ftc2re - Mathbox for Thierry Arnoux

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Theorem ftc2re 35107

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 13461	. . . . . 6 ⊢ (𝐶(,)𝐷) ⊆ ℝ
3	1, 2	eqsstri 3977	. . . . 5 ⊢ 𝐸 ⊆ ℝ
4	3	a1i 11	. . . 4 ⊢ (𝜑 → 𝐸 ⊆ ℝ)
5		ftc2re.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐸)
6	4, 5	sseldd 3932	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ)
7		ftc2re.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐸)
8	4, 7	sseldd 3932	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ)
9		ftc2re.le	. . 3 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
10		ax-resscn 11182	. . . . . . 7 ⊢ ℝ ⊆ ℂ
11	10	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℂ)
12		ftc2re.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐸⟶ℂ)
13		iccssre 13483	. . . . . . 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 25032	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
17	15, 16	dvres 26139	. . . . . 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 25049	. . . . . . 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 5972	. . . . 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 13487	. . . . . . 7 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
24	23	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
25	1, 5, 7	fct2relem 35106	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐸)
26	24, 25	sstrd 3941	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐸)
27		ftc2re.1	. . . . 5 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℂ))
28		rescncf 25126	. . . . 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 25794	. . . . . . 7 ⊢ (𝐴(,)𝐵) ∈ dom vol
32	31	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ dom vol)
33		cnmbf 25888	. . . . . 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 6882	. . . . . 6 ⊢ (vol‘dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))) = (vol‘((𝐴(,)𝐵) ∩ dom (ℝ D 𝐹)))
37		cncff 25122	. . . . . . . . . . . 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 6714	. . . . . . . . . 10 ⊢ (𝜑 → dom (ℝ D 𝐹) = 𝐸)
40	39	ineq2d 4166	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴(,)𝐵) ∩ dom (ℝ D 𝐹)) = ((𝐴(,)𝐵) ∩ 𝐸))
41		dfss2 3917	. . . . . . . . . 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 6883	. . . . . . 7 ⊢ (𝜑 → (vol‘((𝐴(,)𝐵) ∩ dom (ℝ D 𝐹))) = (vol‘(𝐴(,)𝐵)))
45		volioo 25798	. . . . . . . . 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 11667	. . . . . . . 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 25126	. . . . . . . . 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 25690	. . . . . . 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 3965	. . . . . . . . . 10 ⊢ (𝜑 → dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) ⊆ (𝐴[,]𝐵))
58		ssralv 4000	. . . . . . . . . 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 3931	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))) → 𝑦 ∈ (𝐴[,]𝐵))
63		fvres 6898	. . . . . . . . . . . . . 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 6898	. . . . . . . . . . . . . 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 6883	. . . . . . . . . . 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 5113	. . . . . . . . . 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 26068	. . . . 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 26149	. . . . 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 25126	. . . . 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 26272	. 2 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) d𝑡 = (((𝐹 ↾ (𝐴[,]𝐵))‘𝐵) − ((𝐹 ↾ (𝐴[,]𝐵))‘𝐴)))
87	22	fveq1d 6881	. . . . 5 ⊢ (𝜑 → ((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) = (((ℝ D 𝐹) ↾ (𝐴(,)𝐵))‘𝑡))
88		fvres 6898	. . . . 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 26006	. . 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 11284	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
94	8	rexrd 11284	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
95		ubicc2 13519	. . . . 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 6899	. . 3 ⊢ (𝜑 → ((𝐹 ↾ (𝐴[,]𝐵))‘𝐵) = (𝐹‘𝐵))
98		lbicc2 13518	. . . . 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 6899	. . 3 ⊢ (𝜑 → ((𝐹 ↾ (𝐴[,]𝐵))‘𝐴) = (𝐹‘𝐴))
101	97, 100	oveq12d 7432	. 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 1570 ∈ wcel 2145 ∀wral 3076 ∃wrex 3086 ∩ cin 3898 ⊆ wss 3899 class class class wbr 5103 dom cdm 5655 ran crn 5656 ↾ cres 5657 ⟶wf 6529 ‘cfv 6533 (class class class)co 7414 ℂcc 11123 ℝcr 11124 ℝ^*cxr 11267 ≤ cle 11269 − cmin 11466 (,)cioo 13399 [,]cicc 13402 abscabs 15322 TopOpenctopn 17507 topGenctg 17523 ℂ_fldccnfld 21586 intcnt 23243 –cn→ccncf 25105 volcvol 25692 MblFncmbf 25843 𝐿¹cibl 25846 ∫citg 25847 D cdv 26091

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-inf2 9621 ax-cc 10438 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202 ax-pre-sup 11203 ax-addf 11204

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 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 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-symdif 4199 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-disj 5071 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-ofr 7680 df-om 7864 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8456 df-2o 8457 df-oadd 8460 df-omul 8461 df-er 8697 df-map 8829 df-pm 8830 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9333 df-fi 9382 df-sup 9413 df-inf 9414 df-oi 9483 df-dju 9907 df-card 9945 df-acn 9948 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-div 11897 df-nn 12259 df-2 12328 df-3 12329 df-4 12330 df-5 12331 df-6 12332 df-7 12333 df-8 12334 df-9 12335 df-n0 12530 df-z 12617 df-dec 12738 df-uz 12889 df-q 12999 df-rp 13044 df-xneg 13164 df-xadd 13165 df-xmul 13166 df-ioo 13403 df-ioc 13404 df-ico 13405 df-icc 13406 df-fz 13563 df-fzo 13711 df-fl 13854 df-mod 13932 df-seq 14067 df-exp 14127 df-hash 14396 df-cj 15187 df-re 15188 df-im 15189 df-sqrt 15323 df-abs 15324 df-limsup 15559 df-clim 15576 df-rlim 15577 df-sum 15775 df-struct 17240 df-sets 17257 df-slot 17275 df-ndx 17287 df-base 17303 df-ress 17324 df-plusg 17356 df-mulr 17357 df-starv 17358 df-sca 17359 df-vsca 17360 df-ip 17361 df-tset 17362 df-ple 17363 df-ds 17365 df-unif 17366 df-hom 17367 df-cco 17368 df-rest 17508 df-topn 17509 df-0g 17527 df-gsum 17528 df-topgen 17529 df-pt 17530 df-prds 17533 df-xrs 17589 df-qtop 17594 df-imas 17595 df-xps 17597 df-mre 17671 df-mrc 17672 df-acs 17674 df-mgm 18731 df-sgrp 18822 df-mnd 18838 df-submnd 18893 df-mulg 19192 df-cntz 19445 df-cmn 19910 df-psmet 21578 df-xmet 21579 df-met 21580 df-bl 21581 df-mopn 21582 df-fbas 21583 df-fg 21584 df-cnfld 21587 df-top 23120 df-topon 23137 df-topsp 23159 df-bases 23172 df-cld 23245 df-ntr 23246 df-cls 23247 df-nei 23324 df-lp 23362 df-perf 23363 df-cn 23453 df-cnp 23454 df-haus 23541 df-cmp 23613 df-tx 23789 df-hmeo 23982 df-fil 24073 df-fm 24165 df-flim 24166 df-flf 24167 df-xms 24547 df-ms 24548 df-tms 24549 df-cncf 25107 df-ovol 25693 df-vol 25694 df-mbf 25848 df-itg1 25849 df-itg2 25850 df-ibl 25851 df-itg 25852 df-0p 25899 df-limc 26094 df-dv 26095

This theorem is used by: fdvposlt 35108 fdvposle 35110 itgexpif 35115

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