ftc2re - Mathbox for Thierry Arnoux

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Theorem ftc2re 34589

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 13368	. . . . . 6 ⊢ (𝐶(,)𝐷) ⊆ ℝ
3	1, 2	eqsstri 3993	. . . . 5 ⊢ 𝐸 ⊆ ℝ
4	3	a1i 11	. . . 4 ⊢ (𝜑 → 𝐸 ⊆ ℝ)
5		ftc2re.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐸)
6	4, 5	sseldd 3947	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ)
7		ftc2re.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐸)
8	4, 7	sseldd 3947	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ)
9		ftc2re.le	. . 3 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
10		ax-resscn 11125	. . . . . . 7 ⊢ ℝ ⊆ ℂ
11	10	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℂ)
12		ftc2re.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐸⟶ℂ)
13		iccssre 13390	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
14	6, 8, 13	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
15		eqid 2729	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
16		tgioo4 24693	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
17	15, 16	dvres 25812	. . . . . 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 838	. . . . 5 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))))
19		iccntr 24710	. . . . . . 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 5950	. . . . 5 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))) = ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)))
22	18, 21	eqtrd 2764	. . . 4 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) = ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)))
23		ioossicc 13394	. . . . . . 7 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
24	23	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
25	1, 5, 7	fct2relem 34588	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐸)
26	24, 25	sstrd 3957	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐸)
27		ftc2re.1	. . . . 5 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℂ))
28		rescncf 24790	. . . . 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 2828	. . 3 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
31		ioombl 25466	. . . . . . 7 ⊢ (𝐴(,)𝐵) ∈ dom vol
32	31	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ dom vol)
33		cnmbf 25560	. . . . . 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 5983	. . . . . . 7 ⊢ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) = ((𝐴(,)𝐵) ∩ dom (ℝ D 𝐹))
36	35	fveq2i 6861	. . . . . 6 ⊢ (vol‘dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))) = (vol‘((𝐴(,)𝐵) ∩ dom (ℝ D 𝐹)))
37		cncff 24786	. . . . . . . . . . . 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 6698	. . . . . . . . . 10 ⊢ (𝜑 → dom (ℝ D 𝐹) = 𝐸)
40	39	ineq2d 4183	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴(,)𝐵) ∩ dom (ℝ D 𝐹)) = ((𝐴(,)𝐵) ∩ 𝐸))
41		dfss2 3932	. . . . . . . . . 10 ⊢ ((𝐴(,)𝐵) ⊆ 𝐸 ↔ ((𝐴(,)𝐵) ∩ 𝐸) = (𝐴(,)𝐵))
42	26, 41	sylib 218	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴(,)𝐵) ∩ 𝐸) = (𝐴(,)𝐵))
43	40, 42	eqtrd 2764	. . . . . . . 8 ⊢ (𝜑 → ((𝐴(,)𝐵) ∩ dom (ℝ D 𝐹)) = (𝐴(,)𝐵))
44	43	fveq2d 6862	. . . . . . 7 ⊢ (𝜑 → (vol‘((𝐴(,)𝐵) ∩ dom (ℝ D 𝐹))) = (vol‘(𝐴(,)𝐵)))
45		volioo 25470	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴))
46	6, 8, 9, 45	syl3anc 1373	. . . . . . . 8 ⊢ (𝜑 → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴))
47	8, 6	resubcld 11606	. . . . . . . 8 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
48	46, 47	eqeltrd 2828	. . . . . . 7 ⊢ (𝜑 → (vol‘(𝐴(,)𝐵)) ∈ ℝ)
49	44, 48	eqeltrd 2828	. . . . . 6 ⊢ (𝜑 → (vol‘((𝐴(,)𝐵) ∩ dom (ℝ D 𝐹))) ∈ ℝ)
50	36, 49	eqeltrid 2832	. . . . 5 ⊢ (𝜑 → (vol‘dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))) ∈ ℝ)
51		rescncf 24790	. . . . . . . . 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 25362	. . . . . . 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 1373	. . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (𝐴[,]𝐵)(abs‘(((ℝ D 𝐹) ↾ (𝐴[,]𝐵))‘𝑦)) ≤ 𝑥)
56	35, 43	eqtrid 2776	. . . . . . . . . . 11 ⊢ (𝜑 → dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) = (𝐴(,)𝐵))
57	56, 24	eqsstrd 3981	. . . . . . . . . 10 ⊢ (𝜑 → dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) ⊆ (𝐴[,]𝐵))
58		ssralv 4015	. . . . . . . . . 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 480	. . . . . . . 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 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) ⊆ (𝐴[,]𝐵))
62	61	sselda 3946	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))) → 𝑦 ∈ (𝐴[,]𝐵))
63		fvres 6877	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))) → 𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)))
66	56	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))) → dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) = (𝐴(,)𝐵))
67	65, 66	eleqtrd 2830	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ dom ((ℝ D 𝐹) ↾ (𝐴(,)𝐵))) → 𝑦 ∈ (𝐴(,)𝐵))
68		fvres 6877	. . . . . . . . . . . . . 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 2767	. . . . . . . . . . . 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 6862	. . . . . . . . . . 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 5117	. . . . . . . . . 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 229	. . . . . . . . 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 3145	. . . . . . . 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 3146	. . . . . 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 25741	. . . . 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 1373	. . . 4 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴(,)𝐵)) ∈ 𝐿¹)
80	22, 79	eqeltrd 2828	. . 3 ⊢ (𝜑 → (ℝ D (𝐹 ↾ (𝐴[,]𝐵))) ∈ 𝐿¹)
81		dvcn 25823	. . . . 5 ⊢ (((ℝ ⊆ ℂ ∧ 𝐹:𝐸⟶ℂ ∧ 𝐸 ⊆ ℝ) ∧ dom (ℝ D 𝐹) = 𝐸) → 𝐹 ∈ (𝐸–cn→ℂ))
82	11, 12, 4, 39, 81	syl31anc 1375	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐸–cn→ℂ))
83		rescncf 24790	. . . . 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 25951	. 2 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) d𝑡 = (((𝐹 ↾ (𝐴[,]𝐵))‘𝐵) − ((𝐹 ↾ (𝐴[,]𝐵))‘𝐴)))
87	22	fveq1d 6860	. . . . 5 ⊢ (𝜑 → ((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) = (((ℝ D 𝐹) ↾ (𝐴(,)𝐵))‘𝑡))
88		fvres 6877	. . . . 5 ⊢ (𝑡 ∈ (𝐴(,)𝐵) → (((ℝ D 𝐹) ↾ (𝐴(,)𝐵))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
89	87, 88	sylan9eq 2784	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
90	89	ralrimiva 3125	. . 3 ⊢ (𝜑 → ∀𝑡 ∈ (𝐴(,)𝐵)((ℝ D (𝐹 ↾ (𝐴[,]𝐵)))‘𝑡) = ((ℝ D 𝐹)‘𝑡))
91		itgeq2 25679	. . 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 11224	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
94	8	rexrd 11224	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
95		ubicc2 13426	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
96	93, 94, 9, 95	syl3anc 1373	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐵))
97	96	fvresd 6878	. . 3 ⊢ (𝜑 → ((𝐹 ↾ (𝐴[,]𝐵))‘𝐵) = (𝐹‘𝐵))
98		lbicc2 13425	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
99	93, 94, 9, 98	syl3anc 1373	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵))
100	99	fvresd 6878	. . 3 ⊢ (𝜑 → ((𝐹 ↾ (𝐴[,]𝐵))‘𝐴) = (𝐹‘𝐴))
101	97, 100	oveq12d 7405	. 2 ⊢ (𝜑 → (((𝐹 ↾ (𝐴[,]𝐵))‘𝐵) − ((𝐹 ↾ (𝐴[,]𝐵))‘𝐴)) = ((𝐹‘𝐵) − (𝐹‘𝐴)))
102	86, 92, 101	3eqtr3d 2772	1 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 ∩ cin 3913 ⊆ wss 3914 class class class wbr 5107 dom cdm 5638 ran crn 5639 ↾ cres 5640 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 ℂcc 11066 ℝcr 11067 ℝ^*cxr 11207 ≤ cle 11209 − cmin 11405 (,)cioo 13306 [,]cicc 13309 abscabs 15200 TopOpenctopn 17384 topGenctg 17400 ℂ_fldccnfld 21264 intcnt 22904 –cn→ccncf 24769 volcvol 25364 MblFncmbf 25515 𝐿¹cibl 25518 ∫citg 25519 D cdv 25764

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cc 10388 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-symdif 4216 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-disj 5075 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-ofr 7654 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-oadd 8438 df-omul 8439 df-er 8671 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-dju 9854 df-card 9892 df-acn 9895 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ioo 13310 df-ioc 13311 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-limsup 15437 df-clim 15454 df-rlim 15455 df-sum 15653 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17465 df-qtop 17470 df-imas 17471 df-xps 17473 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-mulg 19000 df-cntz 19249 df-cmn 19712 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-fbas 21261 df-fg 21262 df-cnfld 21265 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-nei 22985 df-lp 23023 df-perf 23024 df-cn 23114 df-cnp 23115 df-haus 23202 df-cmp 23274 df-tx 23449 df-hmeo 23642 df-fil 23733 df-fm 23825 df-flim 23826 df-flf 23827 df-xms 24208 df-ms 24209 df-tms 24210 df-cncf 24771 df-ovol 25365 df-vol 25366 df-mbf 25520 df-itg1 25521 df-itg2 25522 df-ibl 25523 df-itg 25524 df-0p 25571 df-limc 25767 df-dv 25768

This theorem is referenced by: fdvposlt 34590 fdvposle 34592 itgexpif 34597

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