ftc1cnnc - Mathbox for Brendan Leahy

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Theorem ftc1cnnc 37676

Description: Choice-free proof of ftc1cn 25948. (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 25806	. . . . 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 6656	. . 3 ⊢ (𝜑 → Fun (ℝ D 𝐺))
4		ax-resscn 11066	. . . . . . 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 3959	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐵))
11		ioossre 13310	. . . . . . . 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 24784	. . . . . . . 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 25941	. . . . . 6 ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ)
18		iccssre 13332	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
19	7, 8, 18	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
20		tgioo4 24691	. . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
21		eqid 2729	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
22	5, 17, 19, 20, 21	dvbssntr 25799	. . . . 5 ⊢ (𝜑 → dom (ℝ D 𝐺) ⊆ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)))
23		iccntr 24708	. . . . . 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 3972	. . . 4 ⊢ (𝜑 → dom (ℝ D 𝐺) ⊆ (𝐴(,)𝐵))
26		retop 24647	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) ∈ Top
27	20, 26	eqeltrri 2825	. . . . . . . . 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 24651	. . . . . . . . . 10 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
31	30, 20	eleqtri 2826	. . . . . . . . 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 13336	. . . . . . . . 9 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
34	33	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
35		uniretop 24648	. . . . . . . . . 10 ⊢ ℝ = ∪ (topGen‘ran (,))
36	20	unieqi 4870	. . . . . . . . . 10 ⊢ ∪ (topGen‘ran (,)) = ∪ ((TopOpen‘ℂ_fld) ↾_t ℝ)
37	35, 36	eqtri 2752	. . . . . . . . 9 ⊢ ℝ = ∪ ((TopOpen‘ℂ_fld) ↾_t ℝ)
38	37	ssntr 22943	. . . . . . . 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 838	. . . . . . 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 3936	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → 𝑐 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t ℝ))‘(𝐴[,]𝐵)))
42	16	ffvelcdmda 7018	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑐) ∈ ℂ)
43		cnxmet 24658	. . . . . . . . . . . 12 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
44	11, 4	sstri 3945	. . . . . . . . . . . 12 ⊢ (𝐴(,)𝐵) ⊆ ℂ
45		xmetres2 24247	. . . . . . . . . . . 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 3958	. . . . . . . . . . . . . . 15 ⊢ ℂ ⊆ ℂ
50		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) = ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵))
51	21	cnfldtopon 24668	. . . . . . . . . . . . . . . . 17 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
52	51	toponrestid 22806	. . . . . . . . . . . . . . . 16 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
53	21, 50, 52	cncfcn 24801	. . . . . . . . . . . . . . 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 2838	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) Cn (TopOpen‘ℂ_fld)))
56		resttopon 23046	. . . . . . . . . . . . . . . . 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 22801	. . . . . . . . . . . . . . 15 ⊢ (𝐴(,)𝐵) = ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵))
59	58	eleq2i 2820	. . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . 14 ⊢ ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) = ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵))
62	61	cncnpi 23163	. . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . 16 ⊢ ((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵))) = ((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))
65	21	cnfldtopn 24667	. . . . . . . . . . . . . . . 16 ⊢ (TopOpen‘ℂ_fld) = (MetOpen‘(abs ∘ − ))
66		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ (MetOpen‘((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))) = (MetOpen‘((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵))))
67	64, 65, 66	metrest 24410	. . . . . . . . . . . . . . 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 7359	. . . . . . . . . . . . 13 ⊢ (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld)) = ((MetOpen‘((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))) CnP (TopOpen‘ℂ_fld))
70	69	fveq1i 6823	. . . . . . . . . . . 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 2838	. . . . . . . . . . 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 24431	. . . . . . . . . 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 838	. . . . . . . . 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 775	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑢 ∈ (𝐴(,)𝐵)) → 𝑐 ∈ (𝐴(,)𝐵))
78	76, 77	ovresd 7516	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑢 ∈ (𝐴(,)𝐵)) → (𝑢((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))𝑐) = (𝑢(abs ∘ − )𝑐))
79		elioore 13278	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ (𝐴(,)𝐵) → 𝑢 ∈ ℝ)
80	79	recnd 11143	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ (𝐴(,)𝐵) → 𝑢 ∈ ℂ)
81	44	sseli 3931	. . . . . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . . 18 ⊢ (abs ∘ − ) = (abs ∘ − )
84	83	cnmetdval 24656	. . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . 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 5102	. . . . . . . . . . . . . 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 7018	. . . . . . . . . . . . . . . 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 24656	. . . . . . . . . . . . . . . 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 5102	. . . . . . . . . . . . . 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 3150	. . . . . . . . . . . 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 778	. . . . . . . . . . . . . . . . . 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 4741	. . . . . . . . . . . . . . . . . 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 4098	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((𝐴[,]𝐵) ∖ {𝑐}) ⊆ ℝ)
100	99	sselda 3935	. . . . . . . . . . . . . . . . . . . . . 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 13278	. . . . . . . . . . . . . . . . . . . . 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 11255	. . . . . . . . . . . . . . . . . . 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 6822	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑠 = 𝑧 → (𝐺‘𝑠) = (𝐺‘𝑧))
108	107	oveq1d 7364	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑠 = 𝑧 → ((𝐺‘𝑠) − (𝐺‘𝑐)) = ((𝐺‘𝑧) − (𝐺‘𝑐)))
109		oveq1 7356	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑠 = 𝑧 → (𝑠 − 𝑐) = (𝑧 − 𝑐))
110	108, 109	oveq12d 7367	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑠 = 𝑧 → (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)) = (((𝐺‘𝑧) − (𝐺‘𝑐)) / (𝑧 − 𝑐)))
111		eqid 2729	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐))) = (𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))
112		ovex 7382	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐺‘𝑧) − (𝐺‘𝑐)) / (𝑧 − 𝑐)) ∈ V
113	110, 111, 112	fvmpt 6930	. . . . . . . . . . . . . . . . . . . . . . . 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 4082	. . . . . . . . . . . . . . . . . . . . . . . . . 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 7019	. . . . . . . . . . . . . . . . . . . . . . 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 3931	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑐 ∈ (𝐴(,)𝐵) → 𝑐 ∈ (𝐴[,]𝐵))
122	17	ffvelcdmda 7018	. . . . . . . . . . . . . . . . . . . . . . . . 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 11143	. . . . . . . . . . . . . . . . . . . . . . 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 11213	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑧 ∈ ℝ ∧ 𝑧 < 𝑐) → 𝑐 ≠ 𝑧)
129	128	necomd 2980	. . . . . . . . . . . . . . . . . . . . . . . 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 11950	. . . . . . . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . . . . . . 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 7371	. . . . . . . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . . . . . . 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 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)) → 𝑤 ∈ ℝ⁺)
141		simplrr 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)) → 𝑣 ∈ ℝ⁺)
142		simprlr 779	. . . . . . . . . . . . . . . . . . . . . 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 7372	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑢 = 𝑦 → (abs‘(𝑢 − 𝑐)) = (abs‘(𝑦 − 𝑐)))
144	143	breq1d 5102	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑢 = 𝑦 → ((abs‘(𝑢 − 𝑐)) < 𝑣 ↔ (abs‘(𝑦 − 𝑐)) < 𝑣))
145	144	imbrov2fvoveq 7374	. . . . . . . . . . . . . . . . . . . . . . 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 3576	. . . . . . . . . . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . . . . . . 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 11143	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑐 ∈ (𝐴(,)𝐵) → 𝑐 ∈ ℂ)
152	151	subidd 11463	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 ∈ (𝐴(,)𝐵) → (𝑐 − 𝑐) = 0)
153	152	abs00bd 15198	. . . . . . . . . . . . . . . . . . . . . . 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 12940	. . . . . . . . . . . . . . . . . . . . . 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 5114	. . . . . . . . . . . . . . . . . . . . 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 37675	. . . . . . . . . . . . . . . . . . . 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 5114	. . . . . . . . . . . . . . . . . . 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 7371	. . . . . . . . . . . . . . . . . . . . . 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 37675	. . . . . . . . . . . . . . . . . . . 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 5114	. . . . . . . . . . . . . . . . . . 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 3129	. . . . . . . . . . . 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 3142	. . . . . . . . 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 3121	. . . . . . 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 3948	. . . . . . . . . . 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 25795	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ 𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐})) → (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)) ∈ ℂ)
181	180	fmpttd 7049	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐))):((𝐴[,]𝐵) ∖ {𝑐})⟶ℂ)
182	177	ssdifssd 4098	. . . . . . . . 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 25778	. . . . . . 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 2729	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℝ) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
188	187, 21, 111, 5, 17, 19	eldv 25797	. . . . . . 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 3440	. . . . . 6 ⊢ 𝑐 ∈ V
192		fvex 6835	. . . . . 6 ⊢ (𝐹‘𝑐) ∈ V
193	191, 192	breldm 5851	. . . . 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 3957	. . 3 ⊢ (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵))
196		df-fn 6485	. . 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 6653	. 2 ⊢ (𝜑 → 𝐹 Fn (𝐴(,)𝐵))
199	3	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → Fun (ℝ D 𝐺))
200		funbrfv 6871	. . 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 6968	1 ⊢ (𝜑 → (ℝ D 𝐺) = 𝐹)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 ∖ cdif 3900 ⊆ wss 3903 {csn 4577 ∪ cuni 4858 class class class wbr 5092 ↦ cmpt 5173 × cxp 5617 dom cdm 5619 ran crn 5620 ↾ cres 5621 ∘ ccom 5623 Fun wfun 6476 Fn wfn 6477 ⟶wf 6478 ‘cfv 6482 (class class class)co 7349 ℂcc 11007 ℝcr 11008 0cc0 11009 < clt 11149 ≤ cle 11150 − cmin 11347 / cdiv 11777 ℝ⁺crp 12893 (,)cioo 13248 [,]cicc 13251 abscabs 15141 ↾_t crest 17324 TopOpenctopn 17325 topGenctg 17341 ∞Metcxmet 21246 MetOpencmopn 21251 ℂ_fldccnfld 21261 Topctop 22778 TopOnctopon 22795 intcnt 22902 Cn ccn 23109 CnP ccnp 23110 –cn→ccncf 24767 𝐿¹cibl 25516 ∫citg 25517 lim_ℂ climc 25761 D cdv 25762

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-symdif 4204 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-disj 5060 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-ofr 7614 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-oadd 8392 df-omul 8393 df-er 8625 df-map 8755 df-pm 8756 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-dju 9797 df-card 9835 df-acn 9838 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-ioo 13252 df-ico 13254 df-icc 13255 df-fz 13411 df-fzo 13558 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-rlim 15396 df-sum 15594 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-submnd 18658 df-mulg 18947 df-cntz 19196 df-cmn 19661 df-psmet 21253 df-xmet 21254 df-met 21255 df-bl 21256 df-mopn 21257 df-fbas 21258 df-fg 21259 df-cnfld 21262 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-cld 22904 df-ntr 22905 df-cls 22906 df-nei 22983 df-lp 23021 df-perf 23022 df-cn 23112 df-cnp 23113 df-haus 23200 df-cmp 23272 df-tx 23447 df-hmeo 23640 df-fil 23731 df-fm 23823 df-flim 23824 df-flf 23825 df-xms 24206 df-ms 24207 df-tms 24208 df-cncf 24769 df-ovol 25363 df-vol 25364 df-mbf 25518 df-itg1 25519 df-itg2 25520 df-ibl 25521 df-itg 25522 df-0p 25569 df-limc 25765 df-dv 25766

This theorem is referenced by: ftc2nc 37686

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