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Theorem ftc1lem6 24630

Description: Lemma for ftc1 24631. (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 24627	. . 3 ⊢ (𝜑 → 𝐹:𝐷⟶ℂ)
14	5, 8	sseldd 3966	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐷)
15	13, 14	ffvelrnd 6845	. 2 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ℂ)
16		cnxmet 23373	. . . . . 6 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
17		ax-resscn 10586	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
18	6, 17	sstrdi 3977	. . . . . . 7 ⊢ (𝜑 → 𝐷 ⊆ ℂ)
19	18	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → 𝐷 ⊆ ℂ)
20		xmetres2 22963	. . . . . 6 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐷 ⊆ ℂ) → ((abs ∘ − ) ↾ (𝐷 × 𝐷)) ∈ (∞Met‘𝐷))
21	16, 19, 20	sylancr 589	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → ((abs ∘ − ) ↾ (𝐷 × 𝐷)) ∈ (∞Met‘𝐷))
22	16	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → (abs ∘ − ) ∈ (∞Met‘ℂ))
23		eqid 2819	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ↾ (𝐷 × 𝐷)) = ((abs ∘ − ) ↾ (𝐷 × 𝐷))
24	12	cnfldtopn 23382	. . . . . . . . . . . 12 ⊢ 𝐿 = (MetOpen‘(abs ∘ − ))
25		eqid 2819	. . . . . . . . . . . 12 ⊢ (MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))) = (MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷)))
26	23, 24, 25	metrest 23126	. . . . . . . . . . 11 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐷 ⊆ ℂ) → (𝐿 ↾_t 𝐷) = (MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))))
27	16, 18, 26	sylancr 589	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿 ↾_t 𝐷) = (MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))))
28	11, 27	syl5eq 2866	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 = (MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))))
29	28	oveq1d 7163	. . . . . . . 8 ⊢ (𝜑 → (𝐾 CnP 𝐿) = ((MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))) CnP 𝐿))
30	29	fveq1d 6665	. . . . . . 7 ⊢ (𝜑 → ((𝐾 CnP 𝐿)‘𝐶) = (((MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))) CnP 𝐿)‘𝐶))
31	9, 30	eleqtrd 2913	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (((MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))) CnP 𝐿)‘𝐶))
32	31	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → 𝐹 ∈ (((MetOpen‘((abs ∘ − ) ↾ (𝐷 × 𝐷))) CnP 𝐿)‘𝐶))
33		simpr 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → 𝑤 ∈ ℝ⁺)
34	25, 24	metcnpi2 23147	. . . . 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 836	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → ∃𝑣 ∈ ℝ⁺ ∀𝑦 ∈ 𝐷 ((𝑦((abs ∘ − ) ↾ (𝐷 × 𝐷))𝐶) < 𝑣 → ((𝐹‘𝑦)(abs ∘ − )(𝐹‘𝐶)) < 𝑤))
36		simpr 487	. . . . . . . . . . . 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 7307	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → (𝑦((abs ∘ − ) ↾ (𝐷 × 𝐷))𝐶) = (𝑦(abs ∘ − )𝐶))
39	18	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) → 𝐷 ⊆ ℂ)
40	39	sselda 3965	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → 𝑦 ∈ ℂ)
41		iccssre 12810	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
42	2, 3, 41	syl2anc 586	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
43	42, 17	sstrdi 3977	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
44		ioossicc 12814	. . . . . . . . . . . . . . 15 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
45	44, 8	sseldi 3963	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵))
46	43, 45	sseldd 3966	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ ℂ)
47	46	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → 𝐶 ∈ ℂ)
48		eqid 2819	. . . . . . . . . . . . 13 ⊢ (abs ∘ − ) = (abs ∘ − )
49	48	cnmetdval 23371	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝑦(abs ∘ − )𝐶) = (abs‘(𝑦 − 𝐶)))
50	40, 47, 49	syl2anc 586	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → (𝑦(abs ∘ − )𝐶) = (abs‘(𝑦 − 𝐶)))
51	38, 50	eqtrd 2854	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → (𝑦((abs ∘ − ) ↾ (𝐷 × 𝐷))𝐶) = (abs‘(𝑦 − 𝐶)))
52	51	breq1d 5067	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → ((𝑦((abs ∘ − ) ↾ (𝐷 × 𝐷))𝐶) < 𝑣 ↔ (abs‘(𝑦 − 𝐶)) < 𝑣))
53	13	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) → 𝐹:𝐷⟶ℂ)
54	53	ffvelrnda 6844	. . . . . . . . . . 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 23371	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑦) ∈ ℂ ∧ (𝐹‘𝐶) ∈ ℂ) → ((𝐹‘𝑦)(abs ∘ − )(𝐹‘𝐶)) = (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))))
57	54, 55, 56	syl2anc 586	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑤 ∈ ℝ⁺ ∧ 𝑣 ∈ ℝ⁺)) ∧ 𝑦 ∈ 𝐷) → ((𝐹‘𝑦)(abs ∘ − )(𝐹‘𝐶)) = (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))))
58	57	breq1d 5067	. . . . . . . . 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 3194	. . . . . . 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 4714	. . . . . . . . . . . . 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 7171	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑢 → (abs‘(𝑦 − 𝐶)) = (abs‘(𝑢 − 𝐶)))
77	76	breq1d 5067	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑢 → ((abs‘(𝑦 − 𝐶)) < 𝑣 ↔ (abs‘(𝑢 − 𝐶)) < 𝑣))
78	77	imbrov2fvoveq 7173	. . . . . . . . . . . . . . 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 582	. . . . . . . . . . . . 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 3946	. . . . . . . . . . . . 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 24629	. . . . . . . . . . . 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 459	. . . . . . . . . 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 493	. . . . . . . . 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 459	. . . . . . . 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 3187	. . . . . . 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 242	. . . . . 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 470	. . . . 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 3272	. . . 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 3180	. 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 24625	. . . . 5 ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ)
95	94, 43, 45	dvlem 24486	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝐶})) → (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶)) ∈ ℂ)
96	95, 72	fmptd 6871	. . 3 ⊢ (𝜑 → 𝐻:((𝐴[,]𝐵) ∖ {𝐶})⟶ℂ)
97	43	ssdifssd 4117	. . 3 ⊢ (𝜑 → ((𝐴[,]𝐵) ∖ {𝐶}) ⊆ ℂ)
98	96, 97, 46	ellimc3 24469	. 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 398 = wceq 1531 ∈ wcel 2108 ≠ wne 3014 ∀wral 3136 ∃wrex 3137 ∖ cdif 3931 ⊆ wss 3934 {csn 4559 class class class wbr 5057 ↦ cmpt 5137 × cxp 5546 ↾ cres 5550 ∘ ccom 5552 ⟶wf 6344 ‘cfv 6348 (class class class)co 7148 ℂcc 10527 ℝcr 10528 < clt 10667 ≤ cle 10668 − cmin 10862 / cdiv 11289 ℝ⁺crp 12381 (,)cioo 12730 [,]cicc 12733 abscabs 14585 ↾_t crest 16686 TopOpenctopn 16687 ∞Metcxmet 20522 MetOpencmopn 20527 ℂ_fldccnfld 20537 CnP ccnp 21825 𝐿¹cibl 24210 ∫citg 24211 lim_ℂ climc 24452

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-inf2 9096 ax-cc 9849 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 ax-addf 10608 ax-mulf 10609

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-fal 1544 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-symdif 4217 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-disj 5023 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-of 7401 df-ofr 7402 df-om 7573 df-1st 7681 df-2nd 7682 df-supp 7823 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-2o 8095 df-oadd 8098 df-omul 8099 df-er 8281 df-map 8400 df-pm 8401 df-ixp 8454 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-fsupp 8826 df-fi 8867 df-sup 8898 df-inf 8899 df-oi 8966 df-dju 9322 df-card 9360 df-acn 9363 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-n0 11890 df-z 11974 df-dec 12091 df-uz 12236 df-q 12341 df-rp 12382 df-xneg 12499 df-xadd 12500 df-xmul 12501 df-ioo 12734 df-ioc 12735 df-ico 12736 df-icc 12737 df-fz 12885 df-fzo 13026 df-fl 13154 df-mod 13230 df-seq 13362 df-exp 13422 df-hash 13683 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-rlim 14838 df-sum 15035 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20529 df-xmet 20530 df-met 20531 df-bl 20532 df-mopn 20533 df-cnfld 20538 df-top 21494 df-topon 21511 df-topsp 21533 df-bases 21546 df-cn 21827 df-cnp 21828 df-cmp 21987 df-tx 22162 df-hmeo 22355 df-xms 22922 df-ms 22923 df-tms 22924 df-cncf 23478 df-ovol 24057 df-vol 24058 df-mbf 24212 df-itg1 24213 df-itg2 24214 df-ibl 24215 df-itg 24216 df-0p 24263 df-limc 24456

This theorem is referenced by: ftc1 24631

W3C validator