ftc1cnnc - Mathbox for Brendan Leahy

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Theorem ftc1cnnc 37832

Description: Choice-free proof of ftc1cn 26004. (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 25862	. . . . 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 6664	. . 3 ⊢ (𝜑 → Fun (ℝ D 𝐺))
4		ax-resscn 11081	. . . . . . 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 3955	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐵))
11		ioossre 13321	. . . . . . . 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 24840	. . . . . . . 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 25997	. . . . . 6 ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ)
18		iccssre 13343	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
19	7, 8, 18	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
20		tgioo4 24747	. . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
21		eqid 2734	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
22	5, 17, 19, 20, 21	dvbssntr 25855	. . . . 5 ⊢ (𝜑 → dom (ℝ D 𝐺) ⊆ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)))
23		iccntr 24764	. . . . . 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 3968	. . . 4 ⊢ (𝜑 → dom (ℝ D 𝐺) ⊆ (𝐴(,)𝐵))
26		retop 24703	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) ∈ Top
27	20, 26	eqeltrri 2831	. . . . . . . . 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 24707	. . . . . . . . . 10 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
31	30, 20	eleqtri 2832	. . . . . . . . 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 13347	. . . . . . . . 9 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
34	33	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
35		uniretop 24704	. . . . . . . . . 10 ⊢ ℝ = ∪ (topGen‘ran (,))
36	20	unieqi 4873	. . . . . . . . . 10 ⊢ ∪ (topGen‘ran (,)) = ∪ ((TopOpen‘ℂ_fld) ↾_t ℝ)
37	35, 36	eqtri 2757	. . . . . . . . 9 ⊢ ℝ = ∪ ((TopOpen‘ℂ_fld) ↾_t ℝ)
38	37	ssntr 23000	. . . . . . . 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 3932	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → 𝑐 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t ℝ))‘(𝐴[,]𝐵)))
42	16	ffvelcdmda 7027	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑐) ∈ ℂ)
43		cnxmet 24714	. . . . . . . . . . . 12 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
44	11, 4	sstri 3941	. . . . . . . . . . . 12 ⊢ (𝐴(,)𝐵) ⊆ ℂ
45		xmetres2 24303	. . . . . . . . . . . 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 3954	. . . . . . . . . . . . . . 15 ⊢ ℂ ⊆ ℂ
50		eqid 2734	. . . . . . . . . . . . . . . 16 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) = ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵))
51	21	cnfldtopon 24724	. . . . . . . . . . . . . . . . 17 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
52	51	toponrestid 22863	. . . . . . . . . . . . . . . 16 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
53	21, 50, 52	cncfcn 24857	. . . . . . . . . . . . . . 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 2844	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) Cn (TopOpen‘ℂ_fld)))
56		resttopon 23103	. . . . . . . . . . . . . . . . 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 22858	. . . . . . . . . . . . . . 15 ⊢ (𝐴(,)𝐵) = ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵))
59	58	eleq2i 2826	. . . . . . . . . . . . . 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 23220	. . . . . . . . . . . . 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	21	cnfldtopn 24723	. . . . . . . . . . . . . . . 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 24466	. . . . . . . . . . . . . . 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 7366	. . . . . . . . . . . . 13 ⊢ (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld)) = ((MetOpen‘((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))) CnP (TopOpen‘ℂ_fld))
70	69	fveq1i 6833	. . . . . . . . . . . 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 2844	. . . . . . . . . . 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 24487	. . . . . . . . . 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 7523	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑢 ∈ (𝐴(,)𝐵)) → (𝑢((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))𝑐) = (𝑢(abs ∘ − )𝑐))
79		elioore 13289	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ (𝐴(,)𝐵) → 𝑢 ∈ ℝ)
80	79	recnd 11158	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ (𝐴(,)𝐵) → 𝑢 ∈ ℂ)
81	44	sseli 3927	. . . . . . . . . . . . . . . . . 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 24712	. . . . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . 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 5106	. . . . . . . . . . . . . 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 7027	. . . . . . . . . . . . . . . 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 24712	. . . . . . . . . . . . . . . 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 5106	. . . . . . . . . . . . . 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 3155	. . . . . . . . . . . 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 4744	. . . . . . . . . . . . . . . . . 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 4097	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((𝐴[,]𝐵) ∖ {𝑐}) ⊆ ℝ)
100	99	sselda 3931	. . . . . . . . . . . . . . . . . . . . . 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 13289	. . . . . . . . . . . . . . . . . . . . 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 11270	. . . . . . . . . . . . . . . . . . 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 6832	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑠 = 𝑧 → (𝐺‘𝑠) = (𝐺‘𝑧))
108	107	oveq1d 7371	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑠 = 𝑧 → ((𝐺‘𝑠) − (𝐺‘𝑐)) = ((𝐺‘𝑧) − (𝐺‘𝑐)))
109		oveq1 7363	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑠 = 𝑧 → (𝑠 − 𝑐) = (𝑧 − 𝑐))
110	108, 109	oveq12d 7374	. . . . . . . . . . . . . . . . . . . . . . . . 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 7389	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐺‘𝑧) − (𝐺‘𝑐)) / (𝑧 − 𝑐)) ∈ V
113	110, 111, 112	fvmpt 6939	. . . . . . . . . . . . . . . . . . . . . . . 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 4081	. . . . . . . . . . . . . . . . . . . . . . . . . 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 7028	. . . . . . . . . . . . . . . . . . . . . . 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 7027	. . . . . . . . . . . . . . . . . . . . . . . . 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 11158	. . . . . . . . . . . . . . . . . . . . . . 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 11228	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑧 ∈ ℝ ∧ 𝑧 < 𝑐) → 𝑐 ≠ 𝑧)
129	128	necomd 2985	. . . . . . . . . . . . . . . . . . . . . . . 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 11965	. . . . . . . . . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . . . . . . 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 7378	. . . . . . . . . . . . . . . . . . . 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 7379	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑢 = 𝑦 → (abs‘(𝑢 − 𝑐)) = (abs‘(𝑦 − 𝑐)))
144	143	breq1d 5106	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑢 = 𝑦 → ((abs‘(𝑢 − 𝑐)) < 𝑣 ↔ (abs‘(𝑦 − 𝑐)) < 𝑣))
145	144	imbrov2fvoveq 7381	. . . . . . . . . . . . . . . . . . . . . . 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 3573	. . . . . . . . . . . . . . . . . . . . . 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 11158	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑐 ∈ (𝐴(,)𝐵) → 𝑐 ∈ ℂ)
152	151	subidd 11478	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 ∈ (𝐴(,)𝐵) → (𝑐 − 𝑐) = 0)
153	152	abs00bd 15212	. . . . . . . . . . . . . . . . . . . . . . 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 12950	. . . . . . . . . . . . . . . . . . . . . 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 5118	. . . . . . . . . . . . . . . . . . . . 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 37831	. . . . . . . . . . . . . . . . . . . 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 5118	. . . . . . . . . . . . . . . . . . 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 7378	. . . . . . . . . . . . . . . . . . . . . 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 37831	. . . . . . . . . . . . . . . . . . . 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 5118	. . . . . . . . . . . . . . . . . . 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 3134	. . . . . . . . . . . 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 3147	. . . . . . . . 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 3126	. . . . . . 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 3944	. . . . . . . . . . 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 25851	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ 𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐})) → (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)) ∈ ℂ)
181	180	fmpttd 7058	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐))):((𝐴[,]𝐵) ∖ {𝑐})⟶ℂ)
182	177	ssdifssd 4097	. . . . . . . . 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 25834	. . . . . . 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, 21, 111, 5, 17, 19	eldv 25853	. . . . . . 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 3442	. . . . . 6 ⊢ 𝑐 ∈ V
192		fvex 6845	. . . . . 6 ⊢ (𝐹‘𝑐) ∈ V
193	191, 192	breldm 5855	. . . . 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 3953	. . 3 ⊢ (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵))
196		df-fn 6493	. . 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 6661	. 2 ⊢ (𝜑 → 𝐹 Fn (𝐴(,)𝐵))
199	3	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → Fun (ℝ D 𝐺))
200		funbrfv 6880	. . 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 6977	1 ⊢ (𝜑 → (ℝ D 𝐺) = 𝐹)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1541 ∈ wcel 2113 ≠ wne 2930 ∀wral 3049 ∃wrex 3058 ∖ cdif 3896 ⊆ wss 3899 {csn 4578 ∪ cuni 4861 class class class wbr 5096 ↦ cmpt 5177 × cxp 5620 dom cdm 5622 ran crn 5623 ↾ cres 5624 ∘ ccom 5626 Fun wfun 6484 Fn wfn 6485 ⟶wf 6486 ‘cfv 6490 (class class class)co 7356 ℂcc 11022 ℝcr 11023 0cc0 11024 < clt 11164 ≤ cle 11165 − cmin 11362 / cdiv 11792 ℝ⁺crp 12903 (,)cioo 13259 [,]cicc 13262 abscabs 15155 ↾_t crest 17338 TopOpenctopn 17339 topGenctg 17355 ∞Metcxmet 21292 MetOpencmopn 21297 ℂ_fldccnfld 21307 Topctop 22835 TopOnctopon 22852 intcnt 22959 Cn ccn 23166 CnP ccnp 23167 –cn→ccncf 24823 𝐿¹cibl 25572 ∫citg 25573 lim_ℂ climc 25817 D cdv 25818

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-inf2 9548 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102 ax-addf 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-symdif 4203 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-iin 4947 df-disj 5064 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-ofr 7621 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-omul 8400 df-er 8633 df-map 8763 df-pm 8764 df-ixp 8834 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-fsupp 9263 df-fi 9312 df-sup 9343 df-inf 9344 df-oi 9413 df-dju 9811 df-card 9849 df-acn 9852 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-z 12487 df-dec 12606 df-uz 12750 df-q 12860 df-rp 12904 df-xneg 13024 df-xadd 13025 df-xmul 13026 df-ioo 13263 df-ico 13265 df-icc 13266 df-fz 13422 df-fzo 13569 df-fl 13710 df-mod 13788 df-seq 13923 df-exp 13983 df-hash 14252 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-clim 15409 df-rlim 15410 df-sum 15608 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-starv 17190 df-sca 17191 df-vsca 17192 df-ip 17193 df-tset 17194 df-ple 17195 df-ds 17197 df-unif 17198 df-hom 17199 df-cco 17200 df-rest 17340 df-topn 17341 df-0g 17359 df-gsum 17360 df-topgen 17361 df-pt 17362 df-prds 17365 df-xrs 17421 df-qtop 17426 df-imas 17427 df-xps 17429 df-mre 17503 df-mrc 17504 df-acs 17506 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18707 df-mulg 18996 df-cntz 19244 df-cmn 19709 df-psmet 21299 df-xmet 21300 df-met 21301 df-bl 21302 df-mopn 21303 df-fbas 21304 df-fg 21305 df-cnfld 21308 df-top 22836 df-topon 22853 df-topsp 22875 df-bases 22888 df-cld 22961 df-ntr 22962 df-cls 22963 df-nei 23040 df-lp 23078 df-perf 23079 df-cn 23169 df-cnp 23170 df-haus 23257 df-cmp 23329 df-tx 23504 df-hmeo 23697 df-fil 23788 df-fm 23880 df-flim 23881 df-flf 23882 df-xms 24262 df-ms 24263 df-tms 24264 df-cncf 24825 df-ovol 25419 df-vol 25420 df-mbf 25574 df-itg1 25575 df-itg2 25576 df-ibl 25577 df-itg 25578 df-0p 25625 df-limc 25821 df-dv 25822

This theorem is referenced by: ftc2nc 37842

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