ftc2re - Mathbox for Thierry Arnoux

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Theorem ftc2re 32578

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 13140	. . . . . 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 3922	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ)
7		ftc2re.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐸)
8	4, 7	sseldd 3922	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ)
9		ftc2re.le	. . 3 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
10		ax-resscn 10928	. . . . . . 7 ⊢ ℝ ⊆ ℂ
11	10	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℂ)
12		ftc2re.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐸⟶ℂ)
13		iccssre 13161	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
14	6, 8, 13	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
15		eqid 2738	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
16	15	tgioo2 23966	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
17	15, 16	dvres 25075	. . . . . 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 836	. . . . 5 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))))
19		iccntr 23984	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
20	6, 8, 19	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
21	20	reseq2d 5891	. . . . 5 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))) = ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)))
22	18, 21	eqtrd 2778	. . . 4 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) = ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)))
23		ioossicc 13165	. . . . . . 7 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
24	23	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
25	1, 5, 7	fct2relem 32577	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐸)
26	24, 25	sstrd 3931	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐸)
27		ftc2re.1	. . . . 5 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℂ))
28		rescncf 24060	. . . . 5 ⊢ ((𝐴(,)𝐵) ⊆ 𝐸 → ((ℝ D 𝐹) ∈ (𝐸–cn→ℂ) → ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) ∈ ((𝐴(,)𝐵)–cn→ℂ)))
29	26, 27, 28	sylc 65	. . . 4 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
30	22, 29	eqeltrd 2839	. . 3 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
31		ioombl 24729	. . . . . . 7 ⊢ (𝐴(,)𝐵) ∈ dom vol
32	31	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ dom vol)
33		cnmbf 24823	. . . . . 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 584	. . . . 5 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) ∈ MblFn)
35		dmres 5913	. . . . . . 7 ⊢ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) = ((𝐴(,)𝐵) ∩ dom (ℝ D 𝐹))
36	35	fveq2i 6777	. . . . . 6 ⊢ (vol‘dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))) = (vol‘((𝐴(,)𝐵) ∩ dom (ℝ D 𝐹)))
37		cncff 24056	. . . . . . . . . . . 12 ⊢ ((ℝ D 𝐹) ∈ (𝐸–cn→ℂ) → (ℝ D 𝐹):𝐸⟶ℂ)
38	27, 37	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D 𝐹):𝐸⟶ℂ)
39	38	fdmd 6611	. . . . . . . . . 10 ⊢ (𝜑 → dom (ℝ D 𝐹) = 𝐸)
40	39	ineq2d 4146	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴(,)𝐵) ∩ dom (ℝ D 𝐹)) = ((𝐴(,)𝐵) ∩ 𝐸))
41		df-ss 3904	. . . . . . . . . 10 ⊢ ((𝐴(,)𝐵) ⊆ 𝐸 ↔ ((𝐴(,)𝐵) ∩ 𝐸) = (𝐴(,)𝐵))
42	26, 41	sylib 217	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴(,)𝐵) ∩ 𝐸) = (𝐴(,)𝐵))
43	40, 42	eqtrd 2778	. . . . . . . 8 ⊢ (𝜑 → ((𝐴(,)𝐵) ∩ dom (ℝ D 𝐹)) = (𝐴(,)𝐵))
44	43	fveq2d 6778	. . . . . . 7 ⊢ (𝜑 → (vol‘((𝐴(,)𝐵) ∩ dom (ℝ D 𝐹))) = (vol‘(𝐴(,)𝐵)))
45		volioo 24733	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴))
46	6, 8, 9, 45	syl3anc 1370	. . . . . . . 8 ⊢ (𝜑 → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴))
47	8, 6	resubcld 11403	. . . . . . . 8 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
48	46, 47	eqeltrd 2839	. . . . . . 7 ⊢ (𝜑 → (vol‘(𝐴(,)𝐵)) ∈ ℝ)
49	44, 48	eqeltrd 2839	. . . . . 6 ⊢ (𝜑 → (vol‘((𝐴(,)𝐵) ∩ dom (ℝ D 𝐹))) ∈ ℝ)
50	36, 49	eqeltrid 2843	. . . . 5 ⊢ (𝜑 → (vol‘dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))) ∈ ℝ)
51		rescncf 24060	. . . . . . . . 9 ⊢ ((𝐴[,]𝐵) ⊆ 𝐸 → ((ℝ D 𝐹) ∈ (𝐸–cn→ℂ) → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ)))
52	25, 51	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D 𝐹) ∈ (𝐸–cn→ℂ) → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ)))
53	27, 52	mpd 15	. . . . . . 7 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
54		cniccbdd 24625	. . . . . . 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 1370	. . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑦)) ≤ 𝑥)
56	35, 43	eqtrid 2790	. . . . . . . . . . 11 ⊢ (𝜑 → dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) = (𝐴(,)𝐵))
57	56, 24	eqsstrd 3959	. . . . . . . . . 10 ⊢ (𝜑 → dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) ⊆ (𝐴[,]𝐵))
58		ssralv 3987	. . . . . . . . . 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 17	. . . . . . . . 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 481	. . . . . . . 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 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) ⊆ (𝐴[,]𝐵))
62	61	sselda 3921	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))) → 𝑦 ∈ (𝐴[,]𝐵))
63		fvres 6793	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (𝐴[,]𝐵) → (((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑦) = ((ℝ D 𝐹)‘𝑦))
64	62, 63	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))) → (((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑦) = ((ℝ D 𝐹)‘𝑦))
65		simpr 485	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))) → 𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)))
66	56	ad2antrr 723	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))) → dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) = (𝐴(,)𝐵))
67	65, 66	eleqtrd 2841	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))) → 𝑦 ∈ (𝐴(,)𝐵))
68		fvres 6793	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (𝐴(,)𝐵) → (((ℝ D 𝐹) ↾ (𝐴(,)𝐵))‘𝑦) = ((ℝ D 𝐹)‘𝑦))
69	67, 68	syl 17	. . . . . . . . . . . . 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 2781	. . . . . . . . . . . 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 6778	. . . . . . . . . . 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 5084	. . . . . . . . . 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 228	. . . . . . . . 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 3108	. . . . . . . 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 47	. . . . . . 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 3203	. . . . . 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 15	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))(abs‘(((ℝ D 𝐹) ↾ (𝐴(,)𝐵))‘𝑦)) ≤ 𝑥)
78		bddibl 25004	. . . . 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 1370	. . . 4 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) ∈ 𝐿¹)
80	22, 79	eqeltrd 2839	. . 3 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) ∈ 𝐿¹)
81		dvcn 25085	. . . . 5 ⊢ (((ℝ ⊆ ℂ ∧ 𝐹:𝐸⟶ℂ ∧ 𝐸 ⊆ ℝ) ∧ dom (ℝ D 𝐹) = 𝐸) → 𝐹 ∈ (𝐸–cn→ℂ))
82	11, 12, 4, 39, 81	syl31anc 1372	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐸–cn→ℂ))
83		rescncf 24060	. . . . 5 ⊢ ((𝐴[,]𝐵) ⊆ 𝐸 → (𝐹 ∈ (𝐸–cn→ℂ) → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ)))
84	25, 83	syl 17	. . . 4 ⊢ (𝜑 → (𝐹 ∈ (𝐸–cn→ℂ) → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ)))
85	82, 84	mpd 15	. . 3 ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
86	6, 8, 9, 30, 80, 85	ftc2 25208	. 2 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) d𝑡 = (((𝐹 ↾ (𝐴[,]𝐵))‘𝐵) − ((𝐹 ↾ (𝐴[,]𝐵))‘𝐴)))
87	22	fveq1d 6776	. . . . 5 ⊢ (𝜑 → ((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) = (((ℝ D 𝐹) ↾ (𝐴(,)𝐵))‘𝑡))
88		fvres 6793	. . . . 5 ⊢ (𝑡 ∈ (𝐴(,)𝐵) → (((ℝ D 𝐹) ↾ (𝐴(,)𝐵))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
89	87, 88	sylan9eq 2798	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
90	89	ralrimiva 3103	. . 3 ⊢ (𝜑 → ∀𝑡 ∈ (𝐴(,)𝐵)((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
91		itgeq2 24942	. . 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 17	. 2 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) d𝑡 = ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡)
93	6	rexrd 11025	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
94	8	rexrd 11025	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
95		ubicc2 13197	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
96	93, 94, 9, 95	syl3anc 1370	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐵))
97	96	fvresd 6794	. . 3 ⊢ (𝜑 → ((𝐹 ↾ (𝐴[,]𝐵))‘𝐵) = (𝐹‘𝐵))
98		lbicc2 13196	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
99	93, 94, 9, 98	syl3anc 1370	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵))
100	99	fvresd 6794	. . 3 ⊢ (𝜑 → ((𝐹 ↾ (𝐴[,]𝐵))‘𝐴) = (𝐹‘𝐴))
101	97, 100	oveq12d 7293	. 2 ⊢ (𝜑 → (((𝐹 ↾ (𝐴[,]𝐵))‘𝐵) − ((𝐹 ↾ (𝐴[,]𝐵))‘𝐴)) = ((𝐹‘𝐵) − (𝐹‘𝐴)))
102	86, 92, 101	3eqtr3d 2786	1 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 ∩ cin 3886 ⊆ wss 3887 class class class wbr 5074 dom cdm 5589 ran crn 5590 ↾ cres 5591 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 ℝcr 10870 ℝ^*cxr 11008 ≤ cle 11010 − cmin 11205 (,)cioo 13079 [,]cicc 13082 abscabs 14945 TopOpenctopn 17132 topGenctg 17148 ℂ_fldccnfld 20597 intcnt 22168 –cn→ccncf 24039 volcvol 24627 MblFncmbf 24778 𝐿¹cibl 24781 ∫citg 24782 D cdv 25027

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cc 10191 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-symdif 4176 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-disj 5040 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-ofr 7534 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-omul 8302 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-dju 9659 df-card 9697 df-acn 9700 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-limsup 15180 df-clim 15197 df-rlim 15198 df-sum 15398 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-cmp 22538 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041 df-ovol 24628 df-vol 24629 df-mbf 24783 df-itg1 24784 df-itg2 24785 df-ibl 24786 df-itg 24787 df-0p 24834 df-limc 25030 df-dv 25031

This theorem is referenced by: fdvposlt 32579 fdvposle 32581 itgexpif 32586

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