ftc1cnnc - Mathbox for Brendan Leahy

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Theorem ftc1cnnc 38610

Description: Choice-free proof of ftc1cn 26363. (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 26227	. . . . 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 6714	. . 3 ⊢ (𝜑 → Fun (ℝ D 𝐺))
4		ax-resscn 11257	. . . . . . 7 ⊢ ℝ ⊆ ℂ
5	4	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℂ)
6		ftc1cnnc.g	. . . . . . 7 ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)
7		ftc1cnnc.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
8		ftc1cnnc.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
9		ftc1cnnc.le	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
10		ssidd 3954	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐵))
11		ioossre 13538	. . . . . . . 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 25214	. . . . . . . 8 ⊢ (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ) → 𝐹:(𝐴(,)𝐵)⟶ℂ)
16	14, 15	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℂ)
17	6, 7, 8, 9, 10, 12, 13, 16	ftc1lem2 26356	. . . . . 6 ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ)
18		iccssre 13560	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
19	7, 8, 18	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
20		tgioo4 25124	. . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
21		eqid 2761	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
22	5, 17, 19, 20, 21	dvbssntr 26220	. . . . 5 ⊢ (𝜑 → dom (ℝ D 𝐺) ⊆ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)))
23		iccntr 25141	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
24	7, 8, 23	syl2anc 596	. . . . 5 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
25	22, 24	sseqtrd 3967	. . . 4 ⊢ (𝜑 → dom (ℝ D 𝐺) ⊆ (𝐴(,)𝐵))
26		retop 25080	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) ∈ Top
27	20, 26	eqeltrri 2858	. . . . . . . . 9 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℝ) ∈ Top
28	27	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → ((TopOpen‘ℂ_fld) ↾_t ℝ) ∈ Top)
29	19	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝐴[,]𝐵) ⊆ ℝ)
30		iooretop 25084	. . . . . . . . . 10 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
31	30, 20	eleqtri 2859	. . . . . . . . 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 13564	. . . . . . . . 9 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
34	33	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
35		uniretop 25081	. . . . . . . . . 10 ⊢ ℝ = ∪ (topGen‘ran (,))
36	20	unieqi 4879	. . . . . . . . . 10 ⊢ ∪ (topGen‘ran (,)) = ∪ ((TopOpen‘ℂ_fld) ↾_t ℝ)
37	35, 36	eqtri 2784	. . . . . . . . 9 ⊢ ℝ = ∪ ((TopOpen‘ℂ_fld) ↾_t ℝ)
38	37	ssntr 23376	. . . . . . . 8 ⊢ (((((TopOpen‘ℂ_fld) ↾_t ℝ) ∈ Top ∧ (𝐴[,]𝐵) ⊆ ℝ) ∧ ((𝐴(,)𝐵) ∈ ((TopOpen‘ℂ_fld) ↾_t ℝ) ∧ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))) → (𝐴(,)𝐵) ⊆ ((int‘((TopOpen‘ℂ_fld) ↾_t ℝ))‘(𝐴[,]𝐵)))
39	28, 29, 32, 34, 38	syl22anc 852	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝐴(,)𝐵) ⊆ ((int‘((TopOpen‘ℂ_fld) ↾_t ℝ))‘(𝐴[,]𝐵)))
40		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → 𝑐 ∈ (𝐴(,)𝐵))
41	39, 40	sseldd 3932	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → 𝑐 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t ℝ))‘(𝐴[,]𝐵)))
42	16	ffvelcdmda 7084	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑐) ∈ ℂ)
43		cnxmet 25091	. . . . . . . . . . . 12 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
44	11, 4	sstri 3940	. . . . . . . . . . . 12 ⊢ (𝐴(,)𝐵) ⊆ ℂ
45		xmetres2 24680	. . . . . . . . . . . 12 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ (𝐴(,)𝐵) ⊆ ℂ) → ((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵))) ∈ (∞Met‘(𝐴(,)𝐵)))
46	43, 44, 45	mp2an 705	. . . . . . . . . . 11 ⊢ ((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵))) ∈ (∞Met‘(𝐴(,)𝐵))
47	46	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ 𝑤 ∈ ℝ⁺) → ((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵))) ∈ (∞Met‘(𝐴(,)𝐵)))
48	43	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ 𝑤 ∈ ℝ⁺) → (abs ∘ − ) ∈ (∞Met‘ℂ))
49		ssid 3953	. . . . . . . . . . . . . . 15 ⊢ ℂ ⊆ ℂ
50		eqid 2761	. . . . . . . . . . . . . . . 16 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) = ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵))
51	21	cnfldtopon 25101	. . . . . . . . . . . . . . . . 17 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
52	51	toponrestid 23239	. . . . . . . . . . . . . . . 16 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
53	21, 50, 52	cncfcn 25231	. . . . . . . . . . . . . . 15 ⊢ (((𝐴(,)𝐵) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴(,)𝐵)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) Cn (TopOpen‘ℂ_fld)))
54	44, 49, 53	mp2an 705	. . . . . . . . . . . . . 14 ⊢ ((𝐴(,)𝐵)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) Cn (TopOpen‘ℂ_fld))
55	14, 54	eleqtrdi 2871	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) Cn (TopOpen‘ℂ_fld)))
56		resttopon 23479	. . . . . . . . . . . . . . . . 17 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ (𝐴(,)𝐵) ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) ∈ (TopOn‘(𝐴(,)𝐵)))
57	51, 44, 56	mp2an 705	. . . . . . . . . . . . . . . 16 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) ∈ (TopOn‘(𝐴(,)𝐵))
58	57	toponunii 23234	. . . . . . . . . . . . . . 15 ⊢ (𝐴(,)𝐵) = ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵))
59	58	eleq2i 2853	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ (𝐴(,)𝐵) ↔ 𝑐 ∈ ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)))
60	59	biimpi 219	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (𝐴(,)𝐵) → 𝑐 ∈ ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)))
61		eqid 2761	. . . . . . . . . . . . . 14 ⊢ ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) = ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵))
62	61	cncnpi 23596	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) Cn (TopOpen‘ℂ_fld)) ∧ 𝑐 ∈ ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵))) → 𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑐))
63	55, 60, 62	syl2an 608	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → 𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑐))
64		eqid 2761	. . . . . . . . . . . . . . . 16 ⊢ ((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵))) = ((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))
65	21	cnfldtopn 25100	. . . . . . . . . . . . . . . 16 ⊢ (TopOpen‘ℂ_fld) = (MetOpen‘(abs ∘ − ))
66		eqid 2761	. . . . . . . . . . . . . . . 16 ⊢ (MetOpen‘((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))) = (MetOpen‘((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵))))
67	64, 65, 66	metrest 24843	. . . . . . . . . . . . . . 15 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ (𝐴(,)𝐵) ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) = (MetOpen‘((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))))
68	43, 44, 67	mp2an 705	. . . . . . . . . . . . . 14 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) = (MetOpen‘((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵))))
69	68	oveq1i 7430	. . . . . . . . . . . . 13 ⊢ (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld)) = ((MetOpen‘((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))) CnP (TopOpen‘ℂ_fld))
70	69	fveq1i 6886	. . . . . . . . . . . 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 2871	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → 𝐹 ∈ (((MetOpen‘((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))) CnP (TopOpen‘ℂ_fld))‘𝑐))
72	71	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ 𝑤 ∈ ℝ⁺) → 𝐹 ∈ (((MetOpen‘((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))) CnP (TopOpen‘ℂ_fld))‘𝑐))
73		simpr 490	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ 𝑤 ∈ ℝ⁺) → 𝑤 ∈ ℝ⁺)
74	66, 65	metcnpi2 24864	. . . . . . . . . 10 ⊢ (((((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵))) ∈ (∞Met‘(𝐴(,)𝐵)) ∧ (abs ∘ − ) ∈ (∞Met‘ℂ)) ∧ (𝐹 ∈ (((MetOpen‘((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))) CnP (TopOpen‘ℂ_fld))‘𝑐) ∧ 𝑤 ∈ ℝ⁺)) → ∃𝑣 ∈ ℝ⁺ ∀𝑢 ∈ (𝐴(,)𝐵)((𝑢((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))𝑐) < 𝑣 → ((𝐹‘𝑢)(abs ∘ − )(𝐹‘𝑐)) < 𝑤))
75	47, 48, 72, 73, 74	syl22anc 852	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ 𝑤 ∈ ℝ⁺) → ∃𝑣 ∈ ℝ⁺ ∀𝑢 ∈ (𝐴(,)𝐵)((𝑢((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))𝑐) < 𝑣 → ((𝐹‘𝑢)(abs ∘ − )(𝐹‘𝑐)) < 𝑤))
76		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑢 ∈ (𝐴(,)𝐵)) → 𝑢 ∈ (𝐴(,)𝐵))
77		simpllr 788	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑢 ∈ (𝐴(,)𝐵)) → 𝑐 ∈ (𝐴(,)𝐵))
78	76, 77	ovresd 7587	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑢 ∈ (𝐴(,)𝐵)) → (𝑢((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))𝑐) = (𝑢(abs ∘ − )𝑐))
79		elioore 13506	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ (𝐴(,)𝐵) → 𝑢 ∈ ℝ)
80	79	recnd 11337	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ (𝐴(,)𝐵) → 𝑢 ∈ ℂ)
81	44	sseli 3927	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 ∈ (𝐴(,)𝐵) → 𝑐 ∈ ℂ)
82	81	ad3antlr 744	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑢 ∈ (𝐴(,)𝐵)) → 𝑐 ∈ ℂ)
83		eqid 2761	. . . . . . . . . . . . . . . . . 18 ⊢ (abs ∘ − ) = (abs ∘ − )
84	83	cnmetdval 25089	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (𝑢(abs ∘ − )𝑐) = (abs‘(𝑢 − 𝑐)))
85	80, 82, 84	syl2an2 699	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑢 ∈ (𝐴(,)𝐵)) → (𝑢(abs ∘ − )𝑐) = (abs‘(𝑢 − 𝑐)))
86	78, 85	eqtrd 2796	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑢 ∈ (𝐴(,)𝐵)) → (𝑢((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))𝑐) = (abs‘(𝑢 − 𝑐)))
87	86	breq1d 5113	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑢 ∈ (𝐴(,)𝐵)) → ((𝑢((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))𝑐) < 𝑣 ↔ (abs‘(𝑢 − 𝑐)) < 𝑣))
88	16	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) → 𝐹:(𝐴(,)𝐵)⟶ℂ)
89	88	ffvelcdmda 7084	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑢 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑢) ∈ ℂ)
90	42	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑢 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑐) ∈ ℂ)
91	83	cnmetdval 25089	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑢) ∈ ℂ ∧ (𝐹‘𝑐) ∈ ℂ) → ((𝐹‘𝑢)(abs ∘ − )(𝐹‘𝑐)) = (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))))
92	89, 90, 91	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑢 ∈ (𝐴(,)𝐵)) → ((𝐹‘𝑢)(abs ∘ − )(𝐹‘𝑐)) = (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))))
93	92	breq1d 5113	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑢 ∈ (𝐴(,)𝐵)) → (((𝐹‘𝑢)(abs ∘ − )(𝐹‘𝑐)) < 𝑤 ↔ (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤))
94	87, 93	imbi12d 347	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑢 ∈ (𝐴(,)𝐵)) → (((𝑢((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))𝑐) < 𝑣 → ((𝐹‘𝑢)(abs ∘ − )(𝐹‘𝑐)) < 𝑤) ↔ ((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)))
95	94	ralbidva 3184	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) → (∀𝑢 ∈ (𝐴(,)𝐵)((𝑢((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))𝑐) < 𝑣 → ((𝐹‘𝑢)(abs ∘ − )(𝐹‘𝑐)) < 𝑤) ↔ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)))
96		simprll 791	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → 𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}))
97		eldifsni 4753	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) → 𝑧 ≠ 𝑐)
98	96, 97	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → 𝑧 ≠ 𝑐)
99	19	ssdifssd 4094	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((𝐴[,]𝐵) ∖ {𝑐}) ⊆ ℝ)
100	99	sselda 3931	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐})) → 𝑧 ∈ ℝ)
101	100	ad2ant2r 760	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤))) → 𝑧 ∈ ℝ)
102	101	ad2ant2r 760	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → 𝑧 ∈ ℝ)
103		elioore 13506	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 ∈ (𝐴(,)𝐵) → 𝑐 ∈ ℝ)
104	103	ad3antlr 744	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → 𝑐 ∈ ℝ)
105	102, 104	lttri2d 11449	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → (𝑧 ≠ 𝑐 ↔ (𝑧 < 𝑐 ∨ 𝑐 < 𝑧)))
106	105	biimpa 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑧 ≠ 𝑐) → (𝑧 < 𝑐 ∨ 𝑐 < 𝑧))
107		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑠 = 𝑧 → (𝐺‘𝑠) = (𝐺‘𝑧))
108	107	oveq1d 7435	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑠 = 𝑧 → ((𝐺‘𝑠) − (𝐺‘𝑐)) = ((𝐺‘𝑧) − (𝐺‘𝑐)))
109		oveq1 7427	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑠 = 𝑧 → (𝑠 − 𝑐) = (𝑧 − 𝑐))
110	108, 109	oveq12d 7438	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑠 = 𝑧 → (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)) = (((𝐺‘𝑧) − (𝐺‘𝑐)) / (𝑧 − 𝑐)))
111		eqid 2761	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐))) = (𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))
112		ovex 7453	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐺‘𝑧) − (𝐺‘𝑐)) / (𝑧 − 𝑐)) ∈ V
113	110, 111, 112	fvmpt 6993	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) → ((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) = (((𝐺‘𝑧) − (𝐺‘𝑐)) / (𝑧 − 𝑐)))
114	113	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣) → ((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) = (((𝐺‘𝑧) − (𝐺‘𝑐)) / (𝑧 − 𝑐)))
115	114	ad2antlr 740	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑧 < 𝑐) → ((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) = (((𝐺‘𝑧) − (𝐺‘𝑐)) / (𝑧 − 𝑐)))
116	17	ad4antr 745	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑧 < 𝑐) → 𝐺:(𝐴[,]𝐵)⟶ℂ)
117		eldifi 4078	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) → 𝑧 ∈ (𝐴[,]𝐵))
118	117	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣) → 𝑧 ∈ (𝐴[,]𝐵))
119	118	ad2antlr 740	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑧 < 𝑐) → 𝑧 ∈ (𝐴[,]𝐵))
120	116, 119	ffvelcdmd 7085	. . . . . . . . . . . . . . . . . . . . . . 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 7084	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴[,]𝐵)) → (𝐺‘𝑐) ∈ ℂ)
123	121, 122	sylan2 605	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑐) ∈ ℂ)
124	123	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑧 < 𝑐) → (𝐺‘𝑐) ∈ ℂ)
125	102	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑧 < 𝑐) → 𝑧 ∈ ℝ)
126	125	recnd 11337	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑧 < 𝑐) → 𝑧 ∈ ℂ)
127	81	ad4antlr 746	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑧 < 𝑐) → 𝑐 ∈ ℂ)
128		ltne 11407	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑧 ∈ ℝ ∧ 𝑧 < 𝑐) → 𝑐 ≠ 𝑧)
129	128	necomd 3011	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑧 ∈ ℝ ∧ 𝑧 < 𝑐) → 𝑧 ≠ 𝑐)
130	102, 129	sylan 592	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑧 < 𝑐) → 𝑧 ≠ 𝑐)
131	120, 124, 126, 127, 130	div2subd 12143	. . . . . . . . . . . . . . . . . . . . . 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 2796	. . . . . . . . . . . . . . . . . . . . 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 7442	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑧 < 𝑐) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) = (abs‘((((𝐺‘𝑐) − (𝐺‘𝑧)) / (𝑐 − 𝑧)) − (𝐹‘𝑐))))
134	7	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → 𝐴 ∈ ℝ)
135	8	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → 𝐵 ∈ ℝ)
136	9	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → 𝐴 ≤ 𝐵)
137	14	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))
138	13	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → 𝐹 ∈ 𝐿¹)
139		simpllr 788	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → 𝑐 ∈ (𝐴(,)𝐵))
140		simplrl 789	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → 𝑤 ∈ ℝ⁺)
141		simplrr 790	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → 𝑣 ∈ ℝ⁺)
142		simprlr 792	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤))
143		fvoveq1 7443	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑢 = 𝑦 → (abs‘(𝑢 − 𝑐)) = (abs‘(𝑦 − 𝑐)))
144	143	breq1d 5113	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑢 = 𝑦 → ((abs‘(𝑢 − 𝑐)) < 𝑣 ↔ (abs‘(𝑦 − 𝑐)) < 𝑣))
145	144	imbrov2fvoveq 7445	. . . . . . . . . . . . . . . . . . . . . . 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 592	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((abs‘(𝑦 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑐))) < 𝑤))
148	96, 117	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → 𝑧 ∈ (𝐴[,]𝐵))
149		simprr 785	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → (abs‘(𝑧 − 𝑐)) < 𝑣)
150	121	ad3antlr 744	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → 𝑐 ∈ (𝐴[,]𝐵))
151	103	recnd 11337	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑐 ∈ (𝐴(,)𝐵) → 𝑐 ∈ ℂ)
152	151	subidd 11657	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 ∈ (𝐴(,)𝐵) → (𝑐 − 𝑐) = 0)
153	152	abs00bd 15458	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑐 ∈ (𝐴(,)𝐵) → (abs‘(𝑐 − 𝑐)) = 0)
154	153	ad3antlr 744	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → (abs‘(𝑐 − 𝑐)) = 0)
155	141	rpgt0d 13167	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → 0 < 𝑣)
156	154, 155	eqbrtrd 5127	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → (abs‘(𝑐 − 𝑐)) < 𝑣)
157	6, 134, 135, 136, 137, 138, 139, 111, 140, 141, 147, 148, 149, 150, 156	ftc1cnnclem 38609	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑧 < 𝑐) → (abs‘((((𝐺‘𝑐) − (𝐺‘𝑧)) / (𝑐 − 𝑧)) − (𝐹‘𝑐))) < 𝑤)
158	133, 157	eqbrtrd 5127	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑧 < 𝑐) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) < 𝑤)
159	113	fvoveq1d 7442	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) = (abs‘((((𝐺‘𝑧) − (𝐺‘𝑐)) / (𝑧 − 𝑐)) − (𝐹‘𝑐))))
160	159	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) = (abs‘((((𝐺‘𝑧) − (𝐺‘𝑐)) / (𝑧 − 𝑐)) − (𝐹‘𝑐))))
161	160	ad2antlr 740	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑐 < 𝑧) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) = (abs‘((((𝐺‘𝑧) − (𝐺‘𝑐)) / (𝑧 − 𝑐)) − (𝐹‘𝑐))))
162	6, 134, 135, 136, 137, 138, 139, 111, 140, 141, 147, 150, 156, 148, 149	ftc1cnnclem 38609	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑐 < 𝑧) → (abs‘((((𝐺‘𝑧) − (𝐺‘𝑐)) / (𝑧 − 𝑐)) − (𝐹‘𝑐))) < 𝑤)
163	161, 162	eqbrtrd 5127	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑐 < 𝑧) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) < 𝑤)
164	158, 163	jaodan 972	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ (𝑧 < 𝑐 ∨ 𝑐 < 𝑧)) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) < 𝑤)
165	106, 164	syldan 603	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) ∧ 𝑧 ≠ 𝑐) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) < 𝑤)
166	98, 165	mpdan 700	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤)) ∧ (abs‘(𝑧 − 𝑐)) < 𝑣)) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) < 𝑤)
167	166	expr 462	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤))) → ((abs‘(𝑧 − 𝑐)) < 𝑣 → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) < 𝑤))
168	167	adantld 496	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ∧ ∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤))) → ((𝑧 ≠ 𝑐 ∧ (abs‘(𝑧 − 𝑐)) < 𝑣) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) < 𝑤))
169	168	expr 462	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐})) → (∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤) → ((𝑧 ≠ 𝑐 ∧ (abs‘(𝑧 − 𝑐)) < 𝑣) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) < 𝑤)))
170	169	ralrimdva 3163	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) → (∀𝑢 ∈ (𝐴(,)𝐵)((abs‘(𝑢 − 𝑐)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝑐))) < 𝑤) → ∀𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐})((𝑧 ≠ 𝑐 ∧ (abs‘(𝑧 − 𝑐)) < 𝑣) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) < 𝑤)))
171	95, 170	sylbid 243	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) → (∀𝑢 ∈ (𝐴(,)𝐵)((𝑢((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))𝑐) < 𝑣 → ((𝐹‘𝑢)(abs ∘ − )(𝐹‘𝑐)) < 𝑤) → ∀𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐})((𝑧 ≠ 𝑐 ∧ (abs‘(𝑧 − 𝑐)) < 𝑣) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) < 𝑤)))
172	171	anassrs 473	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑣 ∈ ℝ⁺) → (∀𝑢 ∈ (𝐴(,)𝐵)((𝑢((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))𝑐) < 𝑣 → ((𝐹‘𝑢)(abs ∘ − )(𝐹‘𝑐)) < 𝑤) → ∀𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐})((𝑧 ≠ 𝑐 ∧ (abs‘(𝑧 − 𝑐)) < 𝑣) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) < 𝑤)))
173	172	reximdva 3176	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ 𝑤 ∈ ℝ⁺) → (∃𝑣 ∈ ℝ⁺ ∀𝑢 ∈ (𝐴(,)𝐵)((𝑢((abs ∘ − ) ↾ ((𝐴(,)𝐵) × (𝐴(,)𝐵)))𝑐) < 𝑣 → ((𝐹‘𝑢)(abs ∘ − )(𝐹‘𝑐)) < 𝑤) → ∃𝑣 ∈ ℝ⁺ ∀𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐})((𝑧 ≠ 𝑐 ∧ (abs‘(𝑧 − 𝑐)) < 𝑣) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) < 𝑤)))
174	75, 173	mpd 16	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ 𝑤 ∈ ℝ⁺) → ∃𝑣 ∈ ℝ⁺ ∀𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐})((𝑧 ≠ 𝑐 ∧ (abs‘(𝑧 − 𝑐)) < 𝑣) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) < 𝑤))
175	174	ralrimiva 3155	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → ∀𝑤 ∈ ℝ⁺ ∃𝑣 ∈ ℝ⁺ ∀𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐})((𝑧 ≠ 𝑐 ∧ (abs‘(𝑧 − 𝑐)) < 𝑣) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) < 𝑤))
176	17	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → 𝐺:(𝐴[,]𝐵)⟶ℂ)
177	19, 4	sstrdi 3943	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
178	177	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝐴[,]𝐵) ⊆ ℂ)
179	121	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → 𝑐 ∈ (𝐴[,]𝐵))
180	176, 178, 179	dvlem 26216	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) ∧ 𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐})) → (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)) ∈ ℂ)
181	180	fmpttd 7115	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐))):((𝐴[,]𝐵) ∖ {𝑐})⟶ℂ)
182	177	ssdifssd 4094	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴[,]𝐵) ∖ {𝑐}) ⊆ ℂ)
183	182	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → ((𝐴[,]𝐵) ∖ {𝑐}) ⊆ ℂ)
184	81	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → 𝑐 ∈ ℂ)
185	181, 183, 184	ellimc3 26199	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → ((𝐹‘𝑐) ∈ ((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐))) lim_ℂ 𝑐) ↔ ((𝐹‘𝑐) ∈ ℂ ∧ ∀𝑤 ∈ ℝ⁺ ∃𝑣 ∈ ℝ⁺ ∀𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐})((𝑧 ≠ 𝑐 ∧ (abs‘(𝑧 − 𝑐)) < 𝑣) → (abs‘(((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐)))‘𝑧) − (𝐹‘𝑐))) < 𝑤))))
186	42, 175, 185	mpbir2and 726	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑐) ∈ ((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐))) lim_ℂ 𝑐))
187		eqid 2761	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℝ) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
188	187, 21, 111, 5, 17, 19	eldv 26218	. . . . . . 7 ⊢ (𝜑 → (𝑐(ℝ D 𝐺)(𝐹‘𝑐) ↔ (𝑐 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t ℝ))‘(𝐴[,]𝐵)) ∧ (𝐹‘𝑐) ∈ ((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐))) lim_ℂ 𝑐))))
189	188	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → (𝑐(ℝ D 𝐺)(𝐹‘𝑐) ↔ (𝑐 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t ℝ))‘(𝐴[,]𝐵)) ∧ (𝐹‘𝑐) ∈ ((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑠) − (𝐺‘𝑐)) / (𝑠 − 𝑐))) lim_ℂ 𝑐))))
190	41, 186, 189	mpbir2and 726	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → 𝑐(ℝ D 𝐺)(𝐹‘𝑐))
191		vex 3455	. . . . . 6 ⊢ 𝑐 ∈ V
192		fvex 6898	. . . . . 6 ⊢ (𝐹‘𝑐) ∈ V
193	191, 192	breldm 5890	. . . . 5 ⊢ (𝑐(ℝ D 𝐺)(𝐹‘𝑐) → 𝑐 ∈ dom (ℝ D 𝐺))
194	190, 193	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → 𝑐 ∈ dom (ℝ D 𝐺))
195	25, 194	eqelssd 3952	. . 3 ⊢ (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵))
196		df-fn 6541	. . 3 ⊢ ((ℝ D 𝐺) Fn (𝐴(,)𝐵) ↔ (Fun (ℝ D 𝐺) ∧ dom (ℝ D 𝐺) = (𝐴(,)𝐵)))
197	3, 195, 196	sylanbrc 595	. 2 ⊢ (𝜑 → (ℝ D 𝐺) Fn (𝐴(,)𝐵))
198	16	ffnd 6710	. 2 ⊢ (𝜑 → 𝐹 Fn (𝐴(,)𝐵))
199	3	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → Fun (ℝ D 𝐺))
200		funbrfv 6933	. . 3 ⊢ (Fun (ℝ D 𝐺) → (𝑐(ℝ D 𝐺)(𝐹‘𝑐) → ((ℝ D 𝐺)‘𝑐) = (𝐹‘𝑐)))
201	199, 190, 200	sylc 66	. 2 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐺)‘𝑐) = (𝐹‘𝑐))
202	197, 198, 201	eqfnfvd 7032	1 ⊢ (𝜑 → (ℝ D 𝐺) = 𝐹)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ∀wral 3077 ∃wrex 3087 ∖ cdif 3896 ⊆ wss 3899 {csn 4584 ∪ cuni 4867 class class class wbr 5103 ↦ cmpt 5186 × cxp 5649 dom cdm 5651 ran crn 5652 ↾ cres 5653 ∘ ccom 5655 Fun wfun 6532 Fn wfn 6533 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 ℂcc 11198 ℝcr 11199 0cc0 11200 < clt 11343 ≤ cle 11344 − cmin 11541 / cdiv 11973 ℝ⁺crp 13120 (,)cioo 13476 [,]cicc 13479 abscabs 15401 ↾_t crest 17591 TopOpenctopn 17592 topGenctg 17608 ∞Metcxmet 21663 MetOpencmopn 21668 ℂ_fldccnfld 21678 Topctop 23211 TopOnctopon 23228 intcnt 23335 Cn ccn 23542 CnP ccnp 23543 –cn→ccncf 25197 𝐿¹cibl 25938 ∫citg 25939 lim_ℂ climc 26182 D cdv 26183

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-inf2 9642 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278 ax-addf 11279

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-symdif 4199 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-disj 5071 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-ofr 7694 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-oadd 8480 df-omul 8481 df-er 8717 df-map 8849 df-pm 8850 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9354 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9504 df-dju 9982 df-card 10020 df-acn 10023 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-z 12694 df-dec 12815 df-uz 12966 df-q 13076 df-rp 13121 df-xneg 13241 df-xadd 13242 df-xmul 13243 df-ioo 13480 df-ico 13482 df-icc 13483 df-fz 13640 df-fzo 13789 df-fl 13932 df-mod 14010 df-seq 14145 df-exp 14205 df-hash 14475 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-clim 15655 df-rlim 15656 df-sum 15854 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-starv 17443 df-sca 17444 df-vsca 17445 df-ip 17446 df-tset 17447 df-ple 17448 df-ds 17450 df-unif 17451 df-hom 17452 df-cco 17453 df-rest 17593 df-topn 17594 df-0g 17612 df-gsum 17613 df-topgen 17614 df-pt 17615 df-prds 17618 df-xrs 17674 df-qtop 17679 df-imas 17680 df-xps 17682 df-mre 17756 df-mrc 17757 df-acs 17759 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-submnd 18979 df-mulg 19278 df-cntz 19531 df-cmn 19996 df-psmet 21670 df-xmet 21671 df-met 21672 df-bl 21673 df-mopn 21674 df-fbas 21675 df-fg 21676 df-cnfld 21679 df-top 23212 df-topon 23229 df-topsp 23251 df-bases 23264 df-cld 23337 df-ntr 23338 df-cls 23339 df-nei 23416 df-lp 23454 df-perf 23455 df-cn 23545 df-cnp 23546 df-haus 23633 df-cmp 23705 df-tx 23881 df-hmeo 24074 df-fil 24165 df-fm 24257 df-flim 24258 df-flf 24259 df-xms 24639 df-ms 24640 df-tms 24641 df-cncf 25199 df-ovol 25785 df-vol 25786 df-mbf 25940 df-itg1 25941 df-itg2 25942 df-ibl 25943 df-itg 25944 df-0p 25991 df-limc 26186 df-dv 26187

This theorem is used by: ftc2nc 38620

W3C validator