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Theorem ftc2nc 37688
Description: Choice-free proof of ftc2 26099. (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 11308 . . . . . 6 (𝜑𝐴 ∈ ℝ*)
3 ftc2nc.b . . . . . . 7 (𝜑𝐵 ∈ ℝ)
43rexrd 11308 . . . . . 6 (𝜑𝐵 ∈ ℝ*)
5 ftc2nc.le . . . . . 6 (𝜑𝐴𝐵)
6 ubicc2 13501 . . . . . 6 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
72, 4, 5, 6syl3anc 1370 . . . . 5 (𝜑𝐵 ∈ (𝐴[,]𝐵))
8 fvex 6919 . . . . . 6 ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))‘𝐴) ∈ V
98fvconst2 7223 . . . . 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 2734 . . . . . . . 8 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
1211subcn 24901 . . . . . . . . 9 − ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld))
1312a1i 11 . . . . . . . 8 (𝜑 → − ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld)))
14 eqid 2734 . . . . . . . . 9 (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡) = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡)
15 ssidd 4018 . . . . . . . . 9 (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐵))
16 ioossre 13444 . . . . . . . . . 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 24932 . . . . . . . . . 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 13483 . . . . . . . . . . . . 13 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
23 ffun 6739 . . . . . . . . . . . . 13 ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → Fun (,))
2422, 23ax-mp 5 . . . . . . . . . . . 12 Fun (,)
25 fvelima 6973 . . . . . . . . . . . 12 ((Fun (,) ∧ 𝑠 ∈ ((,) “ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) → ∃𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))((,)‘𝑥) = 𝑠)
2624, 25mpan 690 . . . . . . . . . . 11 (𝑠 ∈ ((,) “ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ∃𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))((,)‘𝑥) = 𝑠)
27 1st2nd2 8051 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
2827fveq2d 6910 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → ((,)‘𝑥) = ((,)‘⟨(1st𝑥), (2nd𝑥)⟩))
29 df-ov 7433 . . . . . . . . . . . . . . . 16 ((1st𝑥)(,)(2nd𝑥)) = ((,)‘⟨(1st𝑥), (2nd𝑥)⟩)
3028, 29eqtr4di 2792 . . . . . . . . . . . . . . 15 (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → ((,)‘𝑥) = ((1st𝑥)(,)(2nd𝑥)))
3130eqeq1d 2736 . . . . . . . . . . . . . 14 (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → (((,)‘𝑥) = 𝑠 ↔ ((1st𝑥)(,)(2nd𝑥)) = 𝑠))
3231adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (((,)‘𝑥) = 𝑠 ↔ ((1st𝑥)(,)(2nd𝑥)) = 𝑠))
332, 4jca 511 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝐴 ∈ ℝ*𝐵 ∈ ℝ*))
3433adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝐴 ∈ ℝ*𝐵 ∈ ℝ*))
35 xp1st 8044 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → (1st𝑥) ∈ (𝐴[,]𝐵))
36 elicc1 13427 . . . . . . . . . . . . . . . . . . . . . . 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 476 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (1st𝑥) ∈ (𝐴[,]𝐵)) → ((1st𝑥) ∈ ℝ*𝐴 ≤ (1st𝑥) ∧ (1st𝑥) ≤ 𝐵))
3938simp2d 1142 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (1st𝑥) ∈ (𝐴[,]𝐵)) → 𝐴 ≤ (1st𝑥))
4035, 39sylan2 593 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → 𝐴 ≤ (1st𝑥))
41 xp2nd 8045 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → (2nd𝑥) ∈ (𝐴[,]𝐵))
42 iccleub 13438 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ* ∧ (2nd𝑥) ∈ (𝐴[,]𝐵)) → (2nd𝑥) ≤ 𝐵)
43423expa 1117 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (2nd𝑥) ∈ (𝐴[,]𝐵)) → (2nd𝑥) ≤ 𝐵)
4433, 41, 43syl2an 596 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (2nd𝑥) ≤ 𝐵)
45 ioossioo 13477 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐴 ≤ (1st𝑥) ∧ (2nd𝑥) ≤ 𝐵)) → ((1st𝑥)(,)(2nd𝑥)) ⊆ (𝐴(,)𝐵))
4634, 40, 44, 45syl12anc 837 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((1st𝑥)(,)(2nd𝑥)) ⊆ (𝐴(,)𝐵))
4746sselda 3994 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st𝑥)(,)(2nd𝑥))) → 𝑡 ∈ (𝐴(,)𝐵))
4821ffvelcdmda 7103 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ ℂ)
4948adantlr 715 . . . . . . . . . . . . . . . . 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 25613 . . . . . . . . . . . . . . . . . 18 ((1st𝑥)(,)(2nd𝑥)) ∈ dom vol
5251a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((1st𝑥)(,)(2nd𝑥)) ∈ dom vol)
53 fvexd 6921 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ V)
5421feqmptd 6976 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (ℝ D 𝐹) = (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)))
5554, 18eqeltrrd 2839 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿1)
5655adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿1)
5746, 52, 53, 56iblss 25854 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿1)
58 ax-resscn 11209 . . . . . . . . . . . . . . . . . . . . 21 ℝ ⊆ ℂ
59 ssid 4017 . . . . . . . . . . . . . . . . . . . . 21 ℂ ⊆ ℂ
60 cncfss 24938 . . . . . . . . . . . . . . . . . . . . 21 ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℂ–cn→ℝ) ⊆ (ℂ–cn→ℂ))
6158, 59, 60mp2an 692 . . . . . . . . . . . . . . . . . . . 20 (ℂ–cn→ℝ) ⊆ (ℂ–cn→ℂ)
62 abscncf 24940 . . . . . . . . . . . . . . . . . . . 20 abs ∈ (ℂ–cn→ℝ)
6361, 62sselii 3991 . . . . . . . . . . . . . . . . . . 19 abs ∈ (ℂ–cn→ℂ)
6463a1i 11 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → abs ∈ (ℂ–cn→ℂ))
6554reseq1d 5998 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((ℝ D 𝐹) ↾ ((1st𝑥)(,)(2nd𝑥))) = ((𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ↾ ((1st𝑥)(,)(2nd𝑥))))
6665adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((ℝ D 𝐹) ↾ ((1st𝑥)(,)(2nd𝑥))) = ((𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ↾ ((1st𝑥)(,)(2nd𝑥))))
6746resmptd 6059 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ↾ ((1st𝑥)(,)(2nd𝑥))) = (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ ((ℝ D 𝐹)‘𝑡)))
6866, 67eqtrd 2774 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((ℝ D 𝐹) ↾ ((1st𝑥)(,)(2nd𝑥))) = (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ ((ℝ D 𝐹)‘𝑡)))
6919adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (ℝ D 𝐹) ∈ ((𝐴(,)𝐵)–cn→ℂ))
70 rescncf 24936 . . . . . . . . . . . . . . . . . . . 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 2839 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ (((1st𝑥)(,)(2nd𝑥))–cn→ℂ))
7364, 72cncfmpt1f 24953 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ (abs‘((ℝ D 𝐹)‘𝑡))) ∈ (((1st𝑥)(,)(2nd𝑥))–cn→ℂ))
74 cnmbf 25707 . . . . . . . . . . . . . . . . 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 25833 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ)
7776cjcld 15231 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (∗‘∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ∈ ℂ)
78 ioossre 13444 . . . . . . . . . . . . . . . . . . . . 21 ((1st𝑥)(,)(2nd𝑥)) ⊆ ℝ
7978, 58sstri 4004 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑥)(,)(2nd𝑥)) ⊆ ℂ
80 cncfmptc 24951 . . . . . . . . . . . . . . . . . . . 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 1452 . . . . . . . . . . . . . . . . . . 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 2902 . . . . . . . . . . . . . . . . . . . 20 𝑠((ℝ D 𝐹)‘𝑡)
84 nfcsb1v 3932 . . . . . . . . . . . . . . . . . . . 20 𝑡𝑠 / 𝑡((ℝ D 𝐹)‘𝑡)
85 csbeq1a 3921 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑠 → ((ℝ D 𝐹)‘𝑡) = 𝑠 / 𝑡((ℝ D 𝐹)‘𝑡))
8683, 84, 85cbvmpt 5258 . . . . . . . . . . . . . . . . . . 19 (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ ((ℝ D 𝐹)‘𝑡)) = (𝑠 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ 𝑠 / 𝑡((ℝ D 𝐹)‘𝑡))
8786, 72eqeltrrid 2843 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑠 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ 𝑠 / 𝑡((ℝ D 𝐹)‘𝑡)) ∈ (((1st𝑥)(,)(2nd𝑥))–cn→ℂ))
8882, 87mulcncf 25493 . . . . . . . . . . . . . . . . 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 25707 . . . . . . . . . . . . . . . . 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 37675 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (abs‘∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ≤ ∫((1st𝑥)(,)(2nd𝑥))(abs‘((ℝ D 𝐹)‘𝑡)) d𝑡)
9250abscld 15471 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st𝑥)(,)(2nd𝑥))) → (abs‘((ℝ D 𝐹)‘𝑡)) ∈ ℝ)
93 fvexd 6921 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st𝑥)(,)(2nd𝑥))) → ((ℝ D 𝐹)‘𝑡) ∈ V)
9493, 57, 75iblabsnc 37670 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ (abs‘((ℝ D 𝐹)‘𝑡))) ∈ 𝐿1)
9550absge0d 15479 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st𝑥)(,)(2nd𝑥))) → 0 ≤ (abs‘((ℝ D 𝐹)‘𝑡)))
9692, 94, 95itgposval 25845 . . . . . . . . . . . . . . 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 5173 . . . . . . . . . . . . . 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 25822 . . . . . . . . . . . . . . . 16 (((1st𝑥)(,)(2nd𝑥)) = 𝑠 → ∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡 = ∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡)
9998fveq2d 6910 . . . . . . . . . . . . . . 15 (((1st𝑥)(,)(2nd𝑥)) = 𝑠 → (abs‘∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) = (abs‘∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡))
100 eleq2 2827 . . . . . . . . . . . . . . . . . 18 (((1st𝑥)(,)(2nd𝑥)) = 𝑠 → (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↔ 𝑡𝑠))
101100ifbid 4553 . . . . . . . . . . . . . . . . 17 (((1st𝑥)(,)(2nd𝑥)) = 𝑠 → if(𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)), (abs‘((ℝ D 𝐹)‘𝑡)), 0) = if(𝑡𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0))
102101mpteq2dv 5249 . . . . . . . . . . . . . . . 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 6910 . . . . . . . . . . . . . . 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 5160 . . . . . . . . . . . . . 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 245 . . . . . . . . . . . . 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 240 . . . . . . . . . . . 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 3152 . . . . . . . . . . 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 3142 . . . . . . . . 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 37687 . . . . . . . 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 24932 . . . . . . . . . . 11 (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
113111, 112syl 17 . . . . . . . . . 10 (𝜑𝐹:(𝐴[,]𝐵)⟶ℂ)
114113feqmptd 6976 . . . . . . . . 9 (𝜑𝐹 = (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹𝑥)))
115114, 111eqeltrrd 2839 . . . . . . . 8 (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
11611, 13, 110, 115cncfmpt2f 24954 . . . . . . 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 13465 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
1191, 3, 118syl2anc 584 . . . . . . . . . 10 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
120 fvexd 6921 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ 𝑡 ∈ (𝐴(,)𝑥)) → ((ℝ D 𝐹)‘𝑡) ∈ V)
1213adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ)
122121rexrd 11308 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ*)
123 elicc2 13448 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴𝑥𝑥𝐵)))
1241, 3, 123syl2anc 584 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴𝑥𝑥𝐵)))
125124biimpa 476 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝐴𝑥𝑥𝐵))
126125simp3d 1143 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 𝑥𝐵)
127 iooss2 13419 . . . . . . . . . . . . . 14 ((𝐵 ∈ ℝ*𝑥𝐵) → (𝐴(,)𝑥) ⊆ (𝐴(,)𝐵))
128122, 126, 127syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐴(,)𝑥) ⊆ (𝐴(,)𝐵))
129 ioombl 25613 . . . . . . . . . . . . . 14 (𝐴(,)𝑥) ∈ dom vol
130129a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐴(,)𝑥) ∈ dom vol)
131 fvexd 6921 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ V)
13255adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿1)
133128, 130, 131, 132iblss 25854 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ (𝐴(,)𝑥) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿1)
134120, 133itgcl 25833 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ)
135113ffvelcdmda 7103 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℂ)
136134, 135subcld 11617 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)) ∈ ℂ)
13711tgioo2 24838 . . . . . . . . . 10 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
138 iccntr 24856 . . . . . . . . . . 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 26023 . . . . . . . . 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 11244 . . . . . . . . . . 11 ℝ ∈ {ℝ, ℂ}
142141a1i 11 . . . . . . . . . 10 (𝜑 → ℝ ∈ {ℝ, ℂ})
143 ioossicc 13469 . . . . . . . . . . . 12 (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
144143sseli 3990 . . . . . . . . . . 11 (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ (𝐴[,]𝐵))
145144, 134sylan2 593 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ)
14621ffvelcdmda 7103 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
14714, 1, 3, 5, 19, 18ftc1cnnc 37678 . . . . . . . . . . 11 (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡)) = (ℝ D 𝐹))
148117, 119, 134, 137, 11, 139dvmptntr 26023 . . . . . . . . . . 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 6976 . . . . . . . . . . 11 (𝜑 → (ℝ D 𝐹) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥)))
150147, 148, 1493eqtr3d 2782 . . . . . . . . . 10 (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥)))
151144, 135sylan2 593 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → (𝐹𝑥) ∈ ℂ)
152114oveq2d 7446 . . . . . . . . . . 11 (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹𝑥))))
153117, 119, 135, 137, 11, 139dvmptntr 26023 . . . . . . . . . . 11 (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹𝑥))) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹𝑥))))
154152, 149, 1533eqtr3rd 2783 . . . . . . . . . 10 (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥)))
155142, 145, 146, 150, 151, 146, 154dvmptsub 26019 . . . . . . . . 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 11605 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐹)‘𝑥) − ((ℝ D 𝐹)‘𝑥)) = 0)
157156mpteq2dva 5247 . . . . . . . . 9 (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑥) − ((ℝ D 𝐹)‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0))
158140, 155, 1573eqtrd 2778 . . . . . . . 8 (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0))
159 fconstmpt 5750 . . . . . . . 8 ((𝐴(,)𝐵) × {0}) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0)
160158, 159eqtr4di 2792 . . . . . . 7 (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))) = ((𝐴(,)𝐵) × {0}))
1611, 3, 116, 160dveq0 26053 . . . . . 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 6908 . . . . 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 7438 . . . . . . . . 9 (𝑥 = 𝐵 → (𝐴(,)𝑥) = (𝐴(,)𝐵))
164 itgeq1 25822 . . . . . . . . 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 6906 . . . . . . . 8 (𝑥 = 𝐵 → (𝐹𝑥) = (𝐹𝐵))
167165, 166oveq12d 7448 . . . . . . 7 (𝑥 = 𝐵 → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)) = (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝐵)))
168 eqid 2734 . . . . . . 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 7463 . . . . . . 7 (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝐵)) ∈ V
170167, 168, 169fvmpt 7015 . . . . . 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 2776 . . . 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 13500 . . . . . 6 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
1742, 4, 5, 173syl3anc 1370 . . . . 5 (𝜑𝐴 ∈ (𝐴[,]𝐵))
175 oveq2 7438 . . . . . . . . . . 11 (𝑥 = 𝐴 → (𝐴(,)𝑥) = (𝐴(,)𝐴))
176 iooid 13411 . . . . . . . . . . 11 (𝐴(,)𝐴) = ∅
177175, 176eqtrdi 2790 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝐴(,)𝑥) = ∅)
178 itgeq1 25822 . . . . . . . . . 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 25829 . . . . . . . . 9 ∫∅((ℝ D 𝐹)‘𝑡) d𝑡 = 0
181179, 180eqtrdi 2790 . . . . . . . 8 (𝑥 = 𝐴 → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = 0)
182 fveq2 6906 . . . . . . . 8 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
183181, 182oveq12d 7448 . . . . . . 7 (𝑥 = 𝐴 → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)) = (0 − (𝐹𝐴)))
184 df-neg 11492 . . . . . . 7 -(𝐹𝐴) = (0 − (𝐹𝐴))
185183, 184eqtr4di 2792 . . . . . 6 (𝑥 = 𝐴 → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)) = -(𝐹𝐴))
186 negex 11503 . . . . . 6 -(𝐹𝐴) ∈ V
187185, 168, 186fvmpt 7015 . . . . 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 2782 . . 3 (𝜑 → (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝐵)) = -(𝐹𝐴))
190189oveq2d 7446 . 2 (𝜑 → ((𝐹𝐵) + (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝐵))) = ((𝐹𝐵) + -(𝐹𝐴)))
191113, 7ffvelcdmd 7104 . . 3 (𝜑 → (𝐹𝐵) ∈ ℂ)
192 fvexd 6921 . . . 4 ((𝜑𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ V)
193192, 55itgcl 25833 . . 3 (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ)
194191, 193pncan3d 11620 . 2 (𝜑 → ((𝐹𝐵) + (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝐵))) = ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡)
195113, 174ffvelcdmd 7104 . . 3 (𝜑 → (𝐹𝐴) ∈ ℂ)
196191, 195negsubd 11623 . 2 (𝜑 → ((𝐹𝐵) + -(𝐹𝐴)) = ((𝐹𝐵) − (𝐹𝐴)))
197190, 194, 1963eqtr3d 2782 1 (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹𝐵) − (𝐹𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  wrex 3067  Vcvv 3477  csb 3907  wss 3962  c0 4338  ifcif 4530  𝒫 cpw 4604  {csn 4630  {cpr 4632  cop 4636   class class class wbr 5147  cmpt 5230   × cxp 5686  dom cdm 5688  ran crn 5689  cres 5690  cima 5691  Fun wfun 6556  wf 6558  cfv 6562  (class class class)co 7430  1st c1st 8010  2nd c2nd 8011  cc 11150  cr 11151  0cc0 11152   + caddc 11155   · cmul 11157  *cxr 11291  cle 11293  cmin 11489  -cneg 11490  (,)cioo 13383  [,]cicc 13386  ccj 15131  abscabs 15269  TopOpenctopn 17467  topGenctg 17483  fldccnfld 21381  intcnt 23040   Cn ccn 23247   ×t ctx 23583  cnccncf 24915  volcvol 25511  MblFncmbf 25662  2citg2 25664  𝐿1cibl 25665  citg 25666   D cdv 25912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230  ax-addf 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-symdif 4258  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-disj 5115  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-ofr 7697  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-omul 8509  df-er 8743  df-map 8866  df-pm 8867  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-fi 9448  df-sup 9479  df-inf 9480  df-oi 9547  df-dju 9938  df-card 9976  df-acn 9979  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-q 12988  df-rp 13032  df-xneg 13151  df-xadd 13152  df-xmul 13153  df-ioo 13387  df-ico 13389  df-icc 13390  df-fz 13544  df-fzo 13691  df-fl 13828  df-mod 13906  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-rlim 15521  df-sum 15719  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-rest 17468  df-topn 17469  df-0g 17487  df-gsum 17488  df-topgen 17489  df-pt 17490  df-prds 17493  df-xrs 17548  df-qtop 17553  df-imas 17554  df-xps 17556  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-submnd 18809  df-mulg 19098  df-cntz 19347  df-cmn 19814  df-psmet 21373  df-xmet 21374  df-met 21375  df-bl 21376  df-mopn 21377  df-fbas 21378  df-fg 21379  df-cnfld 21382  df-top 22915  df-topon 22932  df-topsp 22954  df-bases 22968  df-cld 23042  df-ntr 23043  df-cls 23044  df-nei 23121  df-lp 23159  df-perf 23160  df-cn 23250  df-cnp 23251  df-haus 23338  df-cmp 23410  df-tx 23585  df-hmeo 23778  df-fil 23869  df-fm 23961  df-flim 23962  df-flf 23963  df-xms 24345  df-ms 24346  df-tms 24347  df-cncf 24917  df-ovol 25512  df-vol 25513  df-mbf 25667  df-itg1 25668  df-itg2 25669  df-ibl 25670  df-itg 25671  df-0p 25718  df-limc 25915  df-dv 25916
This theorem is referenced by:  areacirc  37699
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