ftc2re - Mathbox for Thierry Arnoux

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Theorem ftc2re 35277

Description: The Fundamental Theorem of Calculus, part two, for functions continuous on 𝐷. (Contributed by Thierry Arnoux, 1-Dec-2021.)

**Hypotheses**
Ref	Expression
ftc2re.e	⊢ 𝐸 = (𝐶(,)𝐷)
ftc2re.a	⊢ (𝜑 → 𝐴 ∈ 𝐸)
ftc2re.b	⊢ (𝜑 → 𝐵 ∈ 𝐸)
ftc2re.le	⊢ (𝜑 → 𝐴 ≤ 𝐵)
ftc2re.f	⊢ (𝜑 → 𝐹:𝐸⟶ℂ)
ftc2re.1	⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℂ))

**Assertion**
Ref	Expression
ftc2re	⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴)))

Distinct variable groups: 𝑡,𝐴 𝑡,𝐵 𝑡,𝐹 𝜑,𝑡 Allowed substitution hints: 𝐶(𝑡) 𝐷(𝑡) 𝐸(𝑡)

Proof of Theorem ftc2re Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ftc2re.e	. . . . . 6 ⊢ 𝐸 = (𝐶(,)𝐷)
2		ioossre 13545	. . . . . 6 ⊢ (𝐶(,)𝐷) ⊆ ℝ
3	1, 2	eqsstri 3955	. . . . 5 ⊢ 𝐸 ⊆ ℝ
4	3	a1i 11	. . . 4 ⊢ (𝜑 → 𝐸 ⊆ ℝ)
5		ftc2re.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐸)
6	4, 5	sseldd 3910	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ)
7		ftc2re.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐸)
8	4, 7	sseldd 3910	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ)
9		ftc2re.le	. . 3 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
10		ax-resscn 11264	. . . . . . 7 ⊢ ℝ ⊆ ℂ
11	10	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℂ)
12		ftc2re.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐸⟶ℂ)
13		iccssre 13567	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
14	6, 8, 13	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
15		eqid 2760	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
16		tgioo4 25133	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
17	15, 16	dvres 26240	. . . . . 6 ⊢ (((ℝ ⊆ ℂ ∧ 𝐹:𝐸⟶ℂ) ∧ (𝐸 ⊆ ℝ ∧ (𝐴[,]𝐵) ⊆ ℝ)) → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))))
18	11, 12, 4, 14, 17	syl22anc 852	. . . . 5 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))))
19		iccntr 25150	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
20	6, 8, 19	syl2anc 596	. . . . . 6 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
21	20	reseq2d 5971	. . . . 5 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))) = ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)))
22	18, 21	eqtrd 2795	. . . 4 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) = ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)))
23		ioossicc 13571	. . . . . . 7 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
24	23	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
25	1, 5, 7	fct2relem 35276	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐸)
26	24, 25	sstrd 3919	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐸)
27		ftc2re.1	. . . . 5 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℂ))
28		rescncf 25227	. . . . 5 ⊢ ((𝐴(,)𝐵) ⊆ 𝐸 → ((ℝ D 𝐹) ∈ (𝐸–cn→ℂ) → ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) ∈ ((𝐴(,)𝐵)–cn→ℂ)))
29	26, 27, 28	sylc 66	. . . 4 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
30	22, 29	eqeltrd 2860	. . 3 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
31		ioombl 25895	. . . . . . 7 ⊢ (𝐴(,)𝐵) ∈ dom vol
32	31	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ dom vol)
33		cnmbf 25989	. . . . . 6 ⊢ (((𝐴(,)𝐵) ∈ dom vol ∧ ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) ∈ ((𝐴(,)𝐵)–cn→ℂ)) → ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) ∈ MblFn)
34	32, 29, 33	syl2anc 596	. . . . 5 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) ∈ MblFn)
35		dmres 6005	. . . . . . 7 ⊢ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) = ((𝐴(,)𝐵) ∩ dom (ℝ D 𝐹))
36	35	fveq2i 6893	. . . . . 6 ⊢ (vol‘dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))) = (vol‘((𝐴(,)𝐵) ∩ dom (ℝ D 𝐹)))
37		cncff 25223	. . . . . . . . . . . 12 ⊢ ((ℝ D 𝐹) ∈ (𝐸–cn→ℂ) → (ℝ D 𝐹):𝐸⟶ℂ)
38	27, 37	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D 𝐹):𝐸⟶ℂ)
39	38	fdmd 6725	. . . . . . . . . 10 ⊢ (𝜑 → dom (ℝ D 𝐹) = 𝐸)
40	39	ineq2d 4116	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴(,)𝐵) ∩ dom (ℝ D 𝐹)) = ((𝐴(,)𝐵) ∩ 𝐸))
41		dfss2 4093	. . . . . . . . . 10 ⊢ ((𝐴(,)𝐵) ⊆ 𝐸 ↔ ((𝐴(,)𝐵) ∩ 𝐸) = (𝐴(,)𝐵))
42	26, 41	sylib 221	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴(,)𝐵) ∩ 𝐸) = (𝐴(,)𝐵))
43	40, 42	eqtrd 2795	. . . . . . . 8 ⊢ (𝜑 → ((𝐴(,)𝐵) ∩ dom (ℝ D 𝐹)) = (𝐴(,)𝐵))
44	43	fveq2d 6894	. . . . . . 7 ⊢ (𝜑 → (vol‘((𝐴(,)𝐵) ∩ dom (ℝ D 𝐹))) = (vol‘(𝐴(,)𝐵)))
45		volioo 25899	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴))
46	6, 8, 9, 45	syl3anc 1398	. . . . . . . 8 ⊢ (𝜑 → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴))
47	8, 6	resubcld 11751	. . . . . . . 8 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
48	46, 47	eqeltrd 2860	. . . . . . 7 ⊢ (𝜑 → (vol‘(𝐴(,)𝐵)) ∈ ℝ)
49	44, 48	eqeltrd 2860	. . . . . 6 ⊢ (𝜑 → (vol‘((𝐴(,)𝐵) ∩ dom (ℝ D 𝐹))) ∈ ℝ)
50	36, 49	eqeltrid 2864	. . . . 5 ⊢ (𝜑 → (vol‘dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))) ∈ ℝ)
51		rescncf 25227	. . . . . . . . 9 ⊢ ((𝐴[,]𝐵) ⊆ 𝐸 → ((ℝ D 𝐹) ∈ (𝐸–cn→ℂ) → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ)))
52	25, 51	syl 18	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D 𝐹) ∈ (𝐸–cn→ℂ) → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ)))
53	27, 52	mpd 16	. . . . . . 7 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
54		cniccbdd 25791	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ)) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑦)) ≤ 𝑥)
55	6, 8, 53, 54	syl3anc 1398	. . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑦)) ≤ 𝑥)
56	35, 43	eqtrid 2807	. . . . . . . . . . 11 ⊢ (𝜑 → dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) = (𝐴(,)𝐵))
57	56, 24	eqsstrd 3943	. . . . . . . . . 10 ⊢ (𝜑 → dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) ⊆ (𝐴[,]𝐵))
58		ssralv 3978	. . . . . . . . . 10 ⊢ (dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) ⊆ (𝐴[,]𝐵) → (∀𝑦 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑦)) ≤ 𝑥 → ∀𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑦)) ≤ 𝑥))
59	57, 58	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑦 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑦)) ≤ 𝑥 → ∀𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑦)) ≤ 𝑥))
60	59	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑦 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑦)) ≤ 𝑥 → ∀𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑦)) ≤ 𝑥))
61	57	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) ⊆ (𝐴[,]𝐵))
62	61	sselda 3909	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))) → 𝑦 ∈ (𝐴[,]𝐵))
63		fvres 6909	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (𝐴[,]𝐵) → (((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑦) = ((ℝ D 𝐹)‘𝑦))
64	62, 63	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))) → (((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑦) = ((ℝ D 𝐹)‘𝑦))
65		simpr 490	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))) → 𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)))
66	56	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))) → dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) = (𝐴(,)𝐵))
67	65, 66	eleqtrd 2862	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))) → 𝑦 ∈ (𝐴(,)𝐵))
68		fvres 6909	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (𝐴(,)𝐵) → (((ℝ D 𝐹) ↾ (𝐴(,)𝐵))‘𝑦) = ((ℝ D 𝐹)‘𝑦))
69	67, 68	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))) → (((ℝ D 𝐹) ↾ (𝐴(,)𝐵))‘𝑦) = ((ℝ D 𝐹)‘𝑦))
70	64, 69	eqtr4d 2798	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))) → (((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑦) = (((ℝ D 𝐹) ↾ (𝐴(,)𝐵))‘𝑦))
71	70	fveq2d 6894	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))) → (abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑦)) = (abs‘(((ℝ D 𝐹) ↾ (𝐴(,)𝐵))‘𝑦)))
72	71	breq1d 5112	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))) → ((abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑦)) ≤ 𝑥 ↔ (abs‘(((ℝ D 𝐹) ↾ (𝐴(,)𝐵))‘𝑦)) ≤ 𝑥))
73	72	biimpd 232	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))) → ((abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑦)) ≤ 𝑥 → (abs‘(((ℝ D 𝐹) ↾ (𝐴(,)𝐵))‘𝑦)) ≤ 𝑥))
74	73	ralimdva 3174	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑦)) ≤ 𝑥 → ∀𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))(abs‘(((ℝ D 𝐹) ↾ (𝐴(,)𝐵))‘𝑦)) ≤ 𝑥))
75	60, 74	syld 48	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑦 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑦)) ≤ 𝑥 → ∀𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))(abs‘(((ℝ D 𝐹) ↾ (𝐴(,)𝐵))‘𝑦)) ≤ 𝑥))
76	75	reximdva 3175	. . . . . 6 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑦 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑦)) ≤ 𝑥 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))(abs‘(((ℝ D 𝐹) ↾ (𝐴(,)𝐵))‘𝑦)) ≤ 𝑥))
77	55, 76	mpd 16	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))(abs‘(((ℝ D 𝐹) ↾ (𝐴(,)𝐵))‘𝑦)) ≤ 𝑥)
78		bddibl 26169	. . . . 5 ⊢ ((((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) ∈ MblFn ∧ (vol‘dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))) ∈ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))(abs‘(((ℝ D 𝐹) ↾ (𝐴(,)𝐵))‘𝑦)) ≤ 𝑥) → ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) ∈ 𝐿¹)
79	34, 50, 77, 78	syl3anc 1398	. . . 4 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) ∈ 𝐿¹)
80	22, 79	eqeltrd 2860	. . 3 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) ∈ 𝐿¹)
81		dvcn 26250	. . . . 5 ⊢ (((ℝ ⊆ ℂ ∧ 𝐹:𝐸⟶ℂ ∧ 𝐸 ⊆ ℝ) ∧ dom (ℝ D 𝐹) = 𝐸) → 𝐹 ∈ (𝐸–cn→ℂ))
82	11, 12, 4, 39, 81	syl31anc 1400	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐸–cn→ℂ))
83		rescncf 25227	. . . . 5 ⊢ ((𝐴[,]𝐵) ⊆ 𝐸 → (𝐹 ∈ (𝐸–cn→ℂ) → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ)))
84	25, 83	syl 18	. . . 4 ⊢ (𝜑 → (𝐹 ∈ (𝐸–cn→ℂ) → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ)))
85	82, 84	mpd 16	. . 3 ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
86	6, 8, 9, 30, 80, 85	ftc2 26373	. 2 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) d𝑡 = (((𝐹 ↾ (𝐴[,]𝐵))‘𝐵) − ((𝐹 ↾ (𝐴[,]𝐵))‘𝐴)))
87	22	fveq1d 6892	. . . . 5 ⊢ (𝜑 → ((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) = (((ℝ D 𝐹) ↾ (𝐴(,)𝐵))‘𝑡))
88		fvres 6909	. . . . 5 ⊢ (𝑡 ∈ (𝐴(,)𝐵) → (((ℝ D 𝐹) ↾ (𝐴(,)𝐵))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
89	87, 88	sylan9eq 2815	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
90	89	ralrimiva 3154	. . 3 ⊢ (𝜑 → ∀𝑡 ∈ (𝐴(,)𝐵)((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
91		itgeq2 26107	. . 3 ⊢ (∀𝑡 ∈ (𝐴(,)𝐵)((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) = ((ℝ D 𝐹)‘𝑡) → ∫(𝐴(,)𝐵)((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) d𝑡 = ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡)
92	90, 91	syl 18	. 2 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) d𝑡 = ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡)
93	6	rexrd 11366	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
94	8	rexrd 11366	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
95		ubicc2 13603	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
96	93, 94, 9, 95	syl3anc 1398	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐵))
97	96	fvresd 6910	. . 3 ⊢ (𝜑 → ((𝐹 ↾ (𝐴[,]𝐵))‘𝐵) = (𝐹‘𝐵))
98		lbicc2 13602	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
99	93, 94, 9, 98	syl3anc 1398	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵))
100	99	fvresd 6910	. . 3 ⊢ (𝜑 → ((𝐹 ↾ (𝐴[,]𝐵))‘𝐴) = (𝐹‘𝐴))
101	97, 100	oveq12d 7442	. 2 ⊢ (𝜑 → (((𝐹 ↾ (𝐴[,]𝐵))‘𝐵) − ((𝐹 ↾ (𝐴[,]𝐵))‘𝐴)) = ((𝐹‘𝐵) − (𝐹‘𝐴)))
102	86, 92, 101	3eqtr3d 2803	1 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ∀wral 3076 ∃wrex 3086 ⊆ wss 3896 ∩ cin 4088 class class class wbr 5102 dom cdm 5651 ran crn 5652 ↾ cres 5653 ⟶wf 6539 ‘cfv 6543 (class class class)co 7424 ℂcc 11205 ℝcr 11206 ℝ^*cxr 11349 ≤ cle 11351 − cmin 11548 (,)cioo 13483 [,]cicc 13486 abscabs 15410 TopOpenctopn 17601 topGenctg 17617 ℂ_fldccnfld 21687 intcnt 23344 –cn→ccncf 25206 volcvol 25793 MblFncmbf 25944 𝐿¹cibl 25947 ∫citg 25948 D cdv 26192

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-inf2 9649 ax-cc 10520 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-pre-mulgt0 11284 ax-pre-sup 11285 ax-addf 11286

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-symdif 4197 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-disj 5070 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7697 df-ofr 7698 df-om 7882 df-1st 8005 df-2nd 8006 df-supp 8185 df-frecs 8307 df-wrecs 8338 df-recs 8387 df-rdg 8426 df-1o 8484 df-2o 8485 df-oadd 8488 df-omul 8489 df-er 8725 df-map 8857 df-pm 8858 df-ixp 8934 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-fsupp 9362 df-fi 9410 df-sup 9441 df-inf 9442 df-oi 9511 df-dju 9989 df-card 10027 df-acn 10030 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-div 11981 df-nn 12343 df-2 12412 df-3 12413 df-4 12414 df-5 12415 df-6 12416 df-7 12417 df-8 12418 df-9 12419 df-n0 12614 df-z 12701 df-dec 12822 df-uz 12973 df-q 13083 df-rp 13128 df-xneg 13248 df-xadd 13249 df-xmul 13250 df-ioo 13487 df-ioc 13488 df-ico 13489 df-icc 13490 df-fz 13647 df-fzo 13796 df-fl 13939 df-mod 14017 df-seq 14152 df-exp 14212 df-hash 14482 df-cj 15275 df-re 15276 df-im 15277 df-sqrt 15411 df-abs 15412 df-limsup 15647 df-clim 15664 df-rlim 15665 df-sum 15863 df-struct 17334 df-sets 17351 df-slot 17369 df-ndx 17381 df-base 17397 df-ress 17418 df-plusg 17450 df-mulr 17451 df-starv 17452 df-sca 17453 df-vsca 17454 df-ip 17455 df-tset 17456 df-ple 17457 df-ds 17459 df-unif 17460 df-hom 17461 df-cco 17462 df-rest 17602 df-topn 17603 df-0g 17621 df-gsum 17622 df-topgen 17623 df-pt 17624 df-prds 17627 df-xrs 17683 df-qtop 17688 df-imas 17689 df-xps 17691 df-mre 17765 df-mrc 17766 df-acs 17768 df-mgm 18825 df-sgrp 18917 df-mnd 18933 df-submnd 18988 df-mulg 19287 df-cntz 19540 df-cmn 20005 df-psmet 21679 df-xmet 21680 df-met 21681 df-bl 21682 df-mopn 21683 df-fbas 21684 df-fg 21685 df-cnfld 21688 df-top 23221 df-topon 23238 df-topsp 23260 df-bases 23273 df-cld 23346 df-ntr 23347 df-cls 23348 df-nei 23425 df-lp 23463 df-perf 23464 df-cn 23554 df-cnp 23555 df-haus 23642 df-cmp 23714 df-tx 23890 df-hmeo 24083 df-fil 24174 df-fm 24266 df-flim 24267 df-flf 24268 df-xms 24648 df-ms 24649 df-tms 24650 df-cncf 25208 df-ovol 25794 df-vol 25795 df-mbf 25949 df-itg1 25950 df-itg2 25951 df-ibl 25952 df-itg 25953 df-0p 26000 df-limc 26195 df-dv 26196

This theorem is used by: fdvposlt 35278 fdvposle 35280 itgexpif 35285

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