	sseli 3990	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 ∈ (𝐴(,)𝐵) → 𝑐 ∈ ℂ)
82	81	ad3antlr 731	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑢 ∈ (𝐴(,)𝐵)) → 𝑐 ∈ ℂ)
83		eqid 2734	. . . . . . . . . . . . . . . . . 18 ⊢ (abs ∘ − ) = (abs ∘ − )
84	83	cnmetdval 24806	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (𝑢(abs ∘ − )𝑐) = (abs‘(𝑢 − 𝑐)))
85	80, 82, 84	syl2an2 686	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑢 ∈ (𝐴(,)𝐵)) → (𝑢(abs ∘ − )𝑐) = (abs‘(𝑢 − 𝑐)))
86	78, 85	eqtrd 2774	. . . . . . . . . . . . . . 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 5157	. . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) → 𝐹:(𝐴(,)𝐵)⟶ℂ)
89	88	ffvelcdmda 7103	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑢 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑢) ∈ ℂ)
90	42	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑢 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑐) ∈ ℂ)
91	83	cnmetdval 24806	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑢) ∈ ℂ ∧ (𝐹‘𝑐) ∈ ℂ) → ((𝐹‘𝑢)(abs ∘ − )(𝐹‘𝑐)) = (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))))
92	89, 90, 91	syl2anc 584	. . . . . . . . . . . . . . 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 5157	. . . . . . . . . . . . . 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 344	. . . . . . . . . . . . 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 3173	. . . . . . . . . . . 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 779	. . . . . . . . . . . . . . . . . 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 4794	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) → 𝑧 ≠ 𝑐)
98	96, 97	syl 17	. . . . . . . . . . . . . . . . 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 4156	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((𝐴[,]𝐵) ∖ {𝑐}) ⊆ ℝ)
100	99	sselda 3994	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐})) → 𝑧 ∈ ℝ)
101	100	ad2ant2r 747	. . . . . . . . . . . . . . . . . . . . 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 747	. . . . . . . . . . . . . . . . . . . 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 13413	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 ∈ (𝐴(,)𝐵) → 𝑐 ∈ ℝ)
104	103	ad3antlr 731	. . . . . . . . . . . . . . . . . . . 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 11397	. . . . . . . . . . . . . . . . . . 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 476	. . . . . . . . . . . . . . . . . 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 6906	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑠 = 𝑧 → (𝐺‘𝑠) = (𝐺‘𝑧))
108	107	oveq1d 7445	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑠 = 𝑧 → ((𝐺‘𝑠) − (𝐺‘𝑐)) = ((𝐺‘𝑧) − (𝐺‘𝑐)))
109		oveq1 7437	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑠 = 𝑧 → (𝑠 − 𝑐) = (𝑧 − 𝑐))
110	108, 109	oveq12d 7448	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑠 = 𝑧 → (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)) = (((𝐺‘𝑧) − (𝐺‘𝑐)) / (𝑧 − 𝑐)))
111		eqid 2734	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐))) = (𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))
112		ovex 7463	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐺‘𝑧) − (𝐺‘𝑐)) / (𝑧 − 𝑐)) ∈ V
113	110, 111, 112	fvmpt 7015	. . . . . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . . . . . . . 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 732	. . . . . . . . . . . . . . . . . . . . . . . 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 4140	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) → 𝑧 ∈ (𝐴[,]𝐵))
118	117	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . . . . . . . . . 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 7104	. . . . . . . . . . . . . . . . . . . . . . 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 3990	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑐 ∈ (𝐴(,)𝐵) → 𝑐 ∈ (𝐴[,]𝐵))
122	17	ffvelcdmda 7103	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴[,]𝐵)) → (𝐺‘𝑐) ∈ ℂ)
123	121, 122	sylan2 593	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑐) ∈ ℂ)
124	123	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . . 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 11286	. . . . . . . . . . . . . . . . . . . . . . 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 733	. . . . . . . . . . . . . . . . . . . . . . 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 11355	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑧 ∈ ℝ ∧ 𝑧 < 𝑐) → 𝑐 ≠ 𝑧)
129	128	necomd 2993	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑧 ∈ ℝ ∧ 𝑧 < 𝑐) → 𝑧 ≠ 𝑐)
130	102, 129	sylan 580	. . . . . . . . . . . . . . . . . . . . . . 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 12090	. . . . . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . . . . . 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 7452	. . . . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . . . . . . . . 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 7453	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑢 = 𝑦 → (abs‘(𝑢 − 𝑐)) = (abs‘(𝑦 − 𝑐)))
144	143	breq1d 5157	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑢 = 𝑦 → ((abs‘(𝑢 − 𝑐)) < 𝑣 ↔ (abs‘(𝑦 − 𝑐)) < 𝑣))
145	144	imbrov2fvoveq 7455	. . . . . . . . . . . . . . . . . . . . . . 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 3620	. . . . . . . . . . . . . . . . . . . . . 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 580	. . . . . . . . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . . . . . . . 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 731	. . . . . . . . . . . . . . . . . . . . 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 11286	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑐 ∈ (𝐴(,)𝐵) → 𝑐 ∈ ℂ)
152	151	subidd 11605	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 ∈ (𝐴(,)𝐵) → (𝑐 − 𝑐) = 0)
153	152	abs00bd 15326	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑐 ∈ (𝐴(,)𝐵) → (abs‘(𝑐 − 𝑐)) = 0)
154	153	ad3antlr 731	. . . . . . . . . . . . . . . . . . . . . 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 13077	. . . . . . . . . . . . . . . . . . . . . 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 5169	. . . . . . . . . . . . . . . . . . . . 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 37677	. . . . . . . . . . . . . . . . . . . 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 5169	. . . . . . . . . . . . . . . . . . 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 7452	. . . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . . . . . 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 37677	. . . . . . . . . . . . . . . . . . . 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 5169	. . . . . . . . . . . . . . . . . . 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 959	. . . . . . . . . . . . . . . . . 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 591	. . . . . . . . . . . . . . . . 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 687	. . . . . . . . . . . . . . . 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 456	. . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . 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 456	. . . . . . . . . . . . 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 3151	. . . . . . . . . . . 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 240	. . . . . . . . . . 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 467	. . . . . . . . . 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 3165	. . . . . . . . 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 15	. . . . . . . 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 3143	. . . . . . 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 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → 𝐺:(𝐴[,]𝐵)⟶ℂ)
177	19, 4	sstrdi 4007	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
178	177	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝐴[,]𝐵) ⊆ ℂ)
179	121	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → 𝑐 ∈ (𝐴[,]𝐵))
180	176, 178, 179	dvlem 25945	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ 𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐})) → (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)) ∈ ℂ)
181	180	fmpttd 7134	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐))):((𝐴[,]𝐵) ∖ {𝑐})⟶ℂ)
182	177	ssdifssd 4156	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴[,]𝐵) ∖ {𝑐}) ⊆ ℂ)
183	182	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → ((𝐴[,]𝐵) ∖ {𝑐}) ⊆ ℂ)
184	81	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → 𝑐 ∈ ℂ)
185	181, 183, 184	ellimc3 25928	. . . . . . 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 713	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑐) ∈ ((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐))) lim_ℂ 𝑐))
187		eqid 2734	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℝ) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
188	187, 20, 111, 5, 17, 19	eldv 25947	. . . . . . 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 480	. . . . . 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 713	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → 𝑐(ℝ D 𝐺)(𝐹‘𝑐))
191		vex 3481	. . . . . 6 ⊢ 𝑐 ∈ V
192		fvex 6919	. . . . . 6 ⊢ (𝐹‘𝑐) ∈ V
193	191, 192	breldm 5921	. . . . 5 ⊢ (𝑐(ℝ D 𝐺)(𝐹‘𝑐) → 𝑐 ∈ dom (ℝ D 𝐺))
194	190, 193	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → 𝑐 ∈ dom (ℝ D 𝐺))
195	25, 194	eqelssd 4016	. . 3 ⊢ (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵))
196		df-fn 6565	. . 3 ⊢ ((ℝ D 𝐺) Fn (𝐴(,)𝐵) ↔ (Fun (ℝ D 𝐺) ∧ dom (ℝ D 𝐺) = (𝐴(,)𝐵)))
197	3, 195, 196	sylanbrc 583	. 2 ⊢ (𝜑 → (ℝ D 𝐺) Fn (𝐴(,)𝐵))
198	16	ffnd 6737	. 2 ⊢ (𝜑 → 𝐹 Fn (𝐴(,)𝐵))
199	3	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → Fun (ℝ D 𝐺))
200		funbrfv 6957	. . 3 ⊢ (Fun (ℝ D 𝐺) → (𝑐(ℝ D 𝐺)(𝐹‘𝑐) → ((ℝ D 𝐺)‘𝑐) = (𝐹‘𝑐)))
201	199, 190, 200	sylc 65	. 2 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐺)‘𝑐) = (𝐹‘𝑐))
202	197, 198, 201	eqfnfvd 7053	1 ⊢ (𝜑 → (ℝ D 𝐺) = 𝐹)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 ∖ cdif 3959 ⊆ wss 3962 {csn 4630 ∪ cuni 4911 class class class wbr 5147 ↦ cmpt 5230 × cxp 5686 dom cdm 5688 ran crn 5689 ↾ cres 5690 ∘ ccom 5692 Fun wfun 6556 Fn wfn 6557 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 ℝcr 11151 0cc0 11152 < clt 11292 ≤ cle 11293 − cmin 11489 / cdiv 11917 ℝ⁺crp 13031 (,)cioo 13383 [,]cicc 13386 abscabs 15269 ↾_t crest 17466 TopOpenctopn 17467 topGenctg 17483 ∞Metcxmet 21366 MetOpencmopn 21371 ℂ_fldccnfld 21381 Topctop 22914 TopOnctopon 22931 intcnt 23040 Cn ccn 23247 CnP ccnp 23248 –cn→ccncf 24915 𝐿¹cibl 25665 ∫citg 25666 lim_ℂ climc 25911 D cdv 25912

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-symdif 4258 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-disj 5115 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-ofr 7697 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-oadd 8508 df-omul 8509 df-er 8743 df-map 8866 df-pm 8867 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-dju 9938 df-card 9976 df-acn 9979 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ioo 13387 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-fl 13828 df-mod 13906 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 df-rlim 15521 df-sum 15719 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-mulg 19098 df-cntz 19347 df-cmn 19814 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-fbas 21378 df-fg 21379 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cld 23042 df-ntr 23043 df-cls 23044 df-nei 23121 df-lp 23159 df-perf 23160 df-cn 23250 df-cnp 23251 df-haus 23338 df-cmp 23410 df-tx 23585 df-hmeo 23778 df-fil 23869 df-fm 23961 df-flim 23962 df-flf 23963 df-xms 24345 df-ms 24346 df-tms 24347 df-cncf 24917 df-ovol 25512 df-vol 25513 df-mbf 25667 df-itg1 25668 df-itg2 25669 df-ibl 25670 df-itg 25671 df-0p 25718 df-limc 25915 df-dv 25916

This theorem is referenced by: ftc2nc 37688

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