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Theorem ftc1lem6 26168

Description: Lemma for ftc1 26169. (Contributed by Mario Carneiro, 14-Aug-2014.) (Proof shortened by Mario Carneiro, 28-Dec-2016.)

**Hypotheses**
Ref	Expression
ftc1.g	⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)
ftc1.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
ftc1.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
ftc1.le	⊢ (𝜑 → 𝐴 ≤ 𝐵)
ftc1.s	⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)
ftc1.d	⊢ (𝜑 → 𝐷 ⊆ ℝ)
ftc1.i	⊢ (𝜑 → 𝐹 ∈ 𝐿¹)
ftc1.c	⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵))
ftc1.f	⊢ (𝜑 → 𝐹 ∈ ((𝐾 CnP 𝐿)‘𝐶))
ftc1.j	⊢ 𝐽 = (𝐿 ↾_t ℝ)
ftc1.k	⊢ 𝐾 = (𝐿 ↾_t 𝐷)
ftc1.l	⊢ 𝐿 = (TopOpen‘ℂ_fld)
ftc1.h	⊢ 𝐻 = (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))

**Assertion**
Ref	Expression
ftc1lem6	⊢ (𝜑 → (𝐹‘𝐶) ∈ (𝐻 lim_ℂ 𝐶))

Distinct variable groups: 𝑥,𝑡,𝑧,𝐶 𝑡,𝐷,𝑥,𝑧 𝑧,𝐺 𝑡,𝐴,𝑥,𝑧 𝑡,𝐵,𝑥,𝑧 𝜑,𝑡,𝑥,𝑧 𝑡,𝐹,𝑥,𝑧 𝑥,𝐿,𝑧 Allowed substitution hints: 𝐺(𝑥,𝑡) 𝐻(𝑥,𝑧,𝑡) 𝐽(𝑥,𝑧,𝑡) 𝐾(𝑥,𝑧,𝑡) 𝐿(𝑡)

Proof of Theorem ftc1lem6 Dummy variables 𝑠 𝑢 𝑣 𝑤 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ftc1.g	. . . 4 ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)
2		ftc1.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ)
3		ftc1.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ)
4		ftc1.le	. . . 4 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
5		ftc1.s	. . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)
6		ftc1.d	. . . 4 ⊢ (𝜑 → 𝐷 ⊆ ℝ)
7		ftc1.i	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐿¹)
8		ftc1.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵))
9		ftc1.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝐾 CnP 𝐿)‘𝐶))
10		ftc1.j	. . . 4 ⊢ 𝐽 = (𝐿 ↾_t ℝ)
11		ftc1.k	. . . 4 ⊢ 𝐾 = (𝐿 ↾_t 𝐷)
12		ftc1.l	. . . 4 ⊢ 𝐿 = (TopOpen‘ℂ_fld)
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	ftc1lem3 26165	. . 3 ⊢ (𝜑 → 𝐹:𝐷⟶ℂ)
14	5, 8	sseldd 3946	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐷)
15	13, 14	ffvelcdmd 7081	. 2 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ℂ)
16		cnxmet 24897	. . . . . 6 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
17		ax-resscn 11156	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
18	6, 17	sstrdi 3957	. . . . . . 7 ⊢ (𝜑 → 𝐷 ⊆ ℂ)
19	18	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → 𝐷 ⊆ ℂ)
20		xmetres2 24486	. . . . . 6 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐷 ⊆ ℂ) → ((abs ∘ − ) ↾ (𝐷 × 𝐷)) ∈ (∞Met‘𝐷))
21	16, 19, 20	sylancr 598	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → ((abs ∘ − ) ↾ (𝐷 × 𝐷)) ∈ (∞Met‘𝐷))
22	16	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → (abs ∘ − ) ∈ (∞Met‘ℂ))
23		eqid 2769	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ↾ (𝐷 × 𝐷)) = ((abs ∘ − ) ↾ (𝐷 × 𝐷))
24	12	cnfldtopn 24906	. . . . . . . . . . . 12 ⊢ 𝐿 = (MetOpen‘(abs ∘ − ))
25		eqid 2769	. . . . . . . . . . . 12 ⊢ (MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))) = (MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷)))
26	23, 24, 25	metrest 24649	. . . . . . . . . . 11 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐷 ⊆ ℂ) → (𝐿 ↾_t 𝐷) = (MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))))
27	16, 18, 26	sylancr 598	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿 ↾_t 𝐷) = (MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))))
28	11, 27	eqtrid 2816	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 = (MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))))
29	28	oveq1d 7426	. . . . . . . 8 ⊢ (𝜑 → (𝐾 CnP 𝐿) = ((MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))) CnP 𝐿))
30	29	fveq1d 6884	. . . . . . 7 ⊢ (𝜑 → ((𝐾 CnP 𝐿)‘𝐶) = (((MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))) CnP 𝐿)‘𝐶))
31	9, 30	eleqtrd 2871	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (((MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))) CnP 𝐿)‘𝐶))
32	31	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → 𝐹 ∈ (((MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))) CnP 𝐿)‘𝐶))
33		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → 𝑤 ∈ ℝ⁺)
34	25, 24	metcnpi2 24670	. . . . 5 ⊢ (((((abs ∘ − ) ↾ (𝐷 × 𝐷)) ∈ (∞Met‘𝐷) ∧ (abs ∘ − ) ∈ (∞Met‘ℂ)) ∧ (𝐹 ∈ (((MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))) CnP 𝐿)‘𝐶) ∧ 𝑤 ∈ ℝ⁺)) → ∃𝑣 ∈ ℝ⁺ ∀𝑦 ∈ 𝐷 ((𝑦((abs ∘ − ) ↾ (𝐷 × 𝐷))𝐶) < 𝑣 → ((𝐹‘𝑦)(abs ∘ − )(𝐹‘𝐶)) < 𝑤))
35	21, 22, 32, 33, 34	syl22anc 851	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → ∃𝑣 ∈ ℝ⁺ ∀𝑦 ∈ 𝐷 ((𝑦((abs ∘ − ) ↾ (𝐷 × 𝐷))𝐶) < 𝑣 → ((𝐹‘𝑦)(abs ∘ − )(𝐹‘𝐶)) < 𝑤))
36		simpr 489	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → 𝑦 ∈ 𝐷)
37	14	ad2antrr 738	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → 𝐶 ∈ 𝐷)
38	36, 37	ovresd 7578	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → (𝑦((abs ∘ − ) ↾ (𝐷 × 𝐷))𝐶) = (𝑦(abs ∘ − )𝐶))
39	18	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) → 𝐷 ⊆ ℂ)
40	39	sselda 3945	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → 𝑦 ∈ ℂ)
41		iccssre 13455	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
42	2, 3, 41	syl2anc 595	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
43	42, 17	sstrdi 3957	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
44		ioossicc 13459	. . . . . . . . . . . . . . 15 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
45	44, 8	sselid 3943	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵))
46	43, 45	sseldd 3946	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ ℂ)
47	46	ad2antrr 738	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → 𝐶 ∈ ℂ)
48		eqid 2769	. . . . . . . . . . . . 13 ⊢ (abs ∘ − ) = (abs ∘ − )
49	48	cnmetdval 24895	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝑦(abs ∘ − )𝐶) = (abs‘(𝑦 − 𝐶)))
50	40, 47, 49	syl2anc 595	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → (𝑦(abs ∘ − )𝐶) = (abs‘(𝑦 − 𝐶)))
51	38, 50	eqtrd 2804	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → (𝑦((abs ∘ − ) ↾ (𝐷 × 𝐷))𝐶) = (abs‘(𝑦 − 𝐶)))
52	51	breq1d 5123	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → ((𝑦((abs ∘ − ) ↾ (𝐷 × 𝐷))𝐶) < 𝑣 ↔ (abs‘(𝑦 − 𝐶)) < 𝑣))
53	13	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) → 𝐹:𝐷⟶ℂ)
54	53	ffvelcdmda 7080	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → (𝐹‘𝑦) ∈ ℂ)
55	15	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → (𝐹‘𝐶) ∈ ℂ)
56	48	cnmetdval 24895	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑦) ∈ ℂ ∧ (𝐹‘𝐶) ∈ ℂ) → ((𝐹‘𝑦)(abs ∘ − )(𝐹‘𝐶)) = (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))))
57	54, 55, 56	syl2anc 595	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → ((𝐹‘𝑦)(abs ∘ − )(𝐹‘𝐶)) = (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))))
58	57	breq1d 5123	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → (((𝐹‘𝑦)(abs ∘ − )(𝐹‘𝐶)) < 𝑤 ↔ (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝑤))
59	52, 58	imbi12d 347	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → (((𝑦((abs ∘ − ) ↾ (𝐷 × 𝐷))𝐶) < 𝑣 → ((𝐹‘𝑦)(abs ∘ − )(𝐹‘𝐶)) < 𝑤) ↔ ((abs‘(𝑦 − 𝐶)) < 𝑣 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝑤)))
60	59	ralbidva 3192	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) → (∀𝑦 ∈ 𝐷 ((𝑦((abs ∘ − ) ↾ (𝐷 × 𝐷))𝐶) < 𝑣 → ((𝐹‘𝑦)(abs ∘ − )(𝐹‘𝐶)) < 𝑤) ↔ ∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑣 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝑤)))
61		simprll 790	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ∧ ∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑣 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝑤)) ∧ (abs‘(𝑠 − 𝐶)) < 𝑣)) → 𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝐶}))
62		eldifsni 4762	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) → 𝑠 ≠ 𝐶)
63	61, 62	syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ∧ ∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑣 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝑤)) ∧ (abs‘(𝑠 − 𝐶)) < 𝑣)) → 𝑠 ≠ 𝐶)
64	2	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ∧ ∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑣 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝑤)) ∧ (abs‘(𝑠 − 𝐶)) < 𝑣)) → 𝐴 ∈ ℝ)
65	3	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ∧ ∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑣 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝑤)) ∧ (abs‘(𝑠 − 𝐶)) < 𝑣)) → 𝐵 ∈ ℝ)
66	4	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ∧ ∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑣 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝑤)) ∧ (abs‘(𝑠 − 𝐶)) < 𝑣)) → 𝐴 ≤ 𝐵)
67	5	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ∧ ∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑣 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝑤)) ∧ (abs‘(𝑠 − 𝐶)) < 𝑣)) → (𝐴(,)𝐵) ⊆ 𝐷)
68	6	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ∧ ∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑣 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝑤)) ∧ (abs‘(𝑠 − 𝐶)) < 𝑣)) → 𝐷 ⊆ ℝ)
69	7	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ∧ ∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑣 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝑤)) ∧ (abs‘(𝑠 − 𝐶)) < 𝑣)) → 𝐹 ∈ 𝐿¹)
70	8	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ∧ ∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑣 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝑤)) ∧ (abs‘(𝑠 − 𝐶)) < 𝑣)) → 𝐶 ∈ (𝐴(,)𝐵))
71	9	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ∧ ∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑣 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝑤)) ∧ (abs‘(𝑠 − 𝐶)) < 𝑣)) → 𝐹 ∈ ((𝐾 CnP 𝐿)‘𝐶))
72		ftc1.h	. . . . . . . . . . . . 13 ⊢ 𝐻 = (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)))
73		simplrl 788	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ∧ ∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑣 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝑤)) ∧ (abs‘(𝑠 − 𝐶)) < 𝑣)) → 𝑤 ∈ ℝ⁺)
74		simplrr 789	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ∧ ∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑣 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝑤)) ∧ (abs‘(𝑠 − 𝐶)) < 𝑣)) → 𝑣 ∈ ℝ⁺)
75		simprlr 791	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ∧ ∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑣 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝑤)) ∧ (abs‘(𝑠 − 𝐶)) < 𝑣)) → ∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑣 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝑤))
76		fvoveq1 7434	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑢 → (abs‘(𝑦 − 𝐶)) = (abs‘(𝑢 − 𝐶)))
77	76	breq1d 5123	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑢 → ((abs‘(𝑦 − 𝐶)) < 𝑣 ↔ (abs‘(𝑢 − 𝐶)) < 𝑣))
78	77	imbrov2fvoveq 7436	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑢 → (((abs‘(𝑦 − 𝐶)) < 𝑣 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝑤) ↔ ((abs‘(𝑢 − 𝐶)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝐶))) < 𝑤)))
79	78	rspccva 3589	. . . . . . . . . . . . . 14 ⊢ ((∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑣 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝑤) ∧ 𝑢 ∈ 𝐷) → ((abs‘(𝑢 − 𝐶)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝐶))) < 𝑤))
80	75, 79	sylan 591	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ∧ ∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑣 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝑤)) ∧ (abs‘(𝑠 − 𝐶)) < 𝑣)) ∧ 𝑢 ∈ 𝐷) → ((abs‘(𝑢 − 𝐶)) < 𝑣 → (abs‘((𝐹‘𝑢) − (𝐹‘𝐶))) < 𝑤))
81	61	eldifad 3925	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ∧ ∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑣 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝑤)) ∧ (abs‘(𝑠 − 𝐶)) < 𝑣)) → 𝑠 ∈ (𝐴[,]𝐵))
82		simprr 784	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ∧ ∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑣 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝑤)) ∧ (abs‘(𝑠 − 𝐶)) < 𝑣)) → (abs‘(𝑠 − 𝐶)) < 𝑣)
83	1, 64, 65, 66, 67, 68, 69, 70, 71, 10, 11, 12, 72, 73, 74, 80, 81, 82	ftc1lem5 26167	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ∧ ∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑣 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝑤)) ∧ (abs‘(𝑠 − 𝐶)) < 𝑣)) ∧ 𝑠 ≠ 𝐶) → (abs‘((𝐻‘𝑠) − (𝐹‘𝐶))) < 𝑤)
84	63, 83	mpdan 699	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ ((𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ∧ ∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑣 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝑤)) ∧ (abs‘(𝑠 − 𝐶)) < 𝑣)) → (abs‘((𝐻‘𝑠) − (𝐹‘𝐶))) < 𝑤)
85	84	expr 461	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ (𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ∧ ∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑣 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝑤))) → ((abs‘(𝑠 − 𝐶)) < 𝑣 → (abs‘((𝐻‘𝑠) − (𝐹‘𝐶))) < 𝑤))
86	85	adantld 495	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ (𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ∧ ∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑣 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝑤))) → ((𝑠 ≠ 𝐶 ∧ (abs‘(𝑠 − 𝐶)) < 𝑣) → (abs‘((𝐻‘𝑠) − (𝐹‘𝐶))) < 𝑤))
87	86	expr 461	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝐶})) → (∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑣 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝑤) → ((𝑠 ≠ 𝐶 ∧ (abs‘(𝑠 − 𝐶)) < 𝑣) → (abs‘((𝐻‘𝑠) − (𝐹‘𝐶))) < 𝑤)))
88	87	ralrimdva 3171	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) → (∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑣 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝑤) → ∀𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝐶})((𝑠 ≠ 𝐶 ∧ (abs‘(𝑠 − 𝐶)) < 𝑣) → (abs‘((𝐻‘𝑠) − (𝐹‘𝐶))) < 𝑤)))
89	60, 88	sylbid 243	. . . . . 6 ⊢ ((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) → (∀𝑦 ∈ 𝐷 ((𝑦((abs ∘ − ) ↾ (𝐷 × 𝐷))𝐶) < 𝑣 → ((𝐹‘𝑦)(abs ∘ − )(𝐹‘𝐶)) < 𝑤) → ∀𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝐶})((𝑠 ≠ 𝐶 ∧ (abs‘(𝑠 − 𝐶)) < 𝑣) → (abs‘((𝐻‘𝑠) − (𝐹‘𝐶))) < 𝑤)))
90	89	anassrs 472	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑣 ∈ ℝ⁺) → (∀𝑦 ∈ 𝐷 ((𝑦((abs ∘ − ) ↾ (𝐷 × 𝐷))𝐶) < 𝑣 → ((𝐹‘𝑦)(abs ∘ − )(𝐹‘𝐶)) < 𝑤) → ∀𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝐶})((𝑠 ≠ 𝐶 ∧ (abs‘(𝑠 − 𝐶)) < 𝑣) → (abs‘((𝐻‘𝑠) − (𝐹‘𝐶))) < 𝑤)))
91	90	reximdva 3184	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → (∃𝑣 ∈ ℝ⁺ ∀𝑦 ∈ 𝐷 ((𝑦((abs ∘ − ) ↾ (𝐷 × 𝐷))𝐶) < 𝑣 → ((𝐹‘𝑦)(abs ∘ − )(𝐹‘𝐶)) < 𝑤) → ∃𝑣 ∈ ℝ⁺ ∀𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝐶})((𝑠 ≠ 𝐶 ∧ (abs‘(𝑠 − 𝐶)) < 𝑣) → (abs‘((𝐻‘𝑠) − (𝐹‘𝐶))) < 𝑤)))
92	35, 91	mpd 16	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → ∃𝑣 ∈ ℝ⁺ ∀𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝐶})((𝑠 ≠ 𝐶 ∧ (abs‘(𝑠 − 𝐶)) < 𝑣) → (abs‘((𝐻‘𝑠) − (𝐹‘𝐶))) < 𝑤))
93	92	ralrimiva 3163	. 2 ⊢ (𝜑 → ∀𝑤 ∈ ℝ⁺ ∃𝑣 ∈ ℝ⁺ ∀𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝐶})((𝑠 ≠ 𝐶 ∧ (abs‘(𝑠 − 𝐶)) < 𝑣) → (abs‘((𝐻‘𝑠) − (𝐹‘𝐶))) < 𝑤))
94	1, 2, 3, 4, 5, 6, 7, 13	ftc1lem2 26163	. . . . 5 ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ)
95	94, 43, 45	dvlem 26023	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝐶})) → (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)) ∈ ℂ)
96	95, 72	fmptd 7110	. . 3 ⊢ (𝜑 → 𝐻:((𝐴[,]𝐵) ∖ {𝐶})⟶ℂ)
97	43	ssdifssd 4109	. . 3 ⊢ (𝜑 → ((𝐴[,]𝐵) ∖ {𝐶}) ⊆ ℂ)
98	96, 97, 46	ellimc3 26006	. 2 ⊢ (𝜑 → ((𝐹‘𝐶) ∈ (𝐻 lim_ℂ 𝐶) ↔ ((𝐹‘𝐶) ∈ ℂ ∧ ∀𝑤 ∈ ℝ⁺ ∃𝑣 ∈ ℝ⁺ ∀𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝐶})((𝑠 ≠ 𝐶 ∧ (abs‘(𝑠 − 𝐶)) < 𝑣) → (abs‘((𝐻‘𝑠) − (𝐹‘𝐶))) < 𝑤))))
99	15, 93, 98	mpbir2and 725	1 ⊢ (𝜑 → (𝐹‘𝐶) ∈ (𝐻 lim_ℂ 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ∀wral 3085 ∃wrex 3095 ∖ cdif 3910 ⊆ wss 3913 {csn 4594 class class class wbr 5113 ↦ cmpt 5196 × cxp 5660 ↾ cres 5664 ∘ ccom 5666 ⟶wf 6533 ‘cfv 6537 (class class class)co 7411 ℂcc 11097 ℝcr 11098 < clt 11242 ≤ cle 11243 − cmin 11440 / cdiv 11870 ℝ⁺crp 13015 (,)cioo 13371 [,]cicc 13374 abscabs 15284 ↾_t crest 17472 TopOpenctopn 17473 ∞Metcxmet 21475 MetOpencmopn 21480 ℂ_fldccnfld 21490 CnP ccnp 23350 𝐿¹cibl 25744 ∫citg 25745 lim_ℂ climc 25989

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-inf2 9609 ax-cc 10418 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176 ax-pre-sup 11177 ax-addf 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-symdif 4214 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-iin 4963 df-disj 5081 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-ofr 7676 df-om 7862 df-1st 7985 df-2nd 7986 df-supp 8156 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-2o 8453 df-oadd 8456 df-omul 8457 df-er 8693 df-map 8825 df-pm 8826 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9321 df-fi 9370 df-sup 9401 df-inf 9402 df-oi 9471 df-dju 9886 df-card 9924 df-acn 9927 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-div 11871 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-5 12305 df-6 12306 df-7 12307 df-8 12308 df-9 12309 df-n0 12504 df-z 12591 df-dec 12711 df-uz 12862 df-q 12972 df-rp 13016 df-xneg 13136 df-xadd 13137 df-xmul 13138 df-ioo 13375 df-ioc 13376 df-ico 13377 df-icc 13378 df-fz 13535 df-fzo 13682 df-fl 13824 df-mod 13902 df-seq 14037 df-exp 14097 df-hash 14366 df-cj 15149 df-re 15150 df-im 15151 df-sqrt 15285 df-abs 15286 df-clim 15538 df-rlim 15539 df-sum 15737 df-struct 17206 df-sets 17223 df-slot 17241 df-ndx 17253 df-base 17269 df-ress 17290 df-plusg 17322 df-mulr 17323 df-starv 17324 df-sca 17325 df-vsca 17326 df-ip 17327 df-tset 17328 df-ple 17329 df-ds 17331 df-unif 17332 df-hom 17333 df-cco 17334 df-rest 17474 df-topn 17475 df-0g 17493 df-gsum 17494 df-topgen 17495 df-pt 17496 df-prds 17499 df-xrs 17555 df-qtop 17560 df-imas 17561 df-xps 17563 df-mre 17637 df-mrc 17638 df-acs 17640 df-mgm 18697 df-sgrp 18776 df-mnd 18792 df-submnd 18841 df-mulg 19133 df-cntz 19386 df-cmn 19851 df-psmet 21482 df-xmet 21483 df-met 21484 df-bl 21485 df-mopn 21486 df-cnfld 21491 df-top 23019 df-topon 23036 df-topsp 23058 df-bases 23071 df-cn 23352 df-cnp 23353 df-cmp 23512 df-tx 23687 df-hmeo 23880 df-xms 24445 df-ms 24446 df-tms 24447 df-cncf 25005 df-ovol 25591 df-vol 25592 df-mbf 25746 df-itg1 25747 df-itg2 25748 df-ibl 25749 df-itg 25750 df-0p 25797 df-limc 25993

This theorem is referenced by: ftc1 26169

W3C validator