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Theorem ftc2nc 37709
Description: Choice-free proof of ftc2 26085. (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 11311 . . . . . 6 (𝜑𝐴 ∈ ℝ*)
3 ftc2nc.b . . . . . . 7 (𝜑𝐵 ∈ ℝ)
43rexrd 11311 . . . . . 6 (𝜑𝐵 ∈ ℝ*)
5 ftc2nc.le . . . . . 6 (𝜑𝐴𝐵)
6 ubicc2 13505 . . . . . 6 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
72, 4, 5, 6syl3anc 1373 . . . . 5 (𝜑𝐵 ∈ (𝐴[,]𝐵))
8 fvex 6919 . . . . . 6 ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))‘𝐴) ∈ V
98fvconst2 7224 . . . . 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 2737 . . . . . . . 8 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
1211subcn 24888 . . . . . . . . 9 − ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld))
1312a1i 11 . . . . . . . 8 (𝜑 → − ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld)))
14 eqid 2737 . . . . . . . . 9 (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡) = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡)
15 ssidd 4007 . . . . . . . . 9 (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐵))
16 ioossre 13448 . . . . . . . . . 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 24919 . . . . . . . . . 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 13487 . . . . . . . . . . . . 13 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
23 ffun 6739 . . . . . . . . . . . . 13 ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → Fun (,))
2422, 23ax-mp 5 . . . . . . . . . . . 12 Fun (,)
25 fvelima 6974 . . . . . . . . . . . 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 8053 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
2827fveq2d 6910 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → ((,)‘𝑥) = ((,)‘⟨(1st𝑥), (2nd𝑥)⟩))
29 df-ov 7434 . . . . . . . . . . . . . . . 16 ((1st𝑥)(,)(2nd𝑥)) = ((,)‘⟨(1st𝑥), (2nd𝑥)⟩)
3028, 29eqtr4di 2795 . . . . . . . . . . . . . . 15 (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → ((,)‘𝑥) = ((1st𝑥)(,)(2nd𝑥)))
3130eqeq1d 2739 . . . . . . . . . . . . . 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 8046 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → (1st𝑥) ∈ (𝐴[,]𝐵))
36 elicc1 13431 . . . . . . . . . . . . . . . . . . . . . . 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 1144 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (1st𝑥) ∈ (𝐴[,]𝐵)) → 𝐴 ≤ (1st𝑥))
4035, 39sylan2 593 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → 𝐴 ≤ (1st𝑥))
41 xp2nd 8047 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → (2nd𝑥) ∈ (𝐴[,]𝐵))
42 iccleub 13442 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ* ∧ (2nd𝑥) ∈ (𝐴[,]𝐵)) → (2nd𝑥) ≤ 𝐵)
43423expa 1119 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (2nd𝑥) ∈ (𝐴[,]𝐵)) → (2nd𝑥) ≤ 𝐵)
4433, 41, 43syl2an 596 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (2nd𝑥) ≤ 𝐵)
45 ioossioo 13481 . . . . . . . . . . . . . . . . . . 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 3983 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st𝑥)(,)(2nd𝑥))) → 𝑡 ∈ (𝐴(,)𝐵))
4821ffvelcdmda 7104 . . . . . . . . . . . . . . . . . 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 25600 . . . . . . . . . . . . . . . . . 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 6977 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (ℝ D 𝐹) = (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)))
5554, 18eqeltrrd 2842 . . . . . . . . . . . . . . . . . 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 25840 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿1)
58 ax-resscn 11212 . . . . . . . . . . . . . . . . . . . . 21 ℝ ⊆ ℂ
59 ssid 4006 . . . . . . . . . . . . . . . . . . . . 21 ℂ ⊆ ℂ
60 cncfss 24925 . . . . . . . . . . . . . . . . . . . . 21 ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℂ–cn→ℝ) ⊆ (ℂ–cn→ℂ))
6158, 59, 60mp2an 692 . . . . . . . . . . . . . . . . . . . 20 (ℂ–cn→ℝ) ⊆ (ℂ–cn→ℂ)
62 abscncf 24927 . . . . . . . . . . . . . . . . . . . 20 abs ∈ (ℂ–cn→ℝ)
6361, 62sselii 3980 . . . . . . . . . . . . . . . . . . 19 abs ∈ (ℂ–cn→ℂ)
6463a1i 11 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → abs ∈ (ℂ–cn→ℂ))
6554reseq1d 5996 . . . . . . . . . . . . . . . . . . . . 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 6058 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ↾ ((1st𝑥)(,)(2nd𝑥))) = (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ ((ℝ D 𝐹)‘𝑡)))
6866, 67eqtrd 2777 . . . . . . . . . . . . . . . . . . 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 24923 . . . . . . . . . . . . . . . . . . . 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 2842 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ (((1st𝑥)(,)(2nd𝑥))–cn→ℂ))
7364, 72cncfmpt1f 24940 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ (abs‘((ℝ D 𝐹)‘𝑡))) ∈ (((1st𝑥)(,)(2nd𝑥))–cn→ℂ))
74 cnmbf 25694 . . . . . . . . . . . . . . . . 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 25819 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ)
7776cjcld 15235 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (∗‘∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ∈ ℂ)
78 ioossre 13448 . . . . . . . . . . . . . . . . . . . . 21 ((1st𝑥)(,)(2nd𝑥)) ⊆ ℝ
7978, 58sstri 3993 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑥)(,)(2nd𝑥)) ⊆ ℂ
80 cncfmptc 24938 . . . . . . . . . . . . . . . . . . . 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 1455 . . . . . . . . . . . . . . . . . . 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 2905 . . . . . . . . . . . . . . . . . . . 20 𝑠((ℝ D 𝐹)‘𝑡)
84 nfcsb1v 3923 . . . . . . . . . . . . . . . . . . . 20 𝑡𝑠 / 𝑡((ℝ D 𝐹)‘𝑡)
85 csbeq1a 3913 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑠 → ((ℝ D 𝐹)‘𝑡) = 𝑠 / 𝑡((ℝ D 𝐹)‘𝑡))
8683, 84, 85cbvmpt 5253 . . . . . . . . . . . . . . . . . . 19 (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ ((ℝ D 𝐹)‘𝑡)) = (𝑠 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ 𝑠 / 𝑡((ℝ D 𝐹)‘𝑡))
8786, 72eqeltrrid 2846 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑠 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ 𝑠 / 𝑡((ℝ D 𝐹)‘𝑡)) ∈ (((1st𝑥)(,)(2nd𝑥))–cn→ℂ))
8882, 87mulcncf 25480 . . . . . . . . . . . . . . . . 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 25694 . . . . . . . . . . . . . . . . 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 37696 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (abs‘∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ≤ ∫((1st𝑥)(,)(2nd𝑥))(abs‘((ℝ D 𝐹)‘𝑡)) d𝑡)
9250abscld 15475 . . . . . . . . . . . . . . . 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 37691 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ (abs‘((ℝ D 𝐹)‘𝑡))) ∈ 𝐿1)
9550absge0d 15483 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st𝑥)(,)(2nd𝑥))) → 0 ≤ (abs‘((ℝ D 𝐹)‘𝑡)))
9692, 94, 95itgposval 25831 . . . . . . . . . . . . . . 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 5169 . . . . . . . . . . . . . 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 25808 . . . . . . . . . . . . . . . 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 2830 . . . . . . . . . . . . . . . . . 18 (((1st𝑥)(,)(2nd𝑥)) = 𝑠 → (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↔ 𝑡𝑠))
101100ifbid 4549 . . . . . . . . . . . . . . . . 17 (((1st𝑥)(,)(2nd𝑥)) = 𝑠 → if(𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)), (abs‘((ℝ D 𝐹)‘𝑡)), 0) = if(𝑡𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0))
102101mpteq2dv 5244 . . . . . . . . . . . . . . . 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 5156 . . . . . . . . . . . . . 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 3155 . . . . . . . . . . 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 3145 . . . . . . . . 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 37708 . . . . . . . 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 24919 . . . . . . . . . . 11 (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
113111, 112syl 17 . . . . . . . . . 10 (𝜑𝐹:(𝐴[,]𝐵)⟶ℂ)
114113feqmptd 6977 . . . . . . . . 9 (𝜑𝐹 = (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹𝑥)))
115114, 111eqeltrrd 2842 . . . . . . . 8 (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
11611, 13, 110, 115cncfmpt2f 24941 . . . . . . 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 13469 . . . . . . . . . . 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 11311 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ*)
123 elicc2 13452 . . . . . . . . . . . . . . . . 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 1145 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 𝑥𝐵)
127 iooss2 13423 . . . . . . . . . . . . . 14 ((𝐵 ∈ ℝ*𝑥𝐵) → (𝐴(,)𝑥) ⊆ (𝐴(,)𝐵))
128122, 126, 127syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐴(,)𝑥) ⊆ (𝐴(,)𝐵))
129 ioombl 25600 . . . . . . . . . . . . . 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 25840 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ (𝐴(,)𝑥) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿1)
134120, 133itgcl 25819 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ)
135113ffvelcdmda 7104 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℂ)
136134, 135subcld 11620 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)) ∈ ℂ)
137 tgioo4 24826 . . . . . . . . . 10 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
138 iccntr 24843 . . . . . . . . . . 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 26009 . . . . . . . . 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 11247 . . . . . . . . . . 11 ℝ ∈ {ℝ, ℂ}
142141a1i 11 . . . . . . . . . 10 (𝜑 → ℝ ∈ {ℝ, ℂ})
143 ioossicc 13473 . . . . . . . . . . . 12 (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
144143sseli 3979 . . . . . . . . . . 11 (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ (𝐴[,]𝐵))
145144, 134sylan2 593 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ)
14621ffvelcdmda 7104 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
14714, 1, 3, 5, 19, 18ftc1cnnc 37699 . . . . . . . . . . 11 (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡)) = (ℝ D 𝐹))
148117, 119, 134, 137, 11, 139dvmptntr 26009 . . . . . . . . . . 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 6977 . . . . . . . . . . 11 (𝜑 → (ℝ D 𝐹) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥)))
150147, 148, 1493eqtr3d 2785 . . . . . . . . . 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 7447 . . . . . . . . . . 11 (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹𝑥))))
153117, 119, 135, 137, 11, 139dvmptntr 26009 . . . . . . . . . . 11 (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹𝑥))) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹𝑥))))
154152, 149, 1533eqtr3rd 2786 . . . . . . . . . 10 (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥)))
155142, 145, 146, 150, 151, 146, 154dvmptsub 26005 . . . . . . . . 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 11608 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐹)‘𝑥) − ((ℝ D 𝐹)‘𝑥)) = 0)
157156mpteq2dva 5242 . . . . . . . . 9 (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑥) − ((ℝ D 𝐹)‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0))
158140, 155, 1573eqtrd 2781 . . . . . . . 8 (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0))
159 fconstmpt 5747 . . . . . . . 8 ((𝐴(,)𝐵) × {0}) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0)
160158, 159eqtr4di 2795 . . . . . . 7 (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))) = ((𝐴(,)𝐵) × {0}))
1611, 3, 116, 160dveq0 26039 . . . . . 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 7439 . . . . . . . . 9 (𝑥 = 𝐵 → (𝐴(,)𝑥) = (𝐴(,)𝐵))
164 itgeq1 25808 . . . . . . . . 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 7449 . . . . . . 7 (𝑥 = 𝐵 → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)) = (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝐵)))
168 eqid 2737 . . . . . . 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 7464 . . . . . . 7 (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝐵)) ∈ V
170167, 168, 169fvmpt 7016 . . . . . 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 2779 . . . 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 13504 . . . . . 6 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
1742, 4, 5, 173syl3anc 1373 . . . . 5 (𝜑𝐴 ∈ (𝐴[,]𝐵))
175 oveq2 7439 . . . . . . . . . . 11 (𝑥 = 𝐴 → (𝐴(,)𝑥) = (𝐴(,)𝐴))
176 iooid 13415 . . . . . . . . . . 11 (𝐴(,)𝐴) = ∅
177175, 176eqtrdi 2793 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝐴(,)𝑥) = ∅)
178 itgeq1 25808 . . . . . . . . . 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 25815 . . . . . . . . 9 ∫∅((ℝ D 𝐹)‘𝑡) d𝑡 = 0
181179, 180eqtrdi 2793 . . . . . . . 8 (𝑥 = 𝐴 → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = 0)
182 fveq2 6906 . . . . . . . 8 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
183181, 182oveq12d 7449 . . . . . . 7 (𝑥 = 𝐴 → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)) = (0 − (𝐹𝐴)))
184 df-neg 11495 . . . . . . 7 -(𝐹𝐴) = (0 − (𝐹𝐴))
185183, 184eqtr4di 2795 . . . . . 6 (𝑥 = 𝐴 → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)) = -(𝐹𝐴))
186 negex 11506 . . . . . 6 -(𝐹𝐴) ∈ V
187185, 168, 186fvmpt 7016 . . . . 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 2785 . . 3 (𝜑 → (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝐵)) = -(𝐹𝐴))
190189oveq2d 7447 . 2 (𝜑 → ((𝐹𝐵) + (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝐵))) = ((𝐹𝐵) + -(𝐹𝐴)))
191113, 7ffvelcdmd 7105 . . 3 (𝜑 → (𝐹𝐵) ∈ ℂ)
192 fvexd 6921 . . . 4 ((𝜑𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ V)
193192, 55itgcl 25819 . . 3 (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ)
194191, 193pncan3d 11623 . 2 (𝜑 → ((𝐹𝐵) + (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝐵))) = ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡)
195113, 174ffvelcdmd 7105 . . 3 (𝜑 → (𝐹𝐴) ∈ ℂ)
196191, 195negsubd 11626 . 2 (𝜑 → ((𝐹𝐵) + -(𝐹𝐴)) = ((𝐹𝐵) − (𝐹𝐴)))
197190, 194, 1963eqtr3d 2785 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 1087   = wceq 1540  wcel 2108  wrex 3070  Vcvv 3480  csb 3899  wss 3951  c0 4333  ifcif 4525  𝒫 cpw 4600  {csn 4626  {cpr 4628  cop 4632   class class class wbr 5143  cmpt 5225   × cxp 5683  dom cdm 5685  ran crn 5686  cres 5687  cima 5688  Fun wfun 6555  wf 6557  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  cc 11153  cr 11154  0cc0 11155   + caddc 11158   · cmul 11160  *cxr 11294  cle 11296  cmin 11492  -cneg 11493  (,)cioo 13387  [,]cicc 13390  ccj 15135  abscabs 15273  TopOpenctopn 17466  topGenctg 17482  fldccnfld 21364  intcnt 23025   Cn ccn 23232   ×t ctx 23568  cnccncf 24902  volcvol 25498  MblFncmbf 25649  2citg2 25651  𝐿1cibl 25652  citg 25653   D cdv 25898
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 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-addf 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-symdif 4253  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-disj 5111  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-acn 9982  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-mod 13910  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-rlim 15525  df-sum 15723  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  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 17467  df-topn 17468  df-0g 17486  df-gsum 17487  df-topgen 17488  df-pt 17489  df-prds 17492  df-xrs 17547  df-qtop 17552  df-imas 17553  df-xps 17555  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-mulg 19086  df-cntz 19335  df-cmn 19800  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-fbas 21361  df-fg 21362  df-cnfld 21365  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-cld 23027  df-ntr 23028  df-cls 23029  df-nei 23106  df-lp 23144  df-perf 23145  df-cn 23235  df-cnp 23236  df-haus 23323  df-cmp 23395  df-tx 23570  df-hmeo 23763  df-fil 23854  df-fm 23946  df-flim 23947  df-flf 23948  df-xms 24330  df-ms 24331  df-tms 24332  df-cncf 24904  df-ovol 25499  df-vol 25500  df-mbf 25654  df-itg1 25655  df-itg2 25656  df-ibl 25657  df-itg 25658  df-0p 25705  df-limc 25901  df-dv 25902
This theorem is referenced by:  areacirc  37720
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