Theorem ftc2nc 37672

Description: Choice-free proof of ftc2 26001. (Contributed by Brendan Leahy, 19-Jun-2018.)

Hypotheses

Ref	Expression
ftc2nc.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
ftc2nc.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
ftc2nc.le	⊢ (𝜑 → 𝐴 ≤ 𝐵)
ftc2nc.c	⊢ (𝜑 → (ℝ D 𝐹) ∈ ((𝐴(,)𝐵)–cn→ℂ))
ftc2nc.i	⊢ (𝜑 → (ℝ D 𝐹) ∈ 𝐿1)
ftc2nc.f	⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))

Assertion

Ref	Expression
ftc2nc	⊢ (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 ftc2nc

Dummy variables 𝑠 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ftc2nc.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2	1	rexrd 11283	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
3		ftc2nc.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
4	3	rexrd 11283	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ*)
5		ftc2nc.le	. . . . . 6 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
6		ubicc2 13480	. . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
7	2, 4, 5, 6	syl3anc 1373	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐵))
8		fvex 6888	. . . . . 6 ⊢ ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴) ∈ V
9	8	fvconst2 7195	. . . . 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 2735	. . . . . . . 8 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
12	11	subcn 24804	. . . . . . . . 9 ⊢ − ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld))
13	12	a1i 11	. . . . . . . 8 ⊢ (𝜑 → − ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld)))
14		eqid 2735	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡) = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡)
15		ssidd 3982	. . . . . . . . 9 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐵))
16		ioossre 13422	. . . . . . . . . 10 ⊢ (𝐴(,)𝐵) ⊆ ℝ
17	16	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
18		ftc2nc.i	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D 𝐹) ∈ 𝐿1)
19		ftc2nc.c	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D 𝐹) ∈ ((𝐴(,)𝐵)–cn→ℂ))
20		cncff 24835	. . . . . . . . . 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		ioof 13462	. . . . . . . . . . . . 13 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ
23		ffun 6708	. . . . . . . . . . . . 13 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → Fun (,))
24	22, 23	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun (,)
25		fvelima 6943	. . . . . . . . . . . 12 ⊢ ((Fun (,) ∧ 𝑠 ∈ ((,) “ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) → ∃𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))((,)‘𝑥) = 𝑠)
26	24, 25	mpan 690	. . . . . . . . . . 11 ⊢ (𝑠 ∈ ((,) “ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ∃𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))((,)‘𝑥) = 𝑠)
27		1st2nd2 8025	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
28	27	fveq2d 6879	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → ((,)‘𝑥) = ((,)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩))
29		df-ov 7406	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘𝑥)(,)(2nd ‘𝑥)) = ((,)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
30	28, 29	eqtr4di 2788	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → ((,)‘𝑥) = ((1st ‘𝑥)(,)(2nd ‘𝑥)))
31	30	eqeq1d 2737	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → (((,)‘𝑥) = 𝑠 ↔ ((1st ‘𝑥)(,)(2nd ‘𝑥)) = 𝑠))
32	31	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (((,)‘𝑥) = 𝑠 ↔ ((1st ‘𝑥)(,)(2nd ‘𝑥)) = 𝑠))
33	2, 4	jca 511	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*))
34	33	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*))
35		xp1st 8018	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → (1st ‘𝑥) ∈ (𝐴[,]𝐵))
36		elicc1 13404	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((1st ‘𝑥) ∈ (𝐴[,]𝐵) ↔ ((1st ‘𝑥) ∈ ℝ* ∧ 𝐴 ≤ (1st ‘𝑥) ∧ (1st ‘𝑥) ≤ 𝐵)))
37	2, 4, 36	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((1st ‘𝑥) ∈ (𝐴[,]𝐵) ↔ ((1st ‘𝑥) ∈ ℝ* ∧ 𝐴 ≤ (1st ‘𝑥) ∧ (1st ‘𝑥) ≤ 𝐵)))
38	37	biimpa 476	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (1st ‘𝑥) ∈ (𝐴[,]𝐵)) → ((1st ‘𝑥) ∈ ℝ* ∧ 𝐴 ≤ (1st ‘𝑥) ∧ (1st ‘𝑥) ≤ 𝐵))
39	38	simp2d 1143	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (1st ‘𝑥) ∈ (𝐴[,]𝐵)) → 𝐴 ≤ (1st ‘𝑥))
40	35, 39	sylan2 593	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → 𝐴 ≤ (1st ‘𝑥))
41		xp2nd 8019	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → (2nd ‘𝑥) ∈ (𝐴[,]𝐵))
42		iccleub 13416	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ (2nd ‘𝑥) ∈ (𝐴[,]𝐵)) → (2nd ‘𝑥) ≤ 𝐵)
43	42	3expa 1118	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (2nd ‘𝑥) ∈ (𝐴[,]𝐵)) → (2nd ‘𝑥) ≤ 𝐵)
44	33, 41, 43	syl2an 596	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (2nd ‘𝑥) ≤ 𝐵)
45		ioossioo 13456	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐴 ≤ (1st ‘𝑥) ∧ (2nd ‘𝑥) ≤ 𝐵)) → ((1st ‘𝑥)(,)(2nd ‘𝑥)) ⊆ (𝐴(,)𝐵))
46	34, 40, 44, 45	syl12anc 836	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((1st ‘𝑥)(,)(2nd ‘𝑥)) ⊆ (𝐴(,)𝐵))
47	46	sselda 3958	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥))) → 𝑡 ∈ (𝐴(,)𝐵))
48	21	ffvelcdmda 7073	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ ℂ)
49	48	adantlr 715	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ ℂ)
50	47, 49	syldan 591	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥))) → ((ℝ D 𝐹)‘𝑡) ∈ ℂ)
51		ioombl 25516	. . . . . . . . . . . . . . . . . 18 ⊢ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ∈ dom vol
52	51	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((1st ‘𝑥)(,)(2nd ‘𝑥)) ∈ dom vol)
53		fvexd 6890	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ V)
54	21	feqmptd 6946	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (ℝ D 𝐹) = (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)))
55	54, 18	eqeltrrd 2835	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿1)
56	55	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿1)
57	46, 52, 53, 56	iblss 25756	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿1)
58		ax-resscn 11184	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℝ ⊆ ℂ
59		ssid 3981	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℂ ⊆ ℂ
60		cncfss 24841	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℂ–cn→ℝ) ⊆ (ℂ–cn→ℂ))
61	58, 59, 60	mp2an 692	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℂ–cn→ℝ) ⊆ (ℂ–cn→ℂ)
62		abscncf 24843	. . . . . . . . . . . . . . . . . . . 20 ⊢ abs ∈ (ℂ–cn→ℝ)
63	61, 62	sselii 3955	. . . . . . . . . . . . . . . . . . 19 ⊢ abs ∈ (ℂ–cn→ℂ)
64	63	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → abs ∈ (ℂ–cn→ℂ))
65	54	reseq1d 5965	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((1st ‘𝑥)(,)(2nd ‘𝑥))) = ((𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ↾ ((1st ‘𝑥)(,)(2nd ‘𝑥))))
66	65	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((ℝ D 𝐹) ↾ ((1st ‘𝑥)(,)(2nd ‘𝑥))) = ((𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ↾ ((1st ‘𝑥)(,)(2nd ‘𝑥))))
67	46	resmptd 6027	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ↾ ((1st ‘𝑥)(,)(2nd ‘𝑥))) = (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ ((ℝ D 𝐹)‘𝑡)))
68	66, 67	eqtrd 2770	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((ℝ D 𝐹) ↾ ((1st ‘𝑥)(,)(2nd ‘𝑥))) = (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ ((ℝ D 𝐹)‘𝑡)))
69	19	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (ℝ D 𝐹) ∈ ((𝐴(,)𝐵)–cn→ℂ))
70		rescncf 24839	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((1st ‘𝑥)(,)(2nd ‘𝑥)) ⊆ (𝐴(,)𝐵) → ((ℝ D 𝐹) ∈ ((𝐴(,)𝐵)–cn→ℂ) → ((ℝ D 𝐹) ↾ ((1st ‘𝑥)(,)(2nd ‘𝑥))) ∈ (((1st ‘𝑥)(,)(2nd ‘𝑥))–cn→ℂ)))
71	46, 69, 70	sylc 65	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((ℝ D 𝐹) ↾ ((1st ‘𝑥)(,)(2nd ‘𝑥))) ∈ (((1st ‘𝑥)(,)(2nd ‘𝑥))–cn→ℂ))
72	68, 71	eqeltrrd 2835	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ (((1st ‘𝑥)(,)(2nd ‘𝑥))–cn→ℂ))
73	64, 72	cncfmpt1f 24856	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ (abs‘((ℝ D 𝐹)‘𝑡))) ∈ (((1st ‘𝑥)(,)(2nd ‘𝑥))–cn→ℂ))
74		cnmbf 25610	. . . . . . . . . . . . . . . . 17 ⊢ ((((1st ‘𝑥)(,)(2nd ‘𝑥)) ∈ dom vol ∧ (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ (abs‘((ℝ D 𝐹)‘𝑡))) ∈ (((1st ‘𝑥)(,)(2nd ‘𝑥))–cn→ℂ)) → (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ (abs‘((ℝ D 𝐹)‘𝑡))) ∈ MblFn)
75	51, 73, 74	sylancr 587	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ (abs‘((ℝ D 𝐹)‘𝑡))) ∈ MblFn)
76	50, 57	itgcl 25735	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ)
77	76	cjcld 15213	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (∗‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ∈ ℂ)
78		ioossre 13422	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ⊆ ℝ
79	78, 58	sstri 3968	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ⊆ ℂ
80		cncfmptc 24854	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((∗‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ∈ ℂ ∧ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑠 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ (∗‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡)) ∈ (((1st ‘𝑥)(,)(2nd ‘𝑥))–cn→ℂ))
81	79, 59, 80	mp3an23 1455	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∗‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ∈ ℂ → (𝑠 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ (∗‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡)) ∈ (((1st ‘𝑥)(,)(2nd ‘𝑥))–cn→ℂ))
82	77, 81	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑠 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ (∗‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡)) ∈ (((1st ‘𝑥)(,)(2nd ‘𝑥))–cn→ℂ))
83		nfcv 2898	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑠((ℝ D 𝐹)‘𝑡)
84		nfcsb1v 3898	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑡⦋𝑠 / 𝑡⦌((ℝ D 𝐹)‘𝑡)
85		csbeq1a 3888	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑠 → ((ℝ D 𝐹)‘𝑡) = ⦋𝑠 / 𝑡⦌((ℝ D 𝐹)‘𝑡))
86	83, 84, 85	cbvmpt 5223	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ ((ℝ D 𝐹)‘𝑡)) = (𝑠 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ ⦋𝑠 / 𝑡⦌((ℝ D 𝐹)‘𝑡))
87	86, 72	eqeltrrid 2839	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑠 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ ⦋𝑠 / 𝑡⦌((ℝ D 𝐹)‘𝑡)) ∈ (((1st ‘𝑥)(,)(2nd ‘𝑥))–cn→ℂ))
88	82, 87	mulcncf 25396	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑠 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ ((∗‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) · ⦋𝑠 / 𝑡⦌((ℝ D 𝐹)‘𝑡))) ∈ (((1st ‘𝑥)(,)(2nd ‘𝑥))–cn→ℂ))
89		cnmbf 25610	. . . . . . . . . . . . . . . . 17 ⊢ ((((1st ‘𝑥)(,)(2nd ‘𝑥)) ∈ dom vol ∧ (𝑠 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ ((∗‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) · ⦋𝑠 / 𝑡⦌((ℝ D 𝐹)‘𝑡))) ∈ (((1st ‘𝑥)(,)(2nd ‘𝑥))–cn→ℂ)) → (𝑠 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ ((∗‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) · ⦋𝑠 / 𝑡⦌((ℝ D 𝐹)‘𝑡))) ∈ MblFn)
90	51, 88, 89	sylancr 587	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑠 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ ((∗‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) · ⦋𝑠 / 𝑡⦌((ℝ D 𝐹)‘𝑡))) ∈ MblFn)
91	50, 57, 75, 90	itgabsnc 37659	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (abs‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ≤ ∫((1st ‘𝑥)(,)(2nd ‘𝑥))(abs‘((ℝ D 𝐹)‘𝑡)) d𝑡)
92	50	abscld 15453	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥))) → (abs‘((ℝ D 𝐹)‘𝑡)) ∈ ℝ)
93		fvexd 6890	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥))) → ((ℝ D 𝐹)‘𝑡) ∈ V)
94	93, 57, 75	iblabsnc 37654	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ (abs‘((ℝ D 𝐹)‘𝑡))) ∈ 𝐿1)
95	50	absge0d 15461	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥))) → 0 ≤ (abs‘((ℝ D 𝐹)‘𝑡)))
96	92, 94, 95	itgposval 25747	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ∫((1st ‘𝑥)(,)(2nd ‘𝑥))(abs‘((ℝ D 𝐹)‘𝑡)) d𝑡 = (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)), (abs‘((ℝ D 𝐹)‘𝑡)), 0))))
97	91, 96	breqtrd 5145	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (abs‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)), (abs‘((ℝ D 𝐹)‘𝑡)), 0))))
98		itgeq1 25724	. . . . . . . . . . . . . . . 16 ⊢ (((1st ‘𝑥)(,)(2nd ‘𝑥)) = 𝑠 → ∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡 = ∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡)
99	98	fveq2d 6879	. . . . . . . . . . . . . . 15 ⊢ (((1st ‘𝑥)(,)(2nd ‘𝑥)) = 𝑠 → (abs‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) = (abs‘∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡))
100		eleq2 2823	. . . . . . . . . . . . . . . . . 18 ⊢ (((1st ‘𝑥)(,)(2nd ‘𝑥)) = 𝑠 → (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↔ 𝑡 ∈ 𝑠))
101	100	ifbid 4524	. . . . . . . . . . . . . . . . 17 ⊢ (((1st ‘𝑥)(,)(2nd ‘𝑥)) = 𝑠 → if(𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)), (abs‘((ℝ D 𝐹)‘𝑡)), 0) = if(𝑡 ∈ 𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0))
102	101	mpteq2dv 5215	. . . . . . . . . . . . . . . 16 ⊢ (((1st ‘𝑥)(,)(2nd ‘𝑥)) = 𝑠 → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)), (abs‘((ℝ D 𝐹)‘𝑡)), 0)) = (𝑡 ∈ ℝ ↦ if(𝑡 ∈ 𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0)))
103	102	fveq2d 6879	. . . . . . . . . . . . . . 15 ⊢ (((1st ‘𝑥)(,)(2nd ‘𝑥)) = 𝑠 → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)), (abs‘((ℝ D 𝐹)‘𝑡)), 0))) = (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ 𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0))))
104	99, 103	breq12d 5132	. . . . . . . . . . . . . 14 ⊢ (((1st ‘𝑥)(,)(2nd ‘𝑥)) = 𝑠 → ((abs‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)), (abs‘((ℝ D 𝐹)‘𝑡)), 0))) ↔ (abs‘∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ 𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0)))))
105	97, 104	syl5ibcom 245	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (((1st ‘𝑥)(,)(2nd ‘𝑥)) = 𝑠 → (abs‘∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ 𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0)))))
106	32, 105	sylbid 240	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (((,)‘𝑥) = 𝑠 → (abs‘∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ 𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0)))))
107	106	rexlimdva 3141	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))((,)‘𝑥) = 𝑠 → (abs‘∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ 𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0)))))
108	26, 107	syl5 34	. . . . . . . . . 10 ⊢ (𝜑 → (𝑠 ∈ ((,) “ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (abs‘∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ 𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0)))))
109	108	ralrimiv 3131	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑠 ∈ ((,) “ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))(abs‘∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ 𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0))))
110	14, 1, 3, 5, 15, 17, 18, 21, 109	ftc1anc 37671	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡) ∈ ((𝐴[,]𝐵)–cn→ℂ))
111		ftc2nc.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))
112		cncff 24835	. . . . . . . . . . 11 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
113	111, 112	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ)
114	113	feqmptd 6946	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥)))
115	114, 111	eqeltrrd 2835	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
116	11, 13, 110, 115	cncfmpt2f 24857	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥))) ∈ ((𝐴[,]𝐵)–cn→ℂ))
117	58	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ⊆ ℂ)
118		iccssre 13444	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
119	1, 3, 118	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
120		fvexd 6890	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ 𝑡 ∈ (𝐴(,)𝑥)) → ((ℝ D 𝐹)‘𝑡) ∈ V)
121	3	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ)
122	121	rexrd 11283	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ*)
123		elicc2 13426	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
124	1, 3, 123	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
125	124	biimpa 476	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵))
126	125	simp3d 1144	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ 𝐵)
127		iooss2 13396	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑥 ≤ 𝐵) → (𝐴(,)𝑥) ⊆ (𝐴(,)𝐵))
128	122, 126, 127	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐴(,)𝑥) ⊆ (𝐴(,)𝐵))
129		ioombl 25516	. . . . . . . . . . . . . 14 ⊢ (𝐴(,)𝑥) ∈ dom vol
130	129	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐴(,)𝑥) ∈ dom vol)
131		fvexd 6890	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ V)
132	55	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿1)
133	128, 130, 131, 132	iblss 25756	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ (𝐴(,)𝑥) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿1)
134	120, 133	itgcl 25735	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ)
135	113	ffvelcdmda 7073	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℂ)
136	134, 135	subcld 11592	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)) ∈ ℂ)
137		tgioo4 24742	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
138		iccntr 24759	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
139	1, 3, 138	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
140	117, 119, 136, 137, 11, 139	dvmptntr 25925	. . . . . . . . 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)))))
141		reelprrecn 11219	. . . . . . . . . . 11 ⊢ ℝ ∈ {ℝ, ℂ}
142	141	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
143		ioossicc 13448	. . . . . . . . . . . 12 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
144	143	sseli 3954	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ (𝐴[,]𝐵))
145	144, 134	sylan2 593	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ)
146	21	ffvelcdmda 7073	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
147	14, 1, 3, 5, 19, 18	ftc1cnnc 37662	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡)) = (ℝ D 𝐹))
148	117, 119, 134, 137, 11, 139	dvmptntr 25925	. . . . . . . . . . 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)))
149	21	feqmptd 6946	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D 𝐹) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥)))
150	147, 148, 149	3eqtr3d 2778	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥)))
151	144, 135	sylan2 593	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑥) ∈ ℂ)
152	114	oveq2d 7419	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥))))
153	117, 119, 135, 137, 11, 139	dvmptntr 25925	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥))) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥))))
154	152, 149, 153	3eqtr3rd 2779	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥)))
155	142, 145, 146, 150, 151, 146, 154	dvmptsub 25921	. . . . . . . . 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))))
156	146	subidd 11580	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐹)‘𝑥) − ((ℝ D 𝐹)‘𝑥)) = 0)
157	156	mpteq2dva 5214	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑥) − ((ℝ D 𝐹)‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0))
158	140, 155, 157	3eqtrd 2774	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0))
159		fconstmpt 5716	. . . . . . . 8 ⊢ ((𝐴(,)𝐵) × {0}) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0)
160	158, 159	eqtr4di 2788	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))) = ((𝐴(,)𝐵) × {0}))
161	1, 3, 116, 160	dveq0 25955	. . . . . 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)}))
162	161	fveq1d 6877	. . . . 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))
163		oveq2 7411	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝐴(,)𝑥) = (𝐴(,)𝐵))
164		itgeq1 25724	. . . . . . . . 9 ⊢ ((𝐴(,)𝑥) = (𝐴(,)𝐵) → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡)
165	163, 164	syl 17	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡)
166		fveq2 6875	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
167	165, 166	oveq12d 7421	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)) = (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵)))
168		eqid 2735	. . . . . . 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)))
169		ovex 7436	. . . . . . 7 ⊢ (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵)) ∈ V
170	167, 168, 169	fvmpt 6985	. . . . . 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)))
171	7, 170	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)))
172	162, 171	eqtr3d 2772	. . . 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)))
173		lbicc2 13479	. . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
174	2, 4, 5, 173	syl3anc 1373	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵))
175		oveq2 7411	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (𝐴(,)𝑥) = (𝐴(,)𝐴))
176		iooid 13388	. . . . . . . . . . 11 ⊢ (𝐴(,)𝐴) = ∅
177	175, 176	eqtrdi 2786	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝐴(,)𝑥) = ∅)
178		itgeq1 25724	. . . . . . . . . 10 ⊢ ((𝐴(,)𝑥) = ∅ → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = ∫∅((ℝ D 𝐹)‘𝑡) d𝑡)
179	177, 178	syl 17	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = ∫∅((ℝ D 𝐹)‘𝑡) d𝑡)
180		itg0 25731	. . . . . . . . 9 ⊢ ∫∅((ℝ D 𝐹)‘𝑡) d𝑡 = 0
181	179, 180	eqtrdi 2786	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = 0)
182		fveq2 6875	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
183	181, 182	oveq12d 7421	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)) = (0 − (𝐹‘𝐴)))
184		df-neg 11467	. . . . . . 7 ⊢ -(𝐹‘𝐴) = (0 − (𝐹‘𝐴))
185	183, 184	eqtr4di 2788	. . . . . 6 ⊢ (𝑥 = 𝐴 → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)) = -(𝐹‘𝐴))
186		negex 11478	. . . . . 6 ⊢ -(𝐹‘𝐴) ∈ V
187	185, 168, 186	fvmpt 6985	. . . . 5 ⊢ (𝐴 ∈ (𝐴[,]𝐵) → ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴) = -(𝐹‘𝐴))
188	174, 187	syl 17	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴) = -(𝐹‘𝐴))
189	10, 172, 188	3eqtr3d 2778	. . 3 ⊢ (𝜑 → (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵)) = -(𝐹‘𝐴))
190	189	oveq2d 7419	. 2 ⊢ (𝜑 → ((𝐹‘𝐵) + (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵))) = ((𝐹‘𝐵) + -(𝐹‘𝐴)))
191	113, 7	ffvelcdmd 7074	. . 3 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ℂ)
192		fvexd 6890	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ V)
193	192, 55	itgcl 25735	. . 3 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ)
194	191, 193	pncan3d 11595	. 2 ⊢ (𝜑 → ((𝐹‘𝐵) + (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵))) = ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡)
195	113, 174	ffvelcdmd 7074	. . 3 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℂ)
196	191, 195	negsubd 11598	. 2 ⊢ (𝜑 → ((𝐹‘𝐵) + -(𝐹‘𝐴)) = ((𝐹‘𝐵) − (𝐹‘𝐴)))
197	190, 194, 196	3eqtr3d 2778	1 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ∃wrex 3060  Vcvv 3459  ⦋csb 3874   ⊆ wss 3926  ∅c0 4308  ifcif 4500  𝒫 cpw 4575  {csn 4601  {cpr 4603  ⟨cop 4607   class class class wbr 5119   ↦ cmpt 5201   × cxp 5652  dom cdm 5654  ran crn 5655   ↾ cres 5656   “ cima 5657  Fun wfun 6524  ⟶wf 6526  ‘cfv 6530  (class class class)co 7403  1^st c1st 7984  2^nd c2nd 7985  ℂcc 11125  ℝcr 11126  0cc0 11127   + caddc 11130   · cmul 11132  ℝ^*cxr 11266   ≤ cle 11268   − cmin 11464  -cneg 11465  (,)cioo 13360  [,]cicc 13363  ∗ccj 15113  abscabs 15251  TopOpenctopn 17433  topGenctg 17449  ℂ_fldccnfld 21313  intcnt 22953   Cn ccn 23160   ×_t ctx 23496  –cn→ccncf 24818  volcvol 25414  MblFncmbf 25565  ∫₂citg2 25567  𝐿¹cibl 25568  ∫citg 25569   D cdv 25814

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-inf2 9653  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204  ax-pre-sup 11205  ax-addf 11206

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-symdif 4228  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-iin 4970  df-disj 5087  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-isom 6539  df-riota 7360  df-ov 7406  df-oprab 7407  df-mpo 7408  df-of 7669  df-ofr 7670  df-om 7860  df-1st 7986  df-2nd 7987  df-supp 8158  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-2o 8479  df-oadd 8482  df-omul 8483  df-er 8717  df-map 8840  df-pm 8841  df-ixp 8910  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-fsupp 9372  df-fi 9421  df-sup 9452  df-inf 9453  df-oi 9522  df-dju 9913  df-card 9951  df-acn 9954  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11466  df-neg 11467  df-div 11893  df-nn 12239  df-2 12301  df-3 12302  df-4 12303  df-5 12304  df-6 12305  df-7 12306  df-8 12307  df-9 12308  df-n0 12500  df-z 12587  df-dec 12707  df-uz 12851  df-q 12963  df-rp 13007  df-xneg 13126  df-xadd 13127  df-xmul 13128  df-ioo 13364  df-ico 13366  df-icc 13367  df-fz 13523  df-fzo 13670  df-fl 13807  df-mod 13885  df-seq 14018  df-exp 14078  df-hash 14347  df-cj 15116  df-re 15117  df-im 15118  df-sqrt 15252  df-abs 15253  df-clim 15502  df-rlim 15503  df-sum 15701  df-struct 17164  df-sets 17181  df-slot 17199  df-ndx 17211  df-base 17227  df-ress 17250  df-plusg 17282  df-mulr 17283  df-starv 17284  df-sca 17285  df-vsca 17286  df-ip 17287  df-tset 17288  df-ple 17289  df-ds 17291  df-unif 17292  df-hom 17293  df-cco 17294  df-rest 17434  df-topn 17435  df-0g 17453  df-gsum 17454  df-topgen 17455  df-pt 17456  df-prds 17459  df-xrs 17514  df-qtop 17519  df-imas 17520  df-xps 17522  df-mre 17596  df-mrc 17597  df-acs 17599  df-mgm 18616  df-sgrp 18695  df-mnd 18711  df-submnd 18760  df-mulg 19049  df-cntz 19298  df-cmn 19761  df-psmet 21305  df-xmet 21306  df-met 21307  df-bl 21308  df-mopn 21309  df-fbas 21310  df-fg 21311  df-cnfld 21314  df-top 22830  df-topon 22847  df-topsp 22869  df-bases 22882  df-cld 22955  df-ntr 22956  df-cls 22957  df-nei 23034  df-lp 23072  df-perf 23073  df-cn 23163  df-cnp 23164  df-haus 23251  df-cmp 23323  df-tx 23498  df-hmeo 23691  df-fil 23782  df-fm 23874  df-flim 23875  df-flf 23876  df-xms 24257  df-ms 24258  df-tms 24259  df-cncf 24820  df-ovol 25415  df-vol 25416  df-mbf 25570  df-itg1 25571  df-itg2 25572  df-ibl 25573  df-itg 25574  df-0p 25621  df-limc 25817  df-dv 25818

This theorem is referenced by:  areacirc  37683