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Theorem ftc1lem6 26005

Description: Lemma for ftc1 26006. (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 26002	. . 3 ⊢ (𝜑 → 𝐹:𝐷⟶ℂ)
14	5, 8	sseldd 3964	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐷)
15	13, 14	ffvelcdmd 7080	. 2 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ℂ)
16		cnxmet 24716	. . . . . 6 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
17		ax-resscn 11191	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
18	6, 17	sstrdi 3976	. . . . . . 7 ⊢ (𝜑 → 𝐷 ⊆ ℂ)
19	18	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → 𝐷 ⊆ ℂ)
20		xmetres2 24305	. . . . . 6 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐷 ⊆ ℂ) → ((abs ∘ − ) ↾ (𝐷 × 𝐷)) ∈ (∞Met‘𝐷))
21	16, 19, 20	sylancr 587	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → ((abs ∘ − ) ↾ (𝐷 × 𝐷)) ∈ (∞Met‘𝐷))
22	16	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → (abs ∘ − ) ∈ (∞Met‘ℂ))
23		eqid 2736	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ↾ (𝐷 × 𝐷)) = ((abs ∘ − ) ↾ (𝐷 × 𝐷))
24	12	cnfldtopn 24725	. . . . . . . . . . . 12 ⊢ 𝐿 = (MetOpen‘(abs ∘ − ))
25		eqid 2736	. . . . . . . . . . . 12 ⊢ (MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))) = (MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷)))
26	23, 24, 25	metrest 24468	. . . . . . . . . . 11 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐷 ⊆ ℂ) → (𝐿 ↾_t 𝐷) = (MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))))
27	16, 18, 26	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿 ↾_t 𝐷) = (MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))))
28	11, 27	eqtrid 2783	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 = (MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))))
29	28	oveq1d 7425	. . . . . . . 8 ⊢ (𝜑 → (𝐾 CnP 𝐿) = ((MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))) CnP 𝐿))
30	29	fveq1d 6883	. . . . . . 7 ⊢ (𝜑 → ((𝐾 CnP 𝐿)‘𝐶) = (((MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))) CnP 𝐿)‘𝐶))
31	9, 30	eleqtrd 2837	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (((MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))) CnP 𝐿)‘𝐶))
32	31	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → 𝐹 ∈ (((MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))) CnP 𝐿)‘𝐶))
33		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → 𝑤 ∈ ℝ⁺)
34	25, 24	metcnpi2 24489	. . . . 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 838	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → ∃𝑣 ∈ ℝ⁺ ∀𝑦 ∈ 𝐷 ((𝑦((abs ∘ − ) ↾ (𝐷 × 𝐷))𝐶) < 𝑣 → ((𝐹‘𝑦)(abs ∘ − )(𝐹‘𝐶)) < 𝑤))
36		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → 𝑦 ∈ 𝐷)
37	14	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → 𝐶 ∈ 𝐷)
38	36, 37	ovresd 7579	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → (𝑦((abs ∘ − ) ↾ (𝐷 × 𝐷))𝐶) = (𝑦(abs ∘ − )𝐶))
39	18	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) → 𝐷 ⊆ ℂ)
40	39	sselda 3963	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → 𝑦 ∈ ℂ)
41		iccssre 13451	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
42	2, 3, 41	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
43	42, 17	sstrdi 3976	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
44		ioossicc 13455	. . . . . . . . . . . . . . 15 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
45	44, 8	sselid 3961	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵))
46	43, 45	sseldd 3964	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ ℂ)
47	46	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → 𝐶 ∈ ℂ)
48		eqid 2736	. . . . . . . . . . . . 13 ⊢ (abs ∘ − ) = (abs ∘ − )
49	48	cnmetdval 24714	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝑦(abs ∘ − )𝐶) = (abs‘(𝑦 − 𝐶)))
50	40, 47, 49	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → (𝑦(abs ∘ − )𝐶) = (abs‘(𝑦 − 𝐶)))
51	38, 50	eqtrd 2771	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → (𝑦((abs ∘ − ) ↾ (𝐷 × 𝐷))𝐶) = (abs‘(𝑦 − 𝐶)))
52	51	breq1d 5134	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → ((𝑦((abs ∘ − ) ↾ (𝐷 × 𝐷))𝐶) < 𝑣 ↔ (abs‘(𝑦 − 𝐶)) < 𝑣))
53	13	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) → 𝐹:𝐷⟶ℂ)
54	53	ffvelcdmda 7079	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → (𝐹‘𝑦) ∈ ℂ)
55	15	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → (𝐹‘𝐶) ∈ ℂ)
56	48	cnmetdval 24714	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑦) ∈ ℂ ∧ (𝐹‘𝐶) ∈ ℂ) → ((𝐹‘𝑦)(abs ∘ − )(𝐹‘𝐶)) = (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))))
57	54, 55, 56	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → ((𝐹‘𝑦)(abs ∘ − )(𝐹‘𝐶)) = (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))))
58	57	breq1d 5134	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → (((𝐹‘𝑦)(abs ∘ − )(𝐹‘𝐶)) < 𝑤 ↔ (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < 𝑤))
59	52, 58	imbi12d 344	. . . . . . . 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 3162	. . . . . . 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 778	. . . . . . . . . . . . 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 4771	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) → 𝑠 ≠ 𝐶)
63	61, 62	syl 17	. . . . . . . . . . . 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 726	. . . . . . . . . . . . 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 726	. . . . . . . . . . . . 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 726	. . . . . . . . . . . . 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 726	. . . . . . . . . . . . 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 726	. . . . . . . . . . . . 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 726	. . . . . . . . . . . . 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 726	. . . . . . . . . . . . 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 726	. . . . . . . . . . . . 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 776	. . . . . . . . . . . . 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 777	. . . . . . . . . . . . 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 779	. . . . . . . . . . . . . 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 7433	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑢 → (abs‘(𝑦 − 𝐶)) = (abs‘(𝑢 − 𝐶)))
77	76	breq1d 5134	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑢 → ((abs‘(𝑦 − 𝐶)) < 𝑣 ↔ (abs‘(𝑢 − 𝐶)) < 𝑣))
78	77	imbrov2fvoveq 7435	. . . . . . . . . . . . . . 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 3605	. . . . . . . . . . . . . 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 580	. . . . . . . . . . . . 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 3943	. . . . . . . . . . . . 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 772	. . . . . . . . . . . . 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 26004	. . . . . . . . . . . 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 687	. . . . . . . . . . 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 456	. . . . . . . . . 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 490	. . . . . . . . 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 456	. . . . . . . 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 3141	. . . . . . 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 240	. . . . . 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 467	. . . . 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 3154	. . . 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 15	. . 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 3133	. 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 26000	. . . . 5 ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ)
95	94, 43, 45	dvlem 25854	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝐶})) → (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)) ∈ ℂ)
96	95, 72	fmptd 7109	. . 3 ⊢ (𝜑 → 𝐻:((𝐴[,]𝐵) ∖ {𝐶})⟶ℂ)
97	43	ssdifssd 4127	. . 3 ⊢ (𝜑 → ((𝐴[,]𝐵) ∖ {𝐶}) ⊆ ℂ)
98	96, 97, 46	ellimc3 25837	. 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 713	1 ⊢ (𝜑 → (𝐹‘𝐶) ∈ (𝐻 lim_ℂ 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2933 ∀wral 3052 ∃wrex 3061 ∖ cdif 3928 ⊆ wss 3931 {csn 4606 class class class wbr 5124 ↦ cmpt 5206 × cxp 5657 ↾ cres 5661 ∘ ccom 5663 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ℂcc 11132 ℝcr 11133 < clt 11274 ≤ cle 11275 − cmin 11471 / cdiv 11899 ℝ⁺crp 13013 (,)cioo 13367 [,]cicc 13370 abscabs 15258 ↾_t crest 17439 TopOpenctopn 17440 ∞Metcxmet 21305 MetOpencmopn 21310 ℂ_fldccnfld 21320 CnP ccnp 23168 𝐿¹cibl 25575 ∫citg 25576 lim_ℂ climc 25820

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 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-inf2 9660 ax-cc 10454 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212 ax-addf 11213

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 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-symdif 4233 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-disj 5092 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7676 df-ofr 7677 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-oadd 8489 df-omul 8490 df-er 8724 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9379 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9529 df-dju 9920 df-card 9958 df-acn 9961 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12858 df-q 12970 df-rp 13014 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-ioo 13371 df-ioc 13372 df-ico 13373 df-icc 13374 df-fz 13530 df-fzo 13677 df-fl 13814 df-mod 13892 df-seq 14025 df-exp 14085 df-hash 14354 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-clim 15509 df-rlim 15510 df-sum 15708 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-hom 17300 df-cco 17301 df-rest 17441 df-topn 17442 df-0g 17460 df-gsum 17461 df-topgen 17462 df-pt 17463 df-prds 17466 df-xrs 17521 df-qtop 17526 df-imas 17527 df-xps 17529 df-mre 17603 df-mrc 17604 df-acs 17606 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-submnd 18767 df-mulg 19056 df-cntz 19305 df-cmn 19768 df-psmet 21312 df-xmet 21313 df-met 21314 df-bl 21315 df-mopn 21316 df-cnfld 21321 df-top 22837 df-topon 22854 df-topsp 22876 df-bases 22889 df-cn 23170 df-cnp 23171 df-cmp 23330 df-tx 23505 df-hmeo 23698 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-ovol 25422 df-vol 25423 df-mbf 25577 df-itg1 25578 df-itg2 25579 df-ibl 25580 df-itg 25581 df-0p 25628 df-limc 25824

This theorem is referenced by: ftc1 26006

W3C validator