Theorem ftc2nc 35786

Description: Choice-free proof of ftc2 25113. (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 10956	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
3		ftc2nc.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
4	3	rexrd 10956	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ*)
5		ftc2nc.le	. . . . . 6 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
6		ubicc2 13126	. . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
7	2, 4, 5, 6	syl3anc 1369	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐵))
8		fvex 6769	. . . . . 6 ⊢ ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴) ∈ V
9	8	fvconst2 7061	. . . . 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 2738	. . . . . . . 8 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
12	11	subcn 23935	. . . . . . . . 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 2738	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡) = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡)
15		ssidd 3940	. . . . . . . . 9 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐵))
16		ioossre 13069	. . . . . . . . . 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 23962	. . . . . . . . . 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 13108	. . . . . . . . . . . . 13 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ
23		ffun 6587	. . . . . . . . . . . . 13 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → Fun (,))
24	22, 23	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun (,)
25		fvelima 6817	. . . . . . . . . . . 12 ⊢ ((Fun (,) ∧ 𝑠 ∈ ((,) “ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) → ∃𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))((,)‘𝑥) = 𝑠)
26	24, 25	mpan 686	. . . . . . . . . . 11 ⊢ (𝑠 ∈ ((,) “ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ∃𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))((,)‘𝑥) = 𝑠)
27		1st2nd2 7843	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
28	27	fveq2d 6760	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → ((,)‘𝑥) = ((,)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩))
29		df-ov 7258	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘𝑥)(,)(2nd ‘𝑥)) = ((,)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
30	28, 29	eqtr4di 2797	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → ((,)‘𝑥) = ((1st ‘𝑥)(,)(2nd ‘𝑥)))
31	30	eqeq1d 2740	. . . . . . . . . . . . . 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 7836	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → (1st ‘𝑥) ∈ (𝐴[,]𝐵))
36		elicc1 13052	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((1st ‘𝑥) ∈ (𝐴[,]𝐵) ↔ ((1st ‘𝑥) ∈ ℝ* ∧ 𝐴 ≤ (1st ‘𝑥) ∧ (1st ‘𝑥) ≤ 𝐵)))
37	2, 4, 36	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . 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 1141	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (1st ‘𝑥) ∈ (𝐴[,]𝐵)) → 𝐴 ≤ (1st ‘𝑥))
40	35, 39	sylan2 592	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → 𝐴 ≤ (1st ‘𝑥))
41		xp2nd 7837	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → (2nd ‘𝑥) ∈ (𝐴[,]𝐵))
42		iccleub 13063	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ (2nd ‘𝑥) ∈ (𝐴[,]𝐵)) → (2nd ‘𝑥) ≤ 𝐵)
43	42	3expa 1116	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (2nd ‘𝑥) ∈ (𝐴[,]𝐵)) → (2nd ‘𝑥) ≤ 𝐵)
44	33, 41, 43	syl2an 595	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (2nd ‘𝑥) ≤ 𝐵)
45		ioossioo 13102	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐴 ≤ (1st ‘𝑥) ∧ (2nd ‘𝑥) ≤ 𝐵)) → ((1st ‘𝑥)(,)(2nd ‘𝑥)) ⊆ (𝐴(,)𝐵))
46	34, 40, 44, 45	syl12anc 833	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((1st ‘𝑥)(,)(2nd ‘𝑥)) ⊆ (𝐴(,)𝐵))
47	46	sselda 3917	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥))) → 𝑡 ∈ (𝐴(,)𝐵))
48	21	ffvelrnda 6943	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ ℂ)
49	48	adantlr 711	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ ℂ)
50	47, 49	syldan 590	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥))) → ((ℝ D 𝐹)‘𝑡) ∈ ℂ)
51		ioombl 24634	. . . . . . . . . . . . . . . . . 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 6771	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ V)
54	21	feqmptd 6819	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (ℝ D 𝐹) = (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)))
55	54, 18	eqeltrrd 2840	. . . . . . . . . . . . . . . . . 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 24874	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿1)
58		ax-resscn 10859	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℝ ⊆ ℂ
59		ssid 3939	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℂ ⊆ ℂ
60		cncfss 23968	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℂ–cn→ℝ) ⊆ (ℂ–cn→ℂ))
61	58, 59, 60	mp2an 688	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℂ–cn→ℝ) ⊆ (ℂ–cn→ℂ)
62		abscncf 23970	. . . . . . . . . . . . . . . . . . . 20 ⊢ abs ∈ (ℂ–cn→ℝ)
63	61, 62	sselii 3914	. . . . . . . . . . . . . . . . . . 19 ⊢ abs ∈ (ℂ–cn→ℂ)
64	63	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → abs ∈ (ℂ–cn→ℂ))
65	54	reseq1d 5879	. . . . . . . . . . . . . . . . . . . . 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 5937	. . . . . . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . . . . . 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 23966	. . . . . . . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . . . . . . 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 23983	. . . . . . . . . . . . . . . . 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 24728	. . . . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ (abs‘((ℝ D 𝐹)‘𝑡))) ∈ MblFn)
76	50, 57	itgcl 24853	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ)
77	76	cjcld 14835	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (∗‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ∈ ℂ)
78		ioossre 13069	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ⊆ ℝ
79	78, 58	sstri 3926	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ⊆ ℂ
80		cncfmptc 23981	. . . . . . . . . . . . . . . . . . . 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 1451	. . . . . . . . . . . . . . . . . . 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 2906	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑠((ℝ D 𝐹)‘𝑡)
84		nfcsb1v 3853	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑡⦋𝑠 / 𝑡⦌((ℝ D 𝐹)‘𝑡)
85		csbeq1a 3842	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑠 → ((ℝ D 𝐹)‘𝑡) = ⦋𝑠 / 𝑡⦌((ℝ D 𝐹)‘𝑡))
86	83, 84, 85	cbvmpt 5181	. . . . . . . . . . . . . . . . . . 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 2844	. . . . . . . . . . . . . . . . . 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 24515	. . . . . . . . . . . . . . . . 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 24728	. . . . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . . . 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 35773	. . . . . . . . . . . . . . 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 15076	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥))) → (abs‘((ℝ D 𝐹)‘𝑡)) ∈ ℝ)
93		fvexd 6771	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥))) → ((ℝ D 𝐹)‘𝑡) ∈ V)
94	93, 57, 75	iblabsnc 35768	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ (abs‘((ℝ D 𝐹)‘𝑡))) ∈ 𝐿1)
95	50	absge0d 15084	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥))) → 0 ≤ (abs‘((ℝ D 𝐹)‘𝑡)))
96	92, 94, 95	itgposval 24865	. . . . . . . . . . . . . . 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 5096	. . . . . . . . . . . . . 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 24842	. . . . . . . . . . . . . . . 16 ⊢ (((1st ‘𝑥)(,)(2nd ‘𝑥)) = 𝑠 → ∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡 = ∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡)
99	98	fveq2d 6760	. . . . . . . . . . . . . . 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 2827	. . . . . . . . . . . . . . . . . 18 ⊢ (((1st ‘𝑥)(,)(2nd ‘𝑥)) = 𝑠 → (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↔ 𝑡 ∈ 𝑠))
101	100	ifbid 4479	. . . . . . . . . . . . . . . . 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 5172	. . . . . . . . . . . . . . . 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 6760	. . . . . . . . . . . . . . 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 5083	. . . . . . . . . . . . . 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 244	. . . . . . . . . . . . 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 239	. . . . . . . . . . . 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 3212	. . . . . . . . . . 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 3106	. . . . . . . . 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 35785	. . . . . . . 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 23962	. . . . . . . . . . 11 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
113	111, 112	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ)
114	113	feqmptd 6819	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥)))
115	114, 111	eqeltrrd 2840	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
116	11, 13, 110, 115	cncfmpt2f 23984	. . . . . . 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 13090	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
119	1, 3, 118	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
120		fvexd 6771	. . . . . . . . . . . 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 10956	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ*)
123		elicc2 13073	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
124	1, 3, 123	syl2anc 583	. . . . . . . . . . . . . . . 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 1142	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ 𝐵)
127		iooss2 13044	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑥 ≤ 𝐵) → (𝐴(,)𝑥) ⊆ (𝐴(,)𝐵))
128	122, 126, 127	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐴(,)𝑥) ⊆ (𝐴(,)𝐵))
129		ioombl 24634	. . . . . . . . . . . . . 14 ⊢ (𝐴(,)𝑥) ∈ dom vol
130	129	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐴(,)𝑥) ∈ dom vol)
131		fvexd 6771	. . . . . . . . . . . . 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 24874	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ (𝐴(,)𝑥) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿1)
134	120, 133	itgcl 24853	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ)
135	113	ffvelrnda 6943	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℂ)
136	134, 135	subcld 11262	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)) ∈ ℂ)
137	11	tgioo2 23872	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
138		iccntr 23890	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
139	1, 3, 138	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
140	117, 119, 136, 137, 11, 139	dvmptntr 25040	. . . . . . . . 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 10894	. . . . . . . . . . 11 ⊢ ℝ ∈ {ℝ, ℂ}
142	141	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
143		ioossicc 13094	. . . . . . . . . . . 12 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
144	143	sseli 3913	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ (𝐴[,]𝐵))
145	144, 134	sylan2 592	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ)
146	21	ffvelrnda 6943	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
147	14, 1, 3, 5, 19, 18	ftc1cnnc 35776	. . . . . . . . . . 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 25040	. . . . . . . . . . 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 6819	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D 𝐹) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥)))
150	147, 148, 149	3eqtr3d 2786	. . . . . . . . . 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 592	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑥) ∈ ℂ)
152	114	oveq2d 7271	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥))))
153	117, 119, 135, 137, 11, 139	dvmptntr 25040	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥))) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥))))
154	152, 149, 153	3eqtr3rd 2787	. . . . . . . . . 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 25036	. . . . . . . . 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 11250	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐹)‘𝑥) − ((ℝ D 𝐹)‘𝑥)) = 0)
157	156	mpteq2dva 5170	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑥) − ((ℝ D 𝐹)‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0))
158	140, 155, 157	3eqtrd 2782	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0))
159		fconstmpt 5640	. . . . . . . 8 ⊢ ((𝐴(,)𝐵) × {0}) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0)
160	158, 159	eqtr4di 2797	. . . . . . 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 25069	. . . . . 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 6758	. . . . 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 7263	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝐴(,)𝑥) = (𝐴(,)𝐵))
164		itgeq1 24842	. . . . . . . . 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 6756	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
167	165, 166	oveq12d 7273	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)) = (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵)))
168		eqid 2738	. . . . . . 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 7288	. . . . . . 7 ⊢ (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵)) ∈ V
170	167, 168, 169	fvmpt 6857	. . . . . 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 2780	. . . 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 13125	. . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
174	2, 4, 5, 173	syl3anc 1369	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵))
175		oveq2 7263	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (𝐴(,)𝑥) = (𝐴(,)𝐴))
176		iooid 13036	. . . . . . . . . . 11 ⊢ (𝐴(,)𝐴) = ∅
177	175, 176	eqtrdi 2795	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝐴(,)𝑥) = ∅)
178		itgeq1 24842	. . . . . . . . . 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 24849	. . . . . . . . 9 ⊢ ∫∅((ℝ D 𝐹)‘𝑡) d𝑡 = 0
181	179, 180	eqtrdi 2795	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = 0)
182		fveq2 6756	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
183	181, 182	oveq12d 7273	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)) = (0 − (𝐹‘𝐴)))
184		df-neg 11138	. . . . . . 7 ⊢ -(𝐹‘𝐴) = (0 − (𝐹‘𝐴))
185	183, 184	eqtr4di 2797	. . . . . 6 ⊢ (𝑥 = 𝐴 → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)) = -(𝐹‘𝐴))
186		negex 11149	. . . . . 6 ⊢ -(𝐹‘𝐴) ∈ V
187	185, 168, 186	fvmpt 6857	. . . . 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 2786	. . 3 ⊢ (𝜑 → (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵)) = -(𝐹‘𝐴))
190	189	oveq2d 7271	. 2 ⊢ (𝜑 → ((𝐹‘𝐵) + (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵))) = ((𝐹‘𝐵) + -(𝐹‘𝐴)))
191	113, 7	ffvelrnd 6944	. . 3 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ℂ)
192		fvexd 6771	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ V)
193	192, 55	itgcl 24853	. . 3 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ)
194	191, 193	pncan3d 11265	. 2 ⊢ (𝜑 → ((𝐹‘𝐵) + (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵))) = ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡)
195	113, 174	ffvelrnd 6944	. . 3 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℂ)
196	191, 195	negsubd 11268	. 2 ⊢ (𝜑 → ((𝐹‘𝐵) + -(𝐹‘𝐴)) = ((𝐹‘𝐵) − (𝐹‘𝐴)))
197	190, 194, 196	3eqtr3d 2786	1 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∃wrex 3064  Vcvv 3422  ⦋csb 3828   ⊆ wss 3883  ∅c0 4253  ifcif 4456  𝒫 cpw 4530  {csn 4558  {cpr 4560  ⟨cop 4564   class class class wbr 5070   ↦ cmpt 5153   × cxp 5578  dom cdm 5580  ran crn 5581   ↾ cres 5582   “ cima 5583  Fun wfun 6412  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  1^st c1st 7802  2^nd c2nd 7803  ℂcc 10800  ℝcr 10801  0cc0 10802   + caddc 10805   · cmul 10807  ℝ^*cxr 10939   ≤ cle 10941   − cmin 11135  -cneg 11136  (,)cioo 13008  [,]cicc 13011  ∗ccj 14735  abscabs 14873  TopOpenctopn 17049  topGenctg 17065  ℂ_fldccnfld 20510  intcnt 22076   Cn ccn 22283   ×_t ctx 22619  –cn→ccncf 23945  volcvol 24532  MblFncmbf 24683  ∫₂citg2 24685  𝐿¹cibl 24686  ∫citg 24687   D cdv 24932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880  ax-addf 10881  ax-mulf 10882

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-symdif 4173  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-disj 5036  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-ofr 7512  df-om 7688  df-1st 7804  df-2nd 7805  df-supp 7949  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-oadd 8271  df-omul 8272  df-er 8456  df-map 8575  df-pm 8576  df-ixp 8644  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fsupp 9059  df-fi 9100  df-sup 9131  df-inf 9132  df-oi 9199  df-dju 9590  df-card 9628  df-acn 9631  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-z 12250  df-dec 12367  df-uz 12512  df-q 12618  df-rp 12660  df-xneg 12777  df-xadd 12778  df-xmul 12779  df-ioo 13012  df-ico 13014  df-icc 13015  df-fz 13169  df-fzo 13312  df-fl 13440  df-mod 13518  df-seq 13650  df-exp 13711  df-hash 13973  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-clim 15125  df-rlim 15126  df-sum 15326  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-mulr 16902  df-starv 16903  df-sca 16904  df-vsca 16905  df-ip 16906  df-tset 16907  df-ple 16908  df-ds 16910  df-unif 16911  df-hom 16912  df-cco 16913  df-rest 17050  df-topn 17051  df-0g 17069  df-gsum 17070  df-topgen 17071  df-pt 17072  df-prds 17075  df-xrs 17130  df-qtop 17135  df-imas 17136  df-xps 17138  df-mre 17212  df-mrc 17213  df-acs 17215  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-submnd 18346  df-mulg 18616  df-cntz 18838  df-cmn 19303  df-psmet 20502  df-xmet 20503  df-met 20504  df-bl 20505  df-mopn 20506  df-fbas 20507  df-fg 20508  df-cnfld 20511  df-top 21951  df-topon 21968  df-topsp 21990  df-bases 22004  df-cld 22078  df-ntr 22079  df-cls 22080  df-nei 22157  df-lp 22195  df-perf 22196  df-cn 22286  df-cnp 22287  df-haus 22374  df-cmp 22446  df-tx 22621  df-hmeo 22814  df-fil 22905  df-fm 22997  df-flim 22998  df-flf 22999  df-xms 23381  df-ms 23382  df-tms 23383  df-cncf 23947  df-ovol 24533  df-vol 24534  df-mbf 24688  df-itg1 24689  df-itg2 24690  df-ibl 24691  df-itg 24692  df-0p 24739  df-limc 24935  df-dv 24936

This theorem is referenced by:  areacirc  35797