ftc2re - Mathbox for Thierry Arnoux

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Theorem ftc2re 32097

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 12840	. . . . . 6 ⊢ (𝐶(,)𝐷) ⊆ ℝ
3	1, 2	eqsstri 3926	. . . . 5 ⊢ 𝐸 ⊆ ℝ
4	3	a1i 11	. . . 4 ⊢ (𝜑 → 𝐸 ⊆ ℝ)
5		ftc2re.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐸)
6	4, 5	sseldd 3893	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ)
7		ftc2re.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐸)
8	4, 7	sseldd 3893	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ)
9		ftc2re.le	. . 3 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
10		ax-resscn 10632	. . . . . . 7 ⊢ ℝ ⊆ ℂ
11	10	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℂ)
12		ftc2re.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐸⟶ℂ)
13		iccssre 12861	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
14	6, 8, 13	syl2anc 587	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
15		eqid 2758	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
16	15	tgioo2 23504	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
17	15, 16	dvres 24610	. . . . . 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 837	. . . . 5 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))))
19		iccntr 23522	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
20	6, 8, 19	syl2anc 587	. . . . . 6 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
21	20	reseq2d 5823	. . . . 5 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))) = ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)))
22	18, 21	eqtrd 2793	. . . 4 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) = ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)))
23		ioossicc 12865	. . . . . . 7 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
24	23	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
25	1, 5, 7	fct2relem 32096	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐸)
26	24, 25	sstrd 3902	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐸)
27		ftc2re.1	. . . . 5 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℂ))
28		rescncf 23598	. . . . 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 2852	. . 3 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
31		ioombl 24265	. . . . . . 7 ⊢ (𝐴(,)𝐵) ∈ dom vol
32	31	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ dom vol)
33		cnmbf 24359	. . . . . 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 587	. . . . 5 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) ∈ MblFn)
35		dmres 5845	. . . . . . 7 ⊢ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) = ((𝐴(,)𝐵) ∩ dom (ℝ D 𝐹))
36	35	fveq2i 6661	. . . . . 6 ⊢ (vol‘dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))) = (vol‘((𝐴(,)𝐵) ∩ dom (ℝ D 𝐹)))
37		cncff 23594	. . . . . . . . . . . 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 6508	. . . . . . . . . 10 ⊢ (𝜑 → dom (ℝ D 𝐹) = 𝐸)
40	39	ineq2d 4117	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴(,)𝐵) ∩ dom (ℝ D 𝐹)) = ((𝐴(,)𝐵) ∩ 𝐸))
41		df-ss 3875	. . . . . . . . . 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 2793	. . . . . . . 8 ⊢ (𝜑 → ((𝐴(,)𝐵) ∩ dom (ℝ D 𝐹)) = (𝐴(,)𝐵))
44	43	fveq2d 6662	. . . . . . 7 ⊢ (𝜑 → (vol‘((𝐴(,)𝐵) ∩ dom (ℝ D 𝐹))) = (vol‘(𝐴(,)𝐵)))
45		volioo 24269	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴))
46	6, 8, 9, 45	syl3anc 1368	. . . . . . . 8 ⊢ (𝜑 → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴))
47	8, 6	resubcld 11106	. . . . . . . 8 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
48	46, 47	eqeltrd 2852	. . . . . . 7 ⊢ (𝜑 → (vol‘(𝐴(,)𝐵)) ∈ ℝ)
49	44, 48	eqeltrd 2852	. . . . . 6 ⊢ (𝜑 → (vol‘((𝐴(,)𝐵) ∩ dom (ℝ D 𝐹))) ∈ ℝ)
50	36, 49	eqeltrid 2856	. . . . 5 ⊢ (𝜑 → (vol‘dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))) ∈ ℝ)
51		rescncf 23598	. . . . . . . . 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 24161	. . . . . . 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 1368	. . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑦)) ≤ 𝑥)
56	35, 43	syl5eq 2805	. . . . . . . . . . 11 ⊢ (𝜑 → dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) = (𝐴(,)𝐵))
57	56, 24	eqsstrd 3930	. . . . . . . . . 10 ⊢ (𝜑 → dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) ⊆ (𝐴[,]𝐵))
58		ssralv 3958	. . . . . . . . . 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 484	. . . . . . . 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 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) ⊆ (𝐴[,]𝐵))
62	61	sselda 3892	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))) → 𝑦 ∈ (𝐴[,]𝐵))
63		fvres 6677	. . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))) → 𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)))
66	56	ad2antrr 725	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))) → dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) = (𝐴(,)𝐵))
67	65, 66	eleqtrd 2854	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))) → 𝑦 ∈ (𝐴(,)𝐵))
68		fvres 6677	. . . . . . . . . . . . . 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 2796	. . . . . . . . . . . 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 6662	. . . . . . . . . . 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 5042	. . . . . . . . . 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 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 3198	. . . . . 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 24539	. . . . 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 1368	. . . 4 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) ∈ 𝐿¹)
80	22, 79	eqeltrd 2852	. . 3 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) ∈ 𝐿¹)
81		dvcn 24620	. . . . 5 ⊢ (((ℝ ⊆ ℂ ∧ 𝐹:𝐸⟶ℂ ∧ 𝐸 ⊆ ℝ) ∧ dom (ℝ D 𝐹) = 𝐸) → 𝐹 ∈ (𝐸–cn→ℂ))
82	11, 12, 4, 39, 81	syl31anc 1370	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐸–cn→ℂ))
83		rescncf 23598	. . . . 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 24743	. 2 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) d𝑡 = (((𝐹 ↾ (𝐴[,]𝐵))‘𝐵) − ((𝐹 ↾ (𝐴[,]𝐵))‘𝐴)))
87	22	fveq1d 6660	. . . . 5 ⊢ (𝜑 → ((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) = (((ℝ D 𝐹) ↾ (𝐴(,)𝐵))‘𝑡))
88		fvres 6677	. . . . 5 ⊢ (𝑡 ∈ (𝐴(,)𝐵) → (((ℝ D 𝐹) ↾ (𝐴(,)𝐵))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
89	87, 88	sylan9eq 2813	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
90	89	ralrimiva 3113	. . 3 ⊢ (𝜑 → ∀𝑡 ∈ (𝐴(,)𝐵)((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
91		itgeq2 24477	. . 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 10729	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
94	8	rexrd 10729	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
95		ubicc2 12897	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
96	93, 94, 9, 95	syl3anc 1368	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐵))
97	96	fvresd 6678	. . 3 ⊢ (𝜑 → ((𝐹 ↾ (𝐴[,]𝐵))‘𝐵) = (𝐹‘𝐵))
98		lbicc2 12896	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
99	93, 94, 9, 98	syl3anc 1368	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵))
100	99	fvresd 6678	. . 3 ⊢ (𝜑 → ((𝐹 ↾ (𝐴[,]𝐵))‘𝐴) = (𝐹‘𝐴))
101	97, 100	oveq12d 7168	. 2 ⊢ (𝜑 → (((𝐹 ↾ (𝐴[,]𝐵))‘𝐵) − ((𝐹 ↾ (𝐴[,]𝐵))‘𝐴)) = ((𝐹‘𝐵) − (𝐹‘𝐴)))
102	86, 92, 101	3eqtr3d 2801	1 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3070 ∃wrex 3071 ∩ cin 3857 ⊆ wss 3858 class class class wbr 5032 dom cdm 5524 ran crn 5525 ↾ cres 5526 ⟶wf 6331 ‘cfv 6335 (class class class)co 7150 ℂcc 10573 ℝcr 10574 ℝ^*cxr 10712 ≤ cle 10714 − cmin 10908 (,)cioo 12779 [,]cicc 12782 abscabs 14641 TopOpenctopn 16753 topGenctg 16769 ℂ_fldccnfld 20166 intcnt 21717 –cn→ccncf 23577 volcvol 24163 MblFncmbf 24314 𝐿¹cibl 24317 ∫citg 24318 D cdv 24562

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5156 ax-sep 5169 ax-nul 5176 ax-pow 5234 ax-pr 5298 ax-un 7459 ax-inf2 9137 ax-cc 9895 ax-cnex 10631 ax-resscn 10632 ax-1cn 10633 ax-icn 10634 ax-addcl 10635 ax-addrcl 10636 ax-mulcl 10637 ax-mulrcl 10638 ax-mulcom 10639 ax-addass 10640 ax-mulass 10641 ax-distr 10642 ax-i2m1 10643 ax-1ne0 10644 ax-1rid 10645 ax-rnegex 10646 ax-rrecex 10647 ax-cnre 10648 ax-pre-lttri 10649 ax-pre-lttrn 10650 ax-pre-ltadd 10651 ax-pre-mulgt0 10652 ax-pre-sup 10653 ax-addf 10654 ax-mulf 10655

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3697 df-csb 3806 df-dif 3861 df-un 3863 df-in 3865 df-ss 3875 df-pss 3877 df-symdif 4147 df-nul 4226 df-if 4421 df-pw 4496 df-sn 4523 df-pr 4525 df-tp 4527 df-op 4529 df-uni 4799 df-int 4839 df-iun 4885 df-iin 4886 df-disj 4998 df-br 5033 df-opab 5095 df-mpt 5113 df-tr 5139 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5483 df-se 5484 df-we 5485 df-xp 5530 df-rel 5531 df-cnv 5532 df-co 5533 df-dm 5534 df-rn 5535 df-res 5536 df-ima 5537 df-pred 6126 df-ord 6172 df-on 6173 df-lim 6174 df-suc 6175 df-iota 6294 df-fun 6337 df-fn 6338 df-f 6339 df-f1 6340 df-fo 6341 df-f1o 6342 df-fv 6343 df-isom 6344 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7405 df-ofr 7406 df-om 7580 df-1st 7693 df-2nd 7694 df-supp 7836 df-wrecs 7957 df-recs 8018 df-rdg 8056 df-1o 8112 df-2o 8113 df-oadd 8116 df-omul 8117 df-er 8299 df-map 8418 df-pm 8419 df-ixp 8480 df-en 8528 df-dom 8529 df-sdom 8530 df-fin 8531 df-fsupp 8867 df-fi 8908 df-sup 8939 df-inf 8940 df-oi 9007 df-dju 9363 df-card 9401 df-acn 9404 df-pnf 10715 df-mnf 10716 df-xr 10717 df-ltxr 10718 df-le 10719 df-sub 10910 df-neg 10911 df-div 11336 df-nn 11675 df-2 11737 df-3 11738 df-4 11739 df-5 11740 df-6 11741 df-7 11742 df-8 11743 df-9 11744 df-n0 11935 df-z 12021 df-dec 12138 df-uz 12283 df-q 12389 df-rp 12431 df-xneg 12548 df-xadd 12549 df-xmul 12550 df-ioo 12783 df-ioc 12784 df-ico 12785 df-icc 12786 df-fz 12940 df-fzo 13083 df-fl 13211 df-mod 13287 df-seq 13419 df-exp 13480 df-hash 13741 df-cj 14506 df-re 14507 df-im 14508 df-sqrt 14642 df-abs 14643 df-limsup 14876 df-clim 14893 df-rlim 14894 df-sum 15091 df-struct 16543 df-ndx 16544 df-slot 16545 df-base 16547 df-sets 16548 df-ress 16549 df-plusg 16636 df-mulr 16637 df-starv 16638 df-sca 16639 df-vsca 16640 df-ip 16641 df-tset 16642 df-ple 16643 df-ds 16645 df-unif 16646 df-hom 16647 df-cco 16648 df-rest 16754 df-topn 16755 df-0g 16773 df-gsum 16774 df-topgen 16775 df-pt 16776 df-prds 16779 df-xrs 16833 df-qtop 16838 df-imas 16839 df-xps 16841 df-mre 16915 df-mrc 16916 df-acs 16918 df-mgm 17918 df-sgrp 17967 df-mnd 17978 df-submnd 18023 df-mulg 18292 df-cntz 18514 df-cmn 18975 df-psmet 20158 df-xmet 20159 df-met 20160 df-bl 20161 df-mopn 20162 df-fbas 20163 df-fg 20164 df-cnfld 20167 df-top 21594 df-topon 21611 df-topsp 21633 df-bases 21646 df-cld 21719 df-ntr 21720 df-cls 21721 df-nei 21798 df-lp 21836 df-perf 21837 df-cn 21927 df-cnp 21928 df-haus 22015 df-cmp 22087 df-tx 22262 df-hmeo 22455 df-fil 22546 df-fm 22638 df-flim 22639 df-flf 22640 df-xms 23022 df-ms 23023 df-tms 23024 df-cncf 23579 df-ovol 24164 df-vol 24165 df-mbf 24319 df-itg1 24320 df-itg2 24321 df-ibl 24322 df-itg 24323 df-0p 24370 df-limc 24565 df-dv 24566

This theorem is referenced by: fdvposlt 32098 fdvposle 32100 itgexpif 32105

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