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Theorem ftc1lem6 26083

Description: Lemma for ftc1 26084. (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 26080	. . 3 ⊢ (𝜑 → 𝐹:𝐷⟶ℂ)
14	5, 8	sseldd 3983	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐷)
15	13, 14	ffvelcdmd 7104	. 2 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ℂ)
16		cnxmet 24794	. . . . . 6 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
17		ax-resscn 11213	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
18	6, 17	sstrdi 3995	. . . . . . 7 ⊢ (𝜑 → 𝐷 ⊆ ℂ)
19	18	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → 𝐷 ⊆ ℂ)
20		xmetres2 24372	. . . . . 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 24803	. . . . . . . . . . . 12 ⊢ 𝐿 = (MetOpen‘(abs ∘ − ))
25		eqid 2736	. . . . . . . . . . . 12 ⊢ (MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))) = (MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷)))
26	23, 24, 25	metrest 24538	. . . . . . . . . . 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 2788	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 = (MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))))
29	28	oveq1d 7447	. . . . . . . 8 ⊢ (𝜑 → (𝐾 CnP 𝐿) = ((MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))) CnP 𝐿))
30	29	fveq1d 6907	. . . . . . 7 ⊢ (𝜑 → ((𝐾 CnP 𝐿)‘𝐶) = (((MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))) CnP 𝐿)‘𝐶))
31	9, 30	eleqtrd 2842	. . . . . 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 24559	. . . . 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 7601	. . . . . . . . . . 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 3982	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → 𝑦 ∈ ℂ)
41		iccssre 13470	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
42	2, 3, 41	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
43	42, 17	sstrdi 3995	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
44		ioossicc 13474	. . . . . . . . . . . . . . 15 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
45	44, 8	sselid 3980	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵))
46	43, 45	sseldd 3983	. . . . . . . . . . . . 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 24792	. . . . . . . . . . . 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 2776	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → (𝑦((abs ∘ − ) ↾ (𝐷 × 𝐷))𝐶) = (abs‘(𝑦 − 𝐶)))
52	51	breq1d 5152	. . . . . . . . 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 7103	. . . . . . . . . . 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 24792	. . . . . . . . . . 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 5152	. . . . . . . . 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 3175	. . . . . . 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 4789	. . . . . . . . . . . . 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 7455	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑢 → (abs‘(𝑦 − 𝐶)) = (abs‘(𝑢 − 𝐶)))
77	76	breq1d 5152	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑢 → ((abs‘(𝑦 − 𝐶)) < 𝑣 ↔ (abs‘(𝑢 − 𝐶)) < 𝑣))
78	77	imbrov2fvoveq 7457	. . . . . . . . . . . . . . 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 3620	. . . . . . . . . . . . . 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 3962	. . . . . . . . . . . . 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 26082	. . . . . . . . . . . 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 3153	. . . . . . 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 3167	. . . 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 3145	. 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 26078	. . . . 5 ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ)
95	94, 43, 45	dvlem 25932	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝐶})) → (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)) ∈ ℂ)
96	95, 72	fmptd 7133	. . 3 ⊢ (𝜑 → 𝐻:((𝐴[,]𝐵) ∖ {𝐶})⟶ℂ)
97	43	ssdifssd 4146	. . 3 ⊢ (𝜑 → ((𝐴[,]𝐵) ∖ {𝐶}) ⊆ ℂ)
98	96, 97, 46	ellimc3 25915	. 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 1539 ∈ wcel 2107 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 ∖ cdif 3947 ⊆ wss 3950 {csn 4625 class class class wbr 5142 ↦ cmpt 5224 × cxp 5682 ↾ cres 5686 ∘ ccom 5688 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ℂcc 11154 ℝcr 11155 < clt 11296 ≤ cle 11297 − cmin 11493 / cdiv 11921 ℝ⁺crp 13035 (,)cioo 13388 [,]cicc 13391 abscabs 15274 ↾_t crest 17466 TopOpenctopn 17467 ∞Metcxmet 21350 MetOpencmopn 21355 ℂ_fldccnfld 21365 CnP ccnp 23234 𝐿¹cibl 25653 ∫citg 25654 lim_ℂ climc 25898

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cc 10476 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-symdif 4252 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-disj 5110 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-ofr 7699 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-oadd 8511 df-omul 8512 df-er 8746 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-fi 9452 df-sup 9483 df-inf 9484 df-oi 9551 df-dju 9942 df-card 9980 df-acn 9983 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13392 df-ioc 13393 df-ico 13394 df-icc 13395 df-fz 13549 df-fzo 13696 df-fl 13833 df-mod 13911 df-seq 14044 df-exp 14104 df-hash 14371 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-clim 15525 df-rlim 15526 df-sum 15724 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18798 df-mulg 19087 df-cntz 19336 df-cmn 19801 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-cnfld 21366 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-cn 23236 df-cnp 23237 df-cmp 23396 df-tx 23571 df-hmeo 23764 df-xms 24331 df-ms 24332 df-tms 24333 df-cncf 24905 df-ovol 25500 df-vol 25501 df-mbf 25655 df-itg1 25656 df-itg2 25657 df-ibl 25658 df-itg 25659 df-0p 25706 df-limc 25902

This theorem is referenced by: ftc1 26084

W3C validator