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Theorem ftc1lem6 25405

Description: Lemma for ftc1 25406. (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 25402	. . 3 ⊢ (𝜑 → 𝐹:𝐷⟶ℂ)
14	5, 8	sseldd 3945	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐷)
15	13, 14	ffvelcdmd 7036	. 2 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ℂ)
16		cnxmet 24136	. . . . . 6 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
17		ax-resscn 11108	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
18	6, 17	sstrdi 3956	. . . . . . 7 ⊢ (𝜑 → 𝐷 ⊆ ℂ)
19	18	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → 𝐷 ⊆ ℂ)
20		xmetres2 23714	. . . . . 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 24145	. . . . . . . . . . . 12 ⊢ 𝐿 = (MetOpen‘(abs ∘ − ))
25		eqid 2736	. . . . . . . . . . . 12 ⊢ (MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))) = (MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷)))
26	23, 24, 25	metrest 23880	. . . . . . . . . . 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 7372	. . . . . . . 8 ⊢ (𝜑 → (𝐾 CnP 𝐿) = ((MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))) CnP 𝐿))
30	29	fveq1d 6844	. . . . . . 7 ⊢ (𝜑 → ((𝐾 CnP 𝐿)‘𝐶) = (((MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))) CnP 𝐿)‘𝐶))
31	9, 30	eleqtrd 2840	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (((MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))) CnP 𝐿)‘𝐶))
32	31	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → 𝐹 ∈ (((MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))) CnP 𝐿)‘𝐶))
33		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → 𝑤 ∈ ℝ⁺)
34	25, 24	metcnpi2 23901	. . . . 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 837	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → ∃𝑣 ∈ ℝ⁺ ∀𝑦 ∈ 𝐷 ((𝑦((abs ∘ − ) ↾ (𝐷 × 𝐷))𝐶) < 𝑣 → ((𝐹‘𝑦)(abs ∘ − )(𝐹‘𝐶)) < 𝑤))
36		simpr 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → 𝑦 ∈ 𝐷)
37	14	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → 𝐶 ∈ 𝐷)
38	36, 37	ovresd 7521	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → (𝑦((abs ∘ − ) ↾ (𝐷 × 𝐷))𝐶) = (𝑦(abs ∘ − )𝐶))
39	18	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) → 𝐷 ⊆ ℂ)
40	39	sselda 3944	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → 𝑦 ∈ ℂ)
41		iccssre 13346	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
42	2, 3, 41	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
43	42, 17	sstrdi 3956	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
44		ioossicc 13350	. . . . . . . . . . . . . . 15 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
45	44, 8	sselid 3942	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵))
46	43, 45	sseldd 3945	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ ℂ)
47	46	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → 𝐶 ∈ ℂ)
48		eqid 2736	. . . . . . . . . . . . 13 ⊢ (abs ∘ − ) = (abs ∘ − )
49	48	cnmetdval 24134	. . . . . . . . . . . 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 5115	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → ((𝑦((abs ∘ − ) ↾ (𝐷 × 𝐷))𝐶) < 𝑣 ↔ (abs‘(𝑦 − 𝐶)) < 𝑣))
53	13	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) → 𝐹:𝐷⟶ℂ)
54	53	ffvelcdmda 7035	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → (𝐹‘𝑦) ∈ ℂ)
55	15	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → (𝐹‘𝐶) ∈ ℂ)
56	48	cnmetdval 24134	. . . . . . . . . . 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 5115	. . . . . . . . 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 3172	. . . . . . 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 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)) → 𝑠 ∈ ((𝐴[,]𝐵) ∖ {𝐶}))
62		eldifsni 4750	. . . . . . . . . . . . 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 724	. . . . . . . . . . . . 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 724	. . . . . . . . . . . . 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 724	. . . . . . . . . . . . 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 724	. . . . . . . . . . . . 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 724	. . . . . . . . . . . . 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 724	. . . . . . . . . . . . 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 724	. . . . . . . . . . . . 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 724	. . . . . . . . . . . . 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 775	. . . . . . . . . . . . 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 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)) → 𝑣 ∈ ℝ⁺)
75		simprlr 778	. . . . . . . . . . . . . 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 7380	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑢 → (abs‘(𝑦 − 𝐶)) = (abs‘(𝑢 − 𝐶)))
77	76	breq1d 5115	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑢 → ((abs‘(𝑦 − 𝐶)) < 𝑣 ↔ (abs‘(𝑢 − 𝐶)) < 𝑣))
78	77	imbrov2fvoveq 7382	. . . . . . . . . . . . . . 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 3580	. . . . . . . . . . . . . 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 3922	. . . . . . . . . . . . 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 771	. . . . . . . . . . . . 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 25404	. . . . . . . . . . . 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 685	. . . . . . . . . . 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 457	. . . . . . . . . 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 491	. . . . . . . . 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 457	. . . . . . . 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 3151	. . . . . . 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 239	. . . . . 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 468	. . . . 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 3165	. . . 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 3143	. 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 25400	. . . . 5 ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ)
95	94, 43, 45	dvlem 25260	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝐶})) → (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)) ∈ ℂ)
96	95, 72	fmptd 7062	. . 3 ⊢ (𝜑 → 𝐻:((𝐴[,]𝐵) ∖ {𝐶})⟶ℂ)
97	43	ssdifssd 4102	. . 3 ⊢ (𝜑 → ((𝐴[,]𝐵) ∖ {𝐶}) ⊆ ℂ)
98	96, 97, 46	ellimc3 25243	. 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 711	1 ⊢ (𝜑 → (𝐹‘𝐶) ∈ (𝐻 lim_ℂ 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3073 ∖ cdif 3907 ⊆ wss 3910 {csn 4586 class class class wbr 5105 ↦ cmpt 5188 × cxp 5631 ↾ cres 5635 ∘ ccom 5637 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 ℝcr 11050 < clt 11189 ≤ cle 11190 − cmin 11385 / cdiv 11812 ℝ⁺crp 12915 (,)cioo 13264 [,]cicc 13267 abscabs 15119 ↾_t crest 17302 TopOpenctopn 17303 ∞Metcxmet 20781 MetOpencmopn 20786 ℂ_fldccnfld 20796 CnP ccnp 22576 𝐿¹cibl 24981 ∫citg 24982 lim_ℂ climc 25226

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cc 10371 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-symdif 4202 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-disj 5071 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-ofr 7618 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-oadd 8416 df-omul 8417 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-dju 9837 df-card 9875 df-acn 9878 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ioc 13269 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-rlim 15371 df-sum 15571 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cn 22578 df-cnp 22579 df-cmp 22738 df-tx 22913 df-hmeo 23106 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-ovol 24828 df-vol 24829 df-mbf 24983 df-itg1 24984 df-itg2 24985 df-ibl 24986 df-itg 24987 df-0p 25034 df-limc 25230

This theorem is referenced by: ftc1 25406

W3C validator