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Theorem ftc2nc 34507
Description: Choice-free proof of ftc2 24324. (Contributed by Brendan Leahy, 19-Jun-2018.)
Hypotheses
Ref Expression
ftc2nc.a (𝜑𝐴 ∈ ℝ)
ftc2nc.b (𝜑𝐵 ∈ ℝ)
ftc2nc.le (𝜑𝐴𝐵)
ftc2nc.c (𝜑 → (ℝ D 𝐹) ∈ ((𝐴(,)𝐵)–cn→ℂ))
ftc2nc.i (𝜑 → (ℝ D 𝐹) ∈ 𝐿1)
ftc2nc.f (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))
Assertion
Ref Expression
ftc2nc (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹𝐵) − (𝐹𝐴)))
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.
StepHypRef Expression
1 ftc2nc.a . . . . . . 7 (𝜑𝐴 ∈ ℝ)
21rexrd 10537 . . . . . 6 (𝜑𝐴 ∈ ℝ*)
3 ftc2nc.b . . . . . . 7 (𝜑𝐵 ∈ ℝ)
43rexrd 10537 . . . . . 6 (𝜑𝐵 ∈ ℝ*)
5 ftc2nc.le . . . . . 6 (𝜑𝐴𝐵)
6 ubicc2 12703 . . . . . 6 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
72, 4, 5, 6syl3anc 1364 . . . . 5 (𝜑𝐵 ∈ (𝐴[,]𝐵))
8 fvex 6551 . . . . . 6 ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))‘𝐴) ∈ V
98fvconst2 6833 . . . . 5 (𝐵 ∈ (𝐴[,]𝐵) → (((𝐴[,]𝐵) × {((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))‘𝐴)})‘𝐵) = ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))‘𝐴))
107, 9syl 17 . . . 4 (𝜑 → (((𝐴[,]𝐵) × {((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))‘𝐴)})‘𝐵) = ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))‘𝐴))
11 eqid 2795 . . . . . . . 8 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
1211subcn 23157 . . . . . . . . 9 − ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld))
1312a1i 11 . . . . . . . 8 (𝜑 → − ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld)))
14 eqid 2795 . . . . . . . . 9 (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡) = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡)
15 ssidd 3911 . . . . . . . . 9 (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐵))
16 ioossre 12648 . . . . . . . . . 10 (𝐴(,)𝐵) ⊆ ℝ
1716a1i 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 23184 . . . . . . . . . 10 ((ℝ D 𝐹) ∈ ((𝐴(,)𝐵)–cn→ℂ) → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ)
2119, 20syl 17 . . . . . . . . 9 (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ)
22 ioof 12685 . . . . . . . . . . . . 13 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
23 ffun 6385 . . . . . . . . . . . . 13 ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → Fun (,))
2422, 23ax-mp 5 . . . . . . . . . . . 12 Fun (,)
25 fvelima 6599 . . . . . . . . . . . 12 ((Fun (,) ∧ 𝑠 ∈ ((,) “ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) → ∃𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))((,)‘𝑥) = 𝑠)
2624, 25mpan 686 . . . . . . . . . . 11 (𝑠 ∈ ((,) “ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ∃𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))((,)‘𝑥) = 𝑠)
27 1st2nd2 7584 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
2827fveq2d 6542 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → ((,)‘𝑥) = ((,)‘⟨(1st𝑥), (2nd𝑥)⟩))
29 df-ov 7019 . . . . . . . . . . . . . . . 16 ((1st𝑥)(,)(2nd𝑥)) = ((,)‘⟨(1st𝑥), (2nd𝑥)⟩)
3028, 29syl6eqr 2849 . . . . . . . . . . . . . . 15 (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → ((,)‘𝑥) = ((1st𝑥)(,)(2nd𝑥)))
3130eqeq1d 2797 . . . . . . . . . . . . . 14 (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → (((,)‘𝑥) = 𝑠 ↔ ((1st𝑥)(,)(2nd𝑥)) = 𝑠))
3231adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (((,)‘𝑥) = 𝑠 ↔ ((1st𝑥)(,)(2nd𝑥)) = 𝑠))
332, 4jca 512 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝐴 ∈ ℝ*𝐵 ∈ ℝ*))
3433adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝐴 ∈ ℝ*𝐵 ∈ ℝ*))
35 xp1st 7577 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → (1st𝑥) ∈ (𝐴[,]𝐵))
36 elicc1 12632 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((1st𝑥) ∈ (𝐴[,]𝐵) ↔ ((1st𝑥) ∈ ℝ*𝐴 ≤ (1st𝑥) ∧ (1st𝑥) ≤ 𝐵)))
372, 4, 36syl2anc 584 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((1st𝑥) ∈ (𝐴[,]𝐵) ↔ ((1st𝑥) ∈ ℝ*𝐴 ≤ (1st𝑥) ∧ (1st𝑥) ≤ 𝐵)))
3837biimpa 477 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (1st𝑥) ∈ (𝐴[,]𝐵)) → ((1st𝑥) ∈ ℝ*𝐴 ≤ (1st𝑥) ∧ (1st𝑥) ≤ 𝐵))
3938simp2d 1136 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (1st𝑥) ∈ (𝐴[,]𝐵)) → 𝐴 ≤ (1st𝑥))
4035, 39sylan2 592 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → 𝐴 ≤ (1st𝑥))
41 xp2nd 7578 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → (2nd𝑥) ∈ (𝐴[,]𝐵))
42 iccleub 12642 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ* ∧ (2nd𝑥) ∈ (𝐴[,]𝐵)) → (2nd𝑥) ≤ 𝐵)
43423expa 1111 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (2nd𝑥) ∈ (𝐴[,]𝐵)) → (2nd𝑥) ≤ 𝐵)
4433, 41, 43syl2an 595 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (2nd𝑥) ≤ 𝐵)
45 ioossioo 12679 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐴 ≤ (1st𝑥) ∧ (2nd𝑥) ≤ 𝐵)) → ((1st𝑥)(,)(2nd𝑥)) ⊆ (𝐴(,)𝐵))
4634, 40, 44, 45syl12anc 833 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((1st𝑥)(,)(2nd𝑥)) ⊆ (𝐴(,)𝐵))
4746sselda 3889 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st𝑥)(,)(2nd𝑥))) → 𝑡 ∈ (𝐴(,)𝐵))
4821ffvelrnda 6716 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ ℂ)
4948adantlr 711 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ ℂ)
5047, 49syldan 591 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st𝑥)(,)(2nd𝑥))) → ((ℝ D 𝐹)‘𝑡) ∈ ℂ)
51 ioombl 23849 . . . . . . . . . . . . . . . . . 18 ((1st𝑥)(,)(2nd𝑥)) ∈ dom vol
5251a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((1st𝑥)(,)(2nd𝑥)) ∈ dom vol)
53 fvexd 6553 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ V)
5421feqmptd 6601 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (ℝ D 𝐹) = (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)))
5554, 18eqeltrrd 2884 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿1)
5655adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿1)
5746, 52, 53, 56iblss 24088 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿1)
58 ax-resscn 10440 . . . . . . . . . . . . . . . . . . . . 21 ℝ ⊆ ℂ
59 ssid 3910 . . . . . . . . . . . . . . . . . . . . 21 ℂ ⊆ ℂ
60 cncfss 23190 . . . . . . . . . . . . . . . . . . . . 21 ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℂ–cn→ℝ) ⊆ (ℂ–cn→ℂ))
6158, 59, 60mp2an 688 . . . . . . . . . . . . . . . . . . . 20 (ℂ–cn→ℝ) ⊆ (ℂ–cn→ℂ)
62 abscncf 23192 . . . . . . . . . . . . . . . . . . . 20 abs ∈ (ℂ–cn→ℝ)
6361, 62sselii 3886 . . . . . . . . . . . . . . . . . . 19 abs ∈ (ℂ–cn→ℂ)
6463a1i 11 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → abs ∈ (ℂ–cn→ℂ))
6554reseq1d 5733 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((ℝ D 𝐹) ↾ ((1st𝑥)(,)(2nd𝑥))) = ((𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ↾ ((1st𝑥)(,)(2nd𝑥))))
6665adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((ℝ D 𝐹) ↾ ((1st𝑥)(,)(2nd𝑥))) = ((𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ↾ ((1st𝑥)(,)(2nd𝑥))))
6746resmptd 5789 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ↾ ((1st𝑥)(,)(2nd𝑥))) = (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ ((ℝ D 𝐹)‘𝑡)))
6866, 67eqtrd 2831 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((ℝ D 𝐹) ↾ ((1st𝑥)(,)(2nd𝑥))) = (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ ((ℝ D 𝐹)‘𝑡)))
6919adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (ℝ D 𝐹) ∈ ((𝐴(,)𝐵)–cn→ℂ))
70 rescncf 23188 . . . . . . . . . . . . . . . . . . . 20 (((1st𝑥)(,)(2nd𝑥)) ⊆ (𝐴(,)𝐵) → ((ℝ D 𝐹) ∈ ((𝐴(,)𝐵)–cn→ℂ) → ((ℝ D 𝐹) ↾ ((1st𝑥)(,)(2nd𝑥))) ∈ (((1st𝑥)(,)(2nd𝑥))–cn→ℂ)))
7146, 69, 70sylc 65 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((ℝ D 𝐹) ↾ ((1st𝑥)(,)(2nd𝑥))) ∈ (((1st𝑥)(,)(2nd𝑥))–cn→ℂ))
7268, 71eqeltrrd 2884 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ (((1st𝑥)(,)(2nd𝑥))–cn→ℂ))
7364, 72cncfmpt1f 23204 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ (abs‘((ℝ D 𝐹)‘𝑡))) ∈ (((1st𝑥)(,)(2nd𝑥))–cn→ℂ))
74 cnmbf 23943 . . . . . . . . . . . . . . . . 17 ((((1st𝑥)(,)(2nd𝑥)) ∈ dom vol ∧ (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ (abs‘((ℝ D 𝐹)‘𝑡))) ∈ (((1st𝑥)(,)(2nd𝑥))–cn→ℂ)) → (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ (abs‘((ℝ D 𝐹)‘𝑡))) ∈ MblFn)
7551, 73, 74sylancr 587 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ (abs‘((ℝ D 𝐹)‘𝑡))) ∈ MblFn)
7650, 57itgcl 24067 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ)
7776cjcld 14389 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (∗‘∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ∈ ℂ)
78 ioossre 12648 . . . . . . . . . . . . . . . . . . . . 21 ((1st𝑥)(,)(2nd𝑥)) ⊆ ℝ
7978, 58sstri 3898 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑥)(,)(2nd𝑥)) ⊆ ℂ
80 cncfmptc 23202 . . . . . . . . . . . . . . . . . . . 20 (((∗‘∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ∈ ℂ ∧ ((1st𝑥)(,)(2nd𝑥)) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑠 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ (∗‘∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡)) ∈ (((1st𝑥)(,)(2nd𝑥))–cn→ℂ))
8179, 59, 80mp3an23 1445 . . . . . . . . . . . . . . . . . . 19 ((∗‘∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ∈ ℂ → (𝑠 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ (∗‘∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡)) ∈ (((1st𝑥)(,)(2nd𝑥))–cn→ℂ))
8277, 81syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑠 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ (∗‘∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡)) ∈ (((1st𝑥)(,)(2nd𝑥))–cn→ℂ))
83 nfcv 2949 . . . . . . . . . . . . . . . . . . . 20 𝑠((ℝ D 𝐹)‘𝑡)
84 nfcsb1v 3833 . . . . . . . . . . . . . . . . . . . 20 𝑡𝑠 / 𝑡((ℝ D 𝐹)‘𝑡)
85 csbeq1a 3824 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑠 → ((ℝ D 𝐹)‘𝑡) = 𝑠 / 𝑡((ℝ D 𝐹)‘𝑡))
8683, 84, 85cbvmpt 5060 . . . . . . . . . . . . . . . . . . 19 (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ ((ℝ D 𝐹)‘𝑡)) = (𝑠 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ 𝑠 / 𝑡((ℝ D 𝐹)‘𝑡))
8786, 72syl5eqelr 2888 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑠 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ 𝑠 / 𝑡((ℝ D 𝐹)‘𝑡)) ∈ (((1st𝑥)(,)(2nd𝑥))–cn→ℂ))
8882, 87mulcncf 23730 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑠 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ ((∗‘∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) · 𝑠 / 𝑡((ℝ D 𝐹)‘𝑡))) ∈ (((1st𝑥)(,)(2nd𝑥))–cn→ℂ))
89 cnmbf 23943 . . . . . . . . . . . . . . . . 17 ((((1st𝑥)(,)(2nd𝑥)) ∈ dom vol ∧ (𝑠 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ ((∗‘∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) · 𝑠 / 𝑡((ℝ D 𝐹)‘𝑡))) ∈ (((1st𝑥)(,)(2nd𝑥))–cn→ℂ)) → (𝑠 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ ((∗‘∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) · 𝑠 / 𝑡((ℝ D 𝐹)‘𝑡))) ∈ MblFn)
9051, 88, 89sylancr 587 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑠 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ ((∗‘∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) · 𝑠 / 𝑡((ℝ D 𝐹)‘𝑡))) ∈ MblFn)
9150, 57, 75, 90itgabsnc 34492 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (abs‘∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ≤ ∫((1st𝑥)(,)(2nd𝑥))(abs‘((ℝ D 𝐹)‘𝑡)) d𝑡)
9250abscld 14630 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st𝑥)(,)(2nd𝑥))) → (abs‘((ℝ D 𝐹)‘𝑡)) ∈ ℝ)
93 fvexd 6553 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st𝑥)(,)(2nd𝑥))) → ((ℝ D 𝐹)‘𝑡) ∈ V)
9493, 57, 75iblabsnc 34487 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ (abs‘((ℝ D 𝐹)‘𝑡))) ∈ 𝐿1)
9550absge0d 14638 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st𝑥)(,)(2nd𝑥))) → 0 ≤ (abs‘((ℝ D 𝐹)‘𝑡)))
9692, 94, 95itgposval 24079 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ∫((1st𝑥)(,)(2nd𝑥))(abs‘((ℝ D 𝐹)‘𝑡)) d𝑡 = (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)), (abs‘((ℝ D 𝐹)‘𝑡)), 0))))
9791, 96breqtrd 4988 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (abs‘∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)), (abs‘((ℝ D 𝐹)‘𝑡)), 0))))
98 itgeq1 24056 . . . . . . . . . . . . . . . 16 (((1st𝑥)(,)(2nd𝑥)) = 𝑠 → ∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡 = ∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡)
9998fveq2d 6542 . . . . . . . . . . . . . . 15 (((1st𝑥)(,)(2nd𝑥)) = 𝑠 → (abs‘∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) = (abs‘∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡))
100 eleq2 2871 . . . . . . . . . . . . . . . . . 18 (((1st𝑥)(,)(2nd𝑥)) = 𝑠 → (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↔ 𝑡𝑠))
101100ifbid 4403 . . . . . . . . . . . . . . . . 17 (((1st𝑥)(,)(2nd𝑥)) = 𝑠 → if(𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)), (abs‘((ℝ D 𝐹)‘𝑡)), 0) = if(𝑡𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0))
102101mpteq2dv 5056 . . . . . . . . . . . . . . . 16 (((1st𝑥)(,)(2nd𝑥)) = 𝑠 → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)), (abs‘((ℝ D 𝐹)‘𝑡)), 0)) = (𝑡 ∈ ℝ ↦ if(𝑡𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0)))
103102fveq2d 6542 . . . . . . . . . . . . . . 15 (((1st𝑥)(,)(2nd𝑥)) = 𝑠 → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)), (abs‘((ℝ D 𝐹)‘𝑡)), 0))) = (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0))))
10499, 103breq12d 4975 . . . . . . . . . . . . . 14 (((1st𝑥)(,)(2nd𝑥)) = 𝑠 → ((abs‘∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)), (abs‘((ℝ D 𝐹)‘𝑡)), 0))) ↔ (abs‘∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0)))))
10597, 104syl5ibcom 246 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (((1st𝑥)(,)(2nd𝑥)) = 𝑠 → (abs‘∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0)))))
10632, 105sylbid 241 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (((,)‘𝑥) = 𝑠 → (abs‘∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0)))))
107106rexlimdva 3247 . . . . . . . . . . 11 (𝜑 → (∃𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))((,)‘𝑥) = 𝑠 → (abs‘∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0)))))
10826, 107syl5 34 . . . . . . . . . 10 (𝜑 → (𝑠 ∈ ((,) “ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (abs‘∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0)))))
109108ralrimiv 3148 . . . . . . . . 9 (𝜑 → ∀𝑠 ∈ ((,) “ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))(abs‘∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0))))
11014, 1, 3, 5, 15, 17, 18, 21, 109ftc1anc 34506 . . . . . . . 8 (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡) ∈ ((𝐴[,]𝐵)–cn→ℂ))
111 ftc2nc.f . . . . . . . . . . 11 (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))
112 cncff 23184 . . . . . . . . . . 11 (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
113111, 112syl 17 . . . . . . . . . 10 (𝜑𝐹:(𝐴[,]𝐵)⟶ℂ)
114113feqmptd 6601 . . . . . . . . 9 (𝜑𝐹 = (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹𝑥)))
115114, 111eqeltrrd 2884 . . . . . . . 8 (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
11611, 13, 110, 115cncfmpt2f 23205 . . . . . . 7 (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥))) ∈ ((𝐴[,]𝐵)–cn→ℂ))
11758a1i 11 . . . . . . . . . 10 (𝜑 → ℝ ⊆ ℂ)
118 iccssre 12668 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
1191, 3, 118syl2anc 584 . . . . . . . . . 10 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
120 fvexd 6553 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ 𝑡 ∈ (𝐴(,)𝑥)) → ((ℝ D 𝐹)‘𝑡) ∈ V)
1213adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ)
122121rexrd 10537 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ*)
123 elicc2 12651 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴𝑥𝑥𝐵)))
1241, 3, 123syl2anc 584 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴𝑥𝑥𝐵)))
125124biimpa 477 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝐴𝑥𝑥𝐵))
126125simp3d 1137 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 𝑥𝐵)
127 iooss2 12624 . . . . . . . . . . . . . 14 ((𝐵 ∈ ℝ*𝑥𝐵) → (𝐴(,)𝑥) ⊆ (𝐴(,)𝐵))
128122, 126, 127syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐴(,)𝑥) ⊆ (𝐴(,)𝐵))
129 ioombl 23849 . . . . . . . . . . . . . 14 (𝐴(,)𝑥) ∈ dom vol
130129a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐴(,)𝑥) ∈ dom vol)
131 fvexd 6553 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ V)
13255adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿1)
133128, 130, 131, 132iblss 24088 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ (𝐴(,)𝑥) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿1)
134120, 133itgcl 24067 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ)
135113ffvelrnda 6716 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℂ)
136134, 135subcld 10845 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)) ∈ ℂ)
13711tgioo2 23094 . . . . . . . . . 10 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
138 iccntr 23112 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
1391, 3, 138syl2anc 584 . . . . . . . . . 10 (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
140117, 119, 136, 137, 11, 139dvmptntr 24251 . . . . . . . . 9 (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))))
141 reelprrecn 10475 . . . . . . . . . . 11 ℝ ∈ {ℝ, ℂ}
142141a1i 11 . . . . . . . . . 10 (𝜑 → ℝ ∈ {ℝ, ℂ})
143 ioossicc 12672 . . . . . . . . . . . 12 (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
144143sseli 3885 . . . . . . . . . . 11 (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ (𝐴[,]𝐵))
145144, 134sylan2 592 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ)
14621ffvelrnda 6716 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
14714, 1, 3, 5, 19, 18ftc1cnnc 34497 . . . . . . . . . . 11 (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡)) = (ℝ D 𝐹))
148117, 119, 134, 137, 11, 139dvmptntr 24251 . . . . . . . . . . 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)))
14921feqmptd 6601 . . . . . . . . . . 11 (𝜑 → (ℝ D 𝐹) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥)))
150147, 148, 1493eqtr3d 2839 . . . . . . . . . 10 (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥)))
151144, 135sylan2 592 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → (𝐹𝑥) ∈ ℂ)
152114oveq2d 7032 . . . . . . . . . . 11 (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹𝑥))))
153117, 119, 135, 137, 11, 139dvmptntr 24251 . . . . . . . . . . 11 (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹𝑥))) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹𝑥))))
154152, 149, 1533eqtr3rd 2840 . . . . . . . . . 10 (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥)))
155142, 145, 146, 150, 151, 146, 154dvmptsub 24247 . . . . . . . . 9 (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑥) − ((ℝ D 𝐹)‘𝑥))))
156146subidd 10833 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐹)‘𝑥) − ((ℝ D 𝐹)‘𝑥)) = 0)
157156mpteq2dva 5055 . . . . . . . . 9 (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑥) − ((ℝ D 𝐹)‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0))
158140, 155, 1573eqtrd 2835 . . . . . . . 8 (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0))
159 fconstmpt 5500 . . . . . . . 8 ((𝐴(,)𝐵) × {0}) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0)
160158, 159syl6eqr 2849 . . . . . . 7 (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))) = ((𝐴(,)𝐵) × {0}))
1611, 3, 116, 160dveq0 24280 . . . . . 6 (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥))) = ((𝐴[,]𝐵) × {((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))‘𝐴)}))
162161fveq1d 6540 . . . . 5 (𝜑 → ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))‘𝐵) = (((𝐴[,]𝐵) × {((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))‘𝐴)})‘𝐵))
163 oveq2 7024 . . . . . . . . 9 (𝑥 = 𝐵 → (𝐴(,)𝑥) = (𝐴(,)𝐵))
164 itgeq1 24056 . . . . . . . . 9 ((𝐴(,)𝑥) = (𝐴(,)𝐵) → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡)
165163, 164syl 17 . . . . . . . 8 (𝑥 = 𝐵 → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡)
166 fveq2 6538 . . . . . . . 8 (𝑥 = 𝐵 → (𝐹𝑥) = (𝐹𝐵))
167165, 166oveq12d 7034 . . . . . . 7 (𝑥 = 𝐵 → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)) = (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝐵)))
168 eqid 2795 . . . . . . 7 (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥))) = (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))
169 ovex 7048 . . . . . . 7 (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝐵)) ∈ V
170167, 168, 169fvmpt 6635 . . . . . 6 (𝐵 ∈ (𝐴[,]𝐵) → ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))‘𝐵) = (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝐵)))
1717, 170syl 17 . . . . 5 (𝜑 → ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))‘𝐵) = (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝐵)))
172162, 171eqtr3d 2833 . . . 4 (𝜑 → (((𝐴[,]𝐵) × {((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))‘𝐴)})‘𝐵) = (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝐵)))
173 lbicc2 12702 . . . . . 6 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
1742, 4, 5, 173syl3anc 1364 . . . . 5 (𝜑𝐴 ∈ (𝐴[,]𝐵))
175 oveq2 7024 . . . . . . . . . . 11 (𝑥 = 𝐴 → (𝐴(,)𝑥) = (𝐴(,)𝐴))
176 iooid 12616 . . . . . . . . . . 11 (𝐴(,)𝐴) = ∅
177175, 176syl6eq 2847 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝐴(,)𝑥) = ∅)
178 itgeq1 24056 . . . . . . . . . 10 ((𝐴(,)𝑥) = ∅ → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = ∫∅((ℝ D 𝐹)‘𝑡) d𝑡)
179177, 178syl 17 . . . . . . . . 9 (𝑥 = 𝐴 → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = ∫∅((ℝ D 𝐹)‘𝑡) d𝑡)
180 itg0 24063 . . . . . . . . 9 ∫∅((ℝ D 𝐹)‘𝑡) d𝑡 = 0
181179, 180syl6eq 2847 . . . . . . . 8 (𝑥 = 𝐴 → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = 0)
182 fveq2 6538 . . . . . . . 8 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
183181, 182oveq12d 7034 . . . . . . 7 (𝑥 = 𝐴 → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)) = (0 − (𝐹𝐴)))
184 df-neg 10720 . . . . . . 7 -(𝐹𝐴) = (0 − (𝐹𝐴))
185183, 184syl6eqr 2849 . . . . . 6 (𝑥 = 𝐴 → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)) = -(𝐹𝐴))
186 negex 10731 . . . . . 6 -(𝐹𝐴) ∈ V
187185, 168, 186fvmpt 6635 . . . . 5 (𝐴 ∈ (𝐴[,]𝐵) → ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))‘𝐴) = -(𝐹𝐴))
188174, 187syl 17 . . . 4 (𝜑 → ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))‘𝐴) = -(𝐹𝐴))
18910, 172, 1883eqtr3d 2839 . . 3 (𝜑 → (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝐵)) = -(𝐹𝐴))
190189oveq2d 7032 . 2 (𝜑 → ((𝐹𝐵) + (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝐵))) = ((𝐹𝐵) + -(𝐹𝐴)))
191113, 7ffvelrnd 6717 . . 3 (𝜑 → (𝐹𝐵) ∈ ℂ)
192 fvexd 6553 . . . 4 ((𝜑𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ V)
193192, 55itgcl 24067 . . 3 (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ)
194191, 193pncan3d 10848 . 2 (𝜑 → ((𝐹𝐵) + (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝐵))) = ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡)
195113, 174ffvelrnd 6717 . . 3 (𝜑 → (𝐹𝐴) ∈ ℂ)
196191, 195negsubd 10851 . 2 (𝜑 → ((𝐹𝐵) + -(𝐹𝐴)) = ((𝐹𝐵) − (𝐹𝐴)))
197190, 194, 1963eqtr3d 2839 1 (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹𝐵) − (𝐹𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1080   = wceq 1522  wcel 2081  wrex 3106  Vcvv 3437  csb 3811  wss 3859  c0 4211  ifcif 4381  𝒫 cpw 4453  {csn 4472  {cpr 4474  cop 4478   class class class wbr 4962  cmpt 5041   × cxp 5441  dom cdm 5443  ran crn 5444  cres 5445  cima 5446  Fun wfun 6219  wf 6221  cfv 6225  (class class class)co 7016  1st c1st 7543  2nd c2nd 7544  cc 10381  cr 10382  0cc0 10383   + caddc 10386   · cmul 10388  *cxr 10520  cle 10522  cmin 10717  -cneg 10718  (,)cioo 12588  [,]cicc 12591  ccj 14289  abscabs 14427  TopOpenctopn 16524  topGenctg 16540  fldccnfld 20227  intcnt 21309   Cn ccn 21516   ×t ctx 21852  cnccncf 23167  volcvol 23747  MblFncmbf 23898  2citg2 23900  𝐿1cibl 23901  citg 23902   D cdv 24144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-inf2 8950  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460  ax-pre-sup 10461  ax-addf 10462  ax-mulf 10463
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-symdif 4139  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-iin 4828  df-disj 4931  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-of 7267  df-ofr 7268  df-om 7437  df-1st 7545  df-2nd 7546  df-supp 7682  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-2o 7954  df-oadd 7957  df-omul 7958  df-er 8139  df-map 8258  df-pm 8259  df-ixp 8311  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-fsupp 8680  df-fi 8721  df-sup 8752  df-inf 8753  df-oi 8820  df-dju 9176  df-card 9214  df-acn 9217  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-div 11146  df-nn 11487  df-2 11548  df-3 11549  df-4 11550  df-5 11551  df-6 11552  df-7 11553  df-8 11554  df-9 11555  df-n0 11746  df-z 11830  df-dec 11948  df-uz 12094  df-q 12198  df-rp 12240  df-xneg 12357  df-xadd 12358  df-xmul 12359  df-ioo 12592  df-ico 12594  df-icc 12595  df-fz 12743  df-fzo 12884  df-fl 13012  df-mod 13088  df-seq 13220  df-exp 13280  df-hash 13541  df-cj 14292  df-re 14293  df-im 14294  df-sqrt 14428  df-abs 14429  df-clim 14679  df-rlim 14680  df-sum 14877  df-struct 16314  df-ndx 16315  df-slot 16316  df-base 16318  df-sets 16319  df-ress 16320  df-plusg 16407  df-mulr 16408  df-starv 16409  df-sca 16410  df-vsca 16411  df-ip 16412  df-tset 16413  df-ple 16414  df-ds 16416  df-unif 16417  df-hom 16418  df-cco 16419  df-rest 16525  df-topn 16526  df-0g 16544  df-gsum 16545  df-topgen 16546  df-pt 16547  df-prds 16550  df-xrs 16604  df-qtop 16609  df-imas 16610  df-xps 16612  df-mre 16686  df-mrc 16687  df-acs 16689  df-mgm 17681  df-sgrp 17723  df-mnd 17734  df-submnd 17775  df-mulg 17982  df-cntz 18188  df-cmn 18635  df-psmet 20219  df-xmet 20220  df-met 20221  df-bl 20222  df-mopn 20223  df-fbas 20224  df-fg 20225  df-cnfld 20228  df-top 21186  df-topon 21203  df-topsp 21225  df-bases 21238  df-cld 21311  df-ntr 21312  df-cls 21313  df-nei 21390  df-lp 21428  df-perf 21429  df-cn 21519  df-cnp 21520  df-haus 21607  df-cmp 21679  df-tx 21854  df-hmeo 22047  df-fil 22138  df-fm 22230  df-flim 22231  df-flf 22232  df-xms 22613  df-ms 22614  df-tms 22615  df-cncf 23169  df-ovol 23748  df-vol 23749  df-mbf 23903  df-itg1 23904  df-itg2 23905  df-ibl 23906  df-itg 23907  df-0p 23954  df-limc 24147  df-dv 24148
This theorem is referenced by:  areacirc  34518
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