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Theorem ftc2nc 36160
Description: Choice-free proof of ftc2 25408. (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 11205 . . . . . 6 (𝜑𝐴 ∈ ℝ*)
3 ftc2nc.b . . . . . . 7 (𝜑𝐵 ∈ ℝ)
43rexrd 11205 . . . . . 6 (𝜑𝐵 ∈ ℝ*)
5 ftc2nc.le . . . . . 6 (𝜑𝐴𝐵)
6 ubicc2 13382 . . . . . 6 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
72, 4, 5, 6syl3anc 1371 . . . . 5 (𝜑𝐵 ∈ (𝐴[,]𝐵))
8 fvex 6855 . . . . . 6 ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))‘𝐴) ∈ V
98fvconst2 7153 . . . . 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 2736 . . . . . . . 8 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
1211subcn 24229 . . . . . . . . 9 − ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld))
1312a1i 11 . . . . . . . 8 (𝜑 → − ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld)))
14 eqid 2736 . . . . . . . . 9 (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡) = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡)
15 ssidd 3967 . . . . . . . . 9 (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐵))
16 ioossre 13325 . . . . . . . . . 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 24256 . . . . . . . . . 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 13364 . . . . . . . . . . . . 13 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
23 ffun 6671 . . . . . . . . . . . . 13 ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → Fun (,))
2422, 23ax-mp 5 . . . . . . . . . . . 12 Fun (,)
25 fvelima 6908 . . . . . . . . . . . 12 ((Fun (,) ∧ 𝑠 ∈ ((,) “ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) → ∃𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))((,)‘𝑥) = 𝑠)
2624, 25mpan 688 . . . . . . . . . . 11 (𝑠 ∈ ((,) “ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ∃𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))((,)‘𝑥) = 𝑠)
27 1st2nd2 7960 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
2827fveq2d 6846 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → ((,)‘𝑥) = ((,)‘⟨(1st𝑥), (2nd𝑥)⟩))
29 df-ov 7360 . . . . . . . . . . . . . . . 16 ((1st𝑥)(,)(2nd𝑥)) = ((,)‘⟨(1st𝑥), (2nd𝑥)⟩)
3028, 29eqtr4di 2794 . . . . . . . . . . . . . . 15 (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → ((,)‘𝑥) = ((1st𝑥)(,)(2nd𝑥)))
3130eqeq1d 2738 . . . . . . . . . . . . . 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 7953 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → (1st𝑥) ∈ (𝐴[,]𝐵))
36 elicc1 13308 . . . . . . . . . . . . . . . . . . . . . . 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 1143 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (1st𝑥) ∈ (𝐴[,]𝐵)) → 𝐴 ≤ (1st𝑥))
4035, 39sylan2 593 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → 𝐴 ≤ (1st𝑥))
41 xp2nd 7954 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → (2nd𝑥) ∈ (𝐴[,]𝐵))
42 iccleub 13319 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ* ∧ (2nd𝑥) ∈ (𝐴[,]𝐵)) → (2nd𝑥) ≤ 𝐵)
43423expa 1118 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (2nd𝑥) ∈ (𝐴[,]𝐵)) → (2nd𝑥) ≤ 𝐵)
4433, 41, 43syl2an 596 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (2nd𝑥) ≤ 𝐵)
45 ioossioo 13358 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐴 ≤ (1st𝑥) ∧ (2nd𝑥) ≤ 𝐵)) → ((1st𝑥)(,)(2nd𝑥)) ⊆ (𝐴(,)𝐵))
4634, 40, 44, 45syl12anc 835 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((1st𝑥)(,)(2nd𝑥)) ⊆ (𝐴(,)𝐵))
4746sselda 3944 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st𝑥)(,)(2nd𝑥))) → 𝑡 ∈ (𝐴(,)𝐵))
4821ffvelcdmda 7035 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ ℂ)
4948adantlr 713 . . . . . . . . . . . . . . . . 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 24929 . . . . . . . . . . . . . . . . . 18 ((1st𝑥)(,)(2nd𝑥)) ∈ dom vol
5251a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((1st𝑥)(,)(2nd𝑥)) ∈ dom vol)
53 fvexd 6857 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ V)
5421feqmptd 6910 . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿1)
5746, 52, 53, 56iblss 25169 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿1)
58 ax-resscn 11108 . . . . . . . . . . . . . . . . . . . . 21 ℝ ⊆ ℂ
59 ssid 3966 . . . . . . . . . . . . . . . . . . . . 21 ℂ ⊆ ℂ
60 cncfss 24262 . . . . . . . . . . . . . . . . . . . . 21 ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℂ–cn→ℝ) ⊆ (ℂ–cn→ℂ))
6158, 59, 60mp2an 690 . . . . . . . . . . . . . . . . . . . 20 (ℂ–cn→ℝ) ⊆ (ℂ–cn→ℂ)
62 abscncf 24264 . . . . . . . . . . . . . . . . . . . 20 abs ∈ (ℂ–cn→ℝ)
6361, 62sselii 3941 . . . . . . . . . . . . . . . . . . 19 abs ∈ (ℂ–cn→ℂ)
6463a1i 11 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → abs ∈ (ℂ–cn→ℂ))
6554reseq1d 5936 . . . . . . . . . . . . . . . . . . . . 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 5994 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ↾ ((1st𝑥)(,)(2nd𝑥))) = (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ ((ℝ D 𝐹)‘𝑡)))
6866, 67eqtrd 2776 . . . . . . . . . . . . . . . . . . 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 24260 . . . . . . . . . . . . . . . . . . . 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 24277 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ (abs‘((ℝ D 𝐹)‘𝑡))) ∈ (((1st𝑥)(,)(2nd𝑥))–cn→ℂ))
74 cnmbf 25023 . . . . . . . . . . . . . . . . 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 25148 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ)
7776cjcld 15081 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (∗‘∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ∈ ℂ)
78 ioossre 13325 . . . . . . . . . . . . . . . . . . . . 21 ((1st𝑥)(,)(2nd𝑥)) ⊆ ℝ
7978, 58sstri 3953 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑥)(,)(2nd𝑥)) ⊆ ℂ
80 cncfmptc 24275 . . . . . . . . . . . . . . . . . . . 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 1453 . . . . . . . . . . . . . . . . . . 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 2907 . . . . . . . . . . . . . . . . . . . 20 𝑠((ℝ D 𝐹)‘𝑡)
84 nfcsb1v 3880 . . . . . . . . . . . . . . . . . . . 20 𝑡𝑠 / 𝑡((ℝ D 𝐹)‘𝑡)
85 csbeq1a 3869 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑠 → ((ℝ D 𝐹)‘𝑡) = 𝑠 / 𝑡((ℝ D 𝐹)‘𝑡))
8683, 84, 85cbvmpt 5216 . . . . . . . . . . . . . . . . . . 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 24810 . . . . . . . . . . . . . . . . 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 25023 . . . . . . . . . . . . . . . . 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 36147 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (abs‘∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ≤ ∫((1st𝑥)(,)(2nd𝑥))(abs‘((ℝ D 𝐹)‘𝑡)) d𝑡)
9250abscld 15321 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st𝑥)(,)(2nd𝑥))) → (abs‘((ℝ D 𝐹)‘𝑡)) ∈ ℝ)
93 fvexd 6857 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st𝑥)(,)(2nd𝑥))) → ((ℝ D 𝐹)‘𝑡) ∈ V)
9493, 57, 75iblabsnc 36142 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ (abs‘((ℝ D 𝐹)‘𝑡))) ∈ 𝐿1)
9550absge0d 15329 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st𝑥)(,)(2nd𝑥))) → 0 ≤ (abs‘((ℝ D 𝐹)‘𝑡)))
9692, 94, 95itgposval 25160 . . . . . . . . . . . . . . 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 5131 . . . . . . . . . . . . . 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 25137 . . . . . . . . . . . . . . . 16 (((1st𝑥)(,)(2nd𝑥)) = 𝑠 → ∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡 = ∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡)
9998fveq2d 6846 . . . . . . . . . . . . . . 15 (((1st𝑥)(,)(2nd𝑥)) = 𝑠 → (abs‘∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) = (abs‘∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡))
100 eleq2 2826 . . . . . . . . . . . . . . . . . 18 (((1st𝑥)(,)(2nd𝑥)) = 𝑠 → (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↔ 𝑡𝑠))
101100ifbid 4509 . . . . . . . . . . . . . . . . 17 (((1st𝑥)(,)(2nd𝑥)) = 𝑠 → if(𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)), (abs‘((ℝ D 𝐹)‘𝑡)), 0) = if(𝑡𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0))
102101mpteq2dv 5207 . . . . . . . . . . . . . . . 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 6846 . . . . . . . . . . . . . . 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 5118 . . . . . . . . . . . . . 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 244 . . . . . . . . . . . . 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 239 . . . . . . . . . . . 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 36159 . . . . . . . 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 24256 . . . . . . . . . . 11 (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
113111, 112syl 17 . . . . . . . . . 10 (𝜑𝐹:(𝐴[,]𝐵)⟶ℂ)
114113feqmptd 6910 . . . . . . . . 9 (𝜑𝐹 = (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹𝑥)))
115114, 111eqeltrrd 2839 . . . . . . . 8 (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
11611, 13, 110, 115cncfmpt2f 24278 . . . . . . 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 13346 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
1191, 3, 118syl2anc 584 . . . . . . . . . 10 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
120 fvexd 6857 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ 𝑡 ∈ (𝐴(,)𝑥)) → ((ℝ D 𝐹)‘𝑡) ∈ V)
1213adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ)
122121rexrd 11205 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ*)
123 elicc2 13329 . . . . . . . . . . . . . . . . 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 1144 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 𝑥𝐵)
127 iooss2 13300 . . . . . . . . . . . . . 14 ((𝐵 ∈ ℝ*𝑥𝐵) → (𝐴(,)𝑥) ⊆ (𝐴(,)𝐵))
128122, 126, 127syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐴(,)𝑥) ⊆ (𝐴(,)𝐵))
129 ioombl 24929 . . . . . . . . . . . . . 14 (𝐴(,)𝑥) ∈ dom vol
130129a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐴(,)𝑥) ∈ dom vol)
131 fvexd 6857 . . . . . . . . . . . . 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 25169 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ (𝐴(,)𝑥) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿1)
134120, 133itgcl 25148 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ)
135113ffvelcdmda 7035 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℂ)
136134, 135subcld 11512 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)) ∈ ℂ)
13711tgioo2 24166 . . . . . . . . . 10 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
138 iccntr 24184 . . . . . . . . . . 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 25335 . . . . . . . . 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 11143 . . . . . . . . . . 11 ℝ ∈ {ℝ, ℂ}
142141a1i 11 . . . . . . . . . 10 (𝜑 → ℝ ∈ {ℝ, ℂ})
143 ioossicc 13350 . . . . . . . . . . . 12 (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
144143sseli 3940 . . . . . . . . . . 11 (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ (𝐴[,]𝐵))
145144, 134sylan2 593 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ)
14621ffvelcdmda 7035 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
14714, 1, 3, 5, 19, 18ftc1cnnc 36150 . . . . . . . . . . 11 (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡)) = (ℝ D 𝐹))
148117, 119, 134, 137, 11, 139dvmptntr 25335 . . . . . . . . . . 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 6910 . . . . . . . . . . 11 (𝜑 → (ℝ D 𝐹) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥)))
150147, 148, 1493eqtr3d 2784 . . . . . . . . . 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 7373 . . . . . . . . . . 11 (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹𝑥))))
153117, 119, 135, 137, 11, 139dvmptntr 25335 . . . . . . . . . . 11 (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹𝑥))) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹𝑥))))
154152, 149, 1533eqtr3rd 2785 . . . . . . . . . 10 (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥)))
155142, 145, 146, 150, 151, 146, 154dvmptsub 25331 . . . . . . . . 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 11500 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐹)‘𝑥) − ((ℝ D 𝐹)‘𝑥)) = 0)
157156mpteq2dva 5205 . . . . . . . . 9 (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑥) − ((ℝ D 𝐹)‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0))
158140, 155, 1573eqtrd 2780 . . . . . . . 8 (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0))
159 fconstmpt 5694 . . . . . . . 8 ((𝐴(,)𝐵) × {0}) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0)
160158, 159eqtr4di 2794 . . . . . . 7 (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))) = ((𝐴(,)𝐵) × {0}))
1611, 3, 116, 160dveq0 25364 . . . . . 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 6844 . . . . 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 7365 . . . . . . . . 9 (𝑥 = 𝐵 → (𝐴(,)𝑥) = (𝐴(,)𝐵))
164 itgeq1 25137 . . . . . . . . 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 6842 . . . . . . . 8 (𝑥 = 𝐵 → (𝐹𝑥) = (𝐹𝐵))
167165, 166oveq12d 7375 . . . . . . 7 (𝑥 = 𝐵 → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)) = (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝐵)))
168 eqid 2736 . . . . . . 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 7390 . . . . . . 7 (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝐵)) ∈ V
170167, 168, 169fvmpt 6948 . . . . . 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 2778 . . . 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 13381 . . . . . 6 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
1742, 4, 5, 173syl3anc 1371 . . . . 5 (𝜑𝐴 ∈ (𝐴[,]𝐵))
175 oveq2 7365 . . . . . . . . . . 11 (𝑥 = 𝐴 → (𝐴(,)𝑥) = (𝐴(,)𝐴))
176 iooid 13292 . . . . . . . . . . 11 (𝐴(,)𝐴) = ∅
177175, 176eqtrdi 2792 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝐴(,)𝑥) = ∅)
178 itgeq1 25137 . . . . . . . . . 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 25144 . . . . . . . . 9 ∫∅((ℝ D 𝐹)‘𝑡) d𝑡 = 0
181179, 180eqtrdi 2792 . . . . . . . 8 (𝑥 = 𝐴 → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = 0)
182 fveq2 6842 . . . . . . . 8 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
183181, 182oveq12d 7375 . . . . . . 7 (𝑥 = 𝐴 → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)) = (0 − (𝐹𝐴)))
184 df-neg 11388 . . . . . . 7 -(𝐹𝐴) = (0 − (𝐹𝐴))
185183, 184eqtr4di 2794 . . . . . 6 (𝑥 = 𝐴 → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)) = -(𝐹𝐴))
186 negex 11399 . . . . . 6 -(𝐹𝐴) ∈ V
187185, 168, 186fvmpt 6948 . . . . 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 2784 . . 3 (𝜑 → (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝐵)) = -(𝐹𝐴))
190189oveq2d 7373 . 2 (𝜑 → ((𝐹𝐵) + (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝐵))) = ((𝐹𝐵) + -(𝐹𝐴)))
191113, 7ffvelcdmd 7036 . . 3 (𝜑 → (𝐹𝐵) ∈ ℂ)
192 fvexd 6857 . . . 4 ((𝜑𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ V)
193192, 55itgcl 25148 . . 3 (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ)
194191, 193pncan3d 11515 . 2 (𝜑 → ((𝐹𝐵) + (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝐵))) = ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡)
195113, 174ffvelcdmd 7036 . . 3 (𝜑 → (𝐹𝐴) ∈ ℂ)
196191, 195negsubd 11518 . 2 (𝜑 → ((𝐹𝐵) + -(𝐹𝐴)) = ((𝐹𝐵) − (𝐹𝐴)))
197190, 194, 1963eqtr3d 2784 1 (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹𝐵) − (𝐹𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wrex 3073  Vcvv 3445  csb 3855  wss 3910  c0 4282  ifcif 4486  𝒫 cpw 4560  {csn 4586  {cpr 4588  cop 4592   class class class wbr 5105  cmpt 5188   × cxp 5631  dom cdm 5633  ran crn 5634  cres 5635  cima 5636  Fun wfun 6490  wf 6492  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  cc 11049  cr 11050  0cc0 11051   + caddc 11054   · cmul 11056  *cxr 11188  cle 11190  cmin 11385  -cneg 11386  (,)cioo 13264  [,]cicc 13267  ccj 14981  abscabs 15119  TopOpenctopn 17303  topGenctg 17319  fldccnfld 20796  intcnt 22368   Cn ccn 22575   ×t ctx 22911  cnccncf 24239  volcvol 24827  MblFncmbf 24978  2citg2 24980  𝐿1cibl 24981  citg 24982   D cdv 25227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-symdif 4202  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-disj 5071  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-omul 8417  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-acn 9878  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-rlim 15371  df-sum 15571  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-rest 17304  df-topn 17305  df-0g 17323  df-gsum 17324  df-topgen 17325  df-pt 17326  df-prds 17329  df-xrs 17384  df-qtop 17389  df-imas 17390  df-xps 17392  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-mulg 18873  df-cntz 19097  df-cmn 19564  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-fbas 20793  df-fg 20794  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cld 22370  df-ntr 22371  df-cls 22372  df-nei 22449  df-lp 22487  df-perf 22488  df-cn 22578  df-cnp 22579  df-haus 22666  df-cmp 22738  df-tx 22913  df-hmeo 23106  df-fil 23197  df-fm 23289  df-flim 23290  df-flf 23291  df-xms 23673  df-ms 23674  df-tms 23675  df-cncf 24241  df-ovol 24828  df-vol 24829  df-mbf 24983  df-itg1 24984  df-itg2 24985  df-ibl 24986  df-itg 24987  df-0p 25034  df-limc 25230  df-dv 25231
This theorem is referenced by:  areacirc  36171
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