ftc2 - Metamath Proof Explorer

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Theorem ftc2 26019

Description: The Fundamental Theorem of Calculus, part two. If 𝐹 is a function continuous on [𝐴, 𝐵] and continuously differentiable on (𝐴, 𝐵), then the integral of the derivative of 𝐹 is equal to 𝐹(𝐵) − 𝐹(𝐴). This is part of Metamath 100 proof #15. (Contributed by Mario Carneiro, 2-Sep-2014.)

**Hypotheses**
Ref	Expression
ftc2.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
ftc2.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
ftc2.le	⊢ (𝜑 → 𝐴 ≤ 𝐵)
ftc2.c	⊢ (𝜑 → (ℝ D 𝐹) ∈ ((𝐴(,)𝐵)–cn→ℂ))
ftc2.i	⊢ (𝜑 → (ℝ D 𝐹) ∈ 𝐿¹)
ftc2.f	⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))

**Assertion**
Ref	Expression
ftc2	⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴)))

Distinct variable groups: 𝑡,𝐴 𝑡,𝐵 𝑡,𝐹 𝜑,𝑡

Proof of Theorem ftc2 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ftc2.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2	1	rexrd 11194	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
3		ftc2.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
4	3	rexrd 11194	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
5		ftc2.le	. . . . . 6 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
6		ubicc2 13393	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
7	2, 4, 5, 6	syl3anc 1374	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐵))
8		fvex 6855	. . . . . 6 ⊢ ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴) ∈ V
9	8	fvconst2 7160	. . . . 5 ⊢ (𝐵 ∈ (𝐴[,]𝐵) → (((𝐴[,]𝐵) × {((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴)})‘𝐵) = ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴))
10	7, 9	syl 17	. . . 4 ⊢ (𝜑 → (((𝐴[,]𝐵) × {((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴)})‘𝐵) = ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴))
11		eqid 2737	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
12	11	subcn 24823	. . . . . . . . 9 ⊢ − ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
13	12	a1i 11	. . . . . . . 8 ⊢ (𝜑 → − ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld)))
14		eqid 2737	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡) = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡)
15		ssidd 3959	. . . . . . . . 9 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐵))
16		ioossre 13335	. . . . . . . . . 10 ⊢ (𝐴(,)𝐵) ⊆ ℝ
17	16	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
18		ftc2.i	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D 𝐹) ∈ 𝐿¹)
19		ftc2.c	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D 𝐹) ∈ ((𝐴(,)𝐵)–cn→ℂ))
20		cncff 24854	. . . . . . . . . 10 ⊢ ((ℝ D 𝐹) ∈ ((𝐴(,)𝐵)–cn→ℂ) → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ)
21	19, 20	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ)
22	14, 1, 3, 5, 15, 17, 18, 21	ftc1a 26012	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡) ∈ ((𝐴[,]𝐵)–cn→ℂ))
23		ftc2.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))
24		cncff 24854	. . . . . . . . . . 11 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
25	23, 24	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ)
26	25	feqmptd 6910	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥)))
27	26, 23	eqeltrrd 2838	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
28	11, 13, 22, 27	cncfmpt2f 24876	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥))) ∈ ((𝐴[,]𝐵)–cn→ℂ))
29		ax-resscn 11095	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℂ
30	29	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ⊆ ℂ)
31		iccssre 13357	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
32	1, 3, 31	syl2anc 585	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
33		fvex 6855	. . . . . . . . . . . . 13 ⊢ ((ℝ D 𝐹)‘𝑡) ∈ V
34	33	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ 𝑡 ∈ (𝐴(,)𝑥)) → ((ℝ D 𝐹)‘𝑡) ∈ V)
35	3	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ)
36	35	rexrd 11194	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ^*)
37		elicc2 13339	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
38	1, 3, 37	syl2anc 585	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
39	38	biimpa 476	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵))
40	39	simp3d 1145	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ 𝐵)
41		iooss2 13309	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝑥 ≤ 𝐵) → (𝐴(,)𝑥) ⊆ (𝐴(,)𝐵))
42	36, 40, 41	syl2anc 585	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐴(,)𝑥) ⊆ (𝐴(,)𝐵))
43		ioombl 25534	. . . . . . . . . . . . . 14 ⊢ (𝐴(,)𝑥) ∈ dom vol
44	43	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐴(,)𝑥) ∈ dom vol)
45	33	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ V)
46	21	feqmptd 6910	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ℝ D 𝐹) = (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)))
47	46, 18	eqeltrrd 2838	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿¹)
48	47	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿¹)
49	42, 44, 45, 48	iblss 25774	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ (𝐴(,)𝑥) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿¹)
50	34, 49	itgcl 25753	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ)
51	25	ffvelcdmda 7038	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℂ)
52	50, 51	subcld 11504	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)) ∈ ℂ)
53		tgioo4 24761	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
54		iccntr 24778	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
55	1, 3, 54	syl2anc 585	. . . . . . . . . 10 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
56	30, 32, 52, 53, 11, 55	dvmptntr 25943	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))))
57		reelprrecn 11130	. . . . . . . . . . 11 ⊢ ℝ ∈ {ℝ, ℂ}
58	57	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
59		ioossicc 13361	. . . . . . . . . . . 12 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
60	59	sseli 3931	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ (𝐴[,]𝐵))
61	60, 50	sylan2 594	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ)
62	21	ffvelcdmda 7038	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
63	14, 1, 3, 5, 19, 18	ftc1cn 26018	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡)) = (ℝ D 𝐹))
64	30, 32, 50, 53, 11, 55	dvmptntr 25943	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡)) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡)))
65	21	feqmptd 6910	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D 𝐹) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥)))
66	63, 64, 65	3eqtr3d 2780	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥)))
67	60, 51	sylan2 594	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑥) ∈ ℂ)
68	30, 32, 51, 53, 11, 55	dvmptntr 25943	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥))) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥))))
69	26	oveq2d 7384	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥))))
70	69, 65	eqtr3d 2774	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥)))
71	68, 70	eqtr3d 2774	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥)))
72	58, 61, 62, 66, 67, 62, 71	dvmptsub 25939	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑥) − ((ℝ D 𝐹)‘𝑥))))
73	62	subidd 11492	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐹)‘𝑥) − ((ℝ D 𝐹)‘𝑥)) = 0)
74	73	mpteq2dva 5193	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑥) − ((ℝ D 𝐹)‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0))
75	56, 72, 74	3eqtrd 2776	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0))
76		fconstmpt 5694	. . . . . . . 8 ⊢ ((𝐴(,)𝐵) × {0}) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0)
77	75, 76	eqtr4di 2790	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))) = ((𝐴(,)𝐵) × {0}))
78	1, 3, 28, 77	dveq0 25973	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥))) = ((𝐴[,]𝐵) × {((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴)}))
79	78	fveq1d 6844	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐵) = (((𝐴[,]𝐵) × {((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴)})‘𝐵))
80		oveq2 7376	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝐴(,)𝑥) = (𝐴(,)𝐵))
81		itgeq1 25742	. . . . . . . . 9 ⊢ ((𝐴(,)𝑥) = (𝐴(,)𝐵) → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡)
82	80, 81	syl 17	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡)
83		fveq2 6842	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
84	82, 83	oveq12d 7386	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)) = (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵)))
85		eqid 2737	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥))) = (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))
86		ovex 7401	. . . . . . 7 ⊢ (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵)) ∈ V
87	84, 85, 86	fvmpt 6949	. . . . . 6 ⊢ (𝐵 ∈ (𝐴[,]𝐵) → ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐵) = (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵)))
88	7, 87	syl 17	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐵) = (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵)))
89	79, 88	eqtr3d 2774	. . . 4 ⊢ (𝜑 → (((𝐴[,]𝐵) × {((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴)})‘𝐵) = (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵)))
90		lbicc2 13392	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
91	2, 4, 5, 90	syl3anc 1374	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵))
92		oveq2 7376	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (𝐴(,)𝑥) = (𝐴(,)𝐴))
93		iooid 13301	. . . . . . . . . . 11 ⊢ (𝐴(,)𝐴) = ∅
94	92, 93	eqtrdi 2788	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝐴(,)𝑥) = ∅)
95		itgeq1 25742	. . . . . . . . . 10 ⊢ ((𝐴(,)𝑥) = ∅ → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = ∫∅((ℝ D 𝐹)‘𝑡) d𝑡)
96	94, 95	syl 17	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = ∫∅((ℝ D 𝐹)‘𝑡) d𝑡)
97		itg0 25749	. . . . . . . . 9 ⊢ ∫∅((ℝ D 𝐹)‘𝑡) d𝑡 = 0
98	96, 97	eqtrdi 2788	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = 0)
99		fveq2 6842	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
100	98, 99	oveq12d 7386	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)) = (0 − (𝐹‘𝐴)))
101		df-neg 11379	. . . . . . 7 ⊢ -(𝐹‘𝐴) = (0 − (𝐹‘𝐴))
102	100, 101	eqtr4di 2790	. . . . . 6 ⊢ (𝑥 = 𝐴 → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)) = -(𝐹‘𝐴))
103		negex 11390	. . . . . 6 ⊢ -(𝐹‘𝐴) ∈ V
104	102, 85, 103	fvmpt 6949	. . . . 5 ⊢ (𝐴 ∈ (𝐴[,]𝐵) → ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴) = -(𝐹‘𝐴))
105	91, 104	syl 17	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴) = -(𝐹‘𝐴))
106	10, 89, 105	3eqtr3d 2780	. . 3 ⊢ (𝜑 → (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵)) = -(𝐹‘𝐴))
107	106	oveq2d 7384	. 2 ⊢ (𝜑 → ((𝐹‘𝐵) + (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵))) = ((𝐹‘𝐵) + -(𝐹‘𝐴)))
108	25, 7	ffvelcdmd 7039	. . 3 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ℂ)
109	33	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ V)
110	109, 47	itgcl 25753	. . 3 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ)
111	108, 110	pncan3d 11507	. 2 ⊢ (𝜑 → ((𝐹‘𝐵) + (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵))) = ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡)
112	25, 91	ffvelcdmd 7039	. . 3 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℂ)
113	108, 112	negsubd 11510	. 2 ⊢ (𝜑 → ((𝐹‘𝐵) + -(𝐹‘𝐴)) = ((𝐹‘𝐵) − (𝐹‘𝐴)))
114	107, 111, 113	3eqtr3d 2780	1 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 Vcvv 3442 ⊆ wss 3903 ∅c0 4287 {csn 4582 {cpr 4584 class class class wbr 5100 ↦ cmpt 5181 × cxp 5630 dom cdm 5632 ran crn 5633 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 ℂcc 11036 ℝcr 11037 0cc0 11038 + caddc 11041 ℝ^*cxr 11177 ≤ cle 11179 − cmin 11376 -cneg 11377 (,)cioo 13273 [,]cicc 13276 TopOpenctopn 17353 topGenctg 17369 ℂ_fldccnfld 21321 intcnt 22973 Cn ccn 23180 ×_t ctx 23516 –cn→ccncf 24837 volcvol 25432 𝐿¹cibl 25586 ∫citg 25587 D cdv 25832

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cc 10357 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-symdif 4207 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-disj 5068 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-ofr 7633 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-oadd 8411 df-omul 8412 df-er 8645 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-fi 9326 df-sup 9357 df-inf 9358 df-oi 9427 df-dju 9825 df-card 9863 df-acn 9866 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13277 df-ioc 13278 df-ico 13279 df-icc 13280 df-fz 13436 df-fzo 13583 df-fl 13724 df-mod 13802 df-seq 13937 df-exp 13997 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-clim 15423 df-rlim 15424 df-sum 15622 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-rest 17354 df-topn 17355 df-0g 17373 df-gsum 17374 df-topgen 17375 df-pt 17376 df-prds 17379 df-xrs 17435 df-qtop 17440 df-imas 17441 df-xps 17443 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-mulg 19010 df-cntz 19258 df-cmn 19723 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-fbas 21318 df-fg 21319 df-cnfld 21322 df-top 22850 df-topon 22867 df-topsp 22889 df-bases 22902 df-cld 22975 df-ntr 22976 df-cls 22977 df-nei 23054 df-lp 23092 df-perf 23093 df-cn 23183 df-cnp 23184 df-haus 23271 df-cmp 23343 df-tx 23518 df-hmeo 23711 df-fil 23802 df-fm 23894 df-flim 23895 df-flf 23896 df-xms 24276 df-ms 24277 df-tms 24278 df-cncf 24839 df-ovol 25433 df-vol 25434 df-mbf 25588 df-itg1 25589 df-itg2 25590 df-ibl 25591 df-itg 25592 df-0p 25639 df-limc 25835 df-dv 25836

This theorem is referenced by: ftc2ditglem 26020 itgparts 26022 itgsubstlem 26023 itgpowd 26025 ftc2re 34776 lcmineqlem12 42410 intlewftc 42431 lhe4.4ex1a 44685 itgsin0pilem1 46308 itgcoscmulx 46327 itgsincmulx 46332 dirkeritg 46460 etransclem46 46638

W3C validator