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Theorem ftc2nc 33126
Description: Choice-free proof of ftc2 23711. (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 10033 . . . . . 6 (𝜑𝐴 ∈ ℝ*)
3 ftc2nc.b . . . . . . 7 (𝜑𝐵 ∈ ℝ)
43rexrd 10033 . . . . . 6 (𝜑𝐵 ∈ ℝ*)
5 ftc2nc.le . . . . . 6 (𝜑𝐴𝐵)
6 ubicc2 12231 . . . . . 6 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
72, 4, 5, 6syl3anc 1323 . . . . 5 (𝜑𝐵 ∈ (𝐴[,]𝐵))
8 fvex 6158 . . . . . 6 ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))‘𝐴) ∈ V
98fvconst2 6423 . . . . 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 2621 . . . . . . . 8 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
1211subcn 22577 . . . . . . . . 9 − ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld))
1312a1i 11 . . . . . . . 8 (𝜑 → − ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld)))
14 eqid 2621 . . . . . . . . 9 (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡) = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡)
15 ssid 3603 . . . . . . . . . 10 (𝐴(,)𝐵) ⊆ (𝐴(,)𝐵)
1615a1i 11 . . . . . . . . 9 (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐵))
17 ioossre 12177 . . . . . . . . . 10 (𝐴(,)𝐵) ⊆ ℝ
1817a1i 11 . . . . . . . . 9 (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
19 ftc2nc.i . . . . . . . . 9 (𝜑 → (ℝ D 𝐹) ∈ 𝐿1)
20 ftc2nc.c . . . . . . . . . 10 (𝜑 → (ℝ D 𝐹) ∈ ((𝐴(,)𝐵)–cn→ℂ))
21 cncff 22604 . . . . . . . . . 10 ((ℝ D 𝐹) ∈ ((𝐴(,)𝐵)–cn→ℂ) → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ)
2220, 21syl 17 . . . . . . . . 9 (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ)
23 ioof 12213 . . . . . . . . . . . . 13 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
24 ffun 6005 . . . . . . . . . . . . 13 ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → Fun (,))
2523, 24ax-mp 5 . . . . . . . . . . . 12 Fun (,)
26 fvelima 6205 . . . . . . . . . . . 12 ((Fun (,) ∧ 𝑠 ∈ ((,) “ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) → ∃𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))((,)‘𝑥) = 𝑠)
2725, 26mpan 705 . . . . . . . . . . 11 (𝑠 ∈ ((,) “ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ∃𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))((,)‘𝑥) = 𝑠)
28 1st2nd2 7150 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
2928fveq2d 6152 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → ((,)‘𝑥) = ((,)‘⟨(1st𝑥), (2nd𝑥)⟩))
30 df-ov 6607 . . . . . . . . . . . . . . . 16 ((1st𝑥)(,)(2nd𝑥)) = ((,)‘⟨(1st𝑥), (2nd𝑥)⟩)
3129, 30syl6eqr 2673 . . . . . . . . . . . . . . 15 (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → ((,)‘𝑥) = ((1st𝑥)(,)(2nd𝑥)))
3231eqeq1d 2623 . . . . . . . . . . . . . 14 (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → (((,)‘𝑥) = 𝑠 ↔ ((1st𝑥)(,)(2nd𝑥)) = 𝑠))
3332adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (((,)‘𝑥) = 𝑠 ↔ ((1st𝑥)(,)(2nd𝑥)) = 𝑠))
342, 4jca 554 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝐴 ∈ ℝ*𝐵 ∈ ℝ*))
3534adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝐴 ∈ ℝ*𝐵 ∈ ℝ*))
36 xp1st 7143 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → (1st𝑥) ∈ (𝐴[,]𝐵))
37 elicc1 12161 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((1st𝑥) ∈ (𝐴[,]𝐵) ↔ ((1st𝑥) ∈ ℝ*𝐴 ≤ (1st𝑥) ∧ (1st𝑥) ≤ 𝐵)))
382, 4, 37syl2anc 692 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((1st𝑥) ∈ (𝐴[,]𝐵) ↔ ((1st𝑥) ∈ ℝ*𝐴 ≤ (1st𝑥) ∧ (1st𝑥) ≤ 𝐵)))
3938biimpa 501 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (1st𝑥) ∈ (𝐴[,]𝐵)) → ((1st𝑥) ∈ ℝ*𝐴 ≤ (1st𝑥) ∧ (1st𝑥) ≤ 𝐵))
4039simp2d 1072 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (1st𝑥) ∈ (𝐴[,]𝐵)) → 𝐴 ≤ (1st𝑥))
4136, 40sylan2 491 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → 𝐴 ≤ (1st𝑥))
42 xp2nd 7144 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → (2nd𝑥) ∈ (𝐴[,]𝐵))
43 iccleub 12171 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ* ∧ (2nd𝑥) ∈ (𝐴[,]𝐵)) → (2nd𝑥) ≤ 𝐵)
44433expa 1262 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (2nd𝑥) ∈ (𝐴[,]𝐵)) → (2nd𝑥) ≤ 𝐵)
4534, 42, 44syl2an 494 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (2nd𝑥) ≤ 𝐵)
46 ioossioo 12207 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐴 ≤ (1st𝑥) ∧ (2nd𝑥) ≤ 𝐵)) → ((1st𝑥)(,)(2nd𝑥)) ⊆ (𝐴(,)𝐵))
4735, 41, 45, 46syl12anc 1321 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((1st𝑥)(,)(2nd𝑥)) ⊆ (𝐴(,)𝐵))
4847sselda 3583 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st𝑥)(,)(2nd𝑥))) → 𝑡 ∈ (𝐴(,)𝐵))
4922ffvelrnda 6315 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ ℂ)
5049adantlr 750 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ ℂ)
5148, 50syldan 487 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st𝑥)(,)(2nd𝑥))) → ((ℝ D 𝐹)‘𝑡) ∈ ℂ)
52 ioombl 23240 . . . . . . . . . . . . . . . . . 18 ((1st𝑥)(,)(2nd𝑥)) ∈ dom vol
5352a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((1st𝑥)(,)(2nd𝑥)) ∈ dom vol)
54 fvex 6158 . . . . . . . . . . . . . . . . . 18 ((ℝ D 𝐹)‘𝑡) ∈ V
5554a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ V)
5622feqmptd 6206 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (ℝ D 𝐹) = (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)))
5756, 19eqeltrrd 2699 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿1)
5857adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿1)
5947, 53, 55, 58iblss 23477 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿1)
60 ax-resscn 9937 . . . . . . . . . . . . . . . . . . . . 21 ℝ ⊆ ℂ
61 ssid 3603 . . . . . . . . . . . . . . . . . . . . 21 ℂ ⊆ ℂ
62 cncfss 22610 . . . . . . . . . . . . . . . . . . . . 21 ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℂ–cn→ℝ) ⊆ (ℂ–cn→ℂ))
6360, 61, 62mp2an 707 . . . . . . . . . . . . . . . . . . . 20 (ℂ–cn→ℝ) ⊆ (ℂ–cn→ℂ)
64 abscncf 22612 . . . . . . . . . . . . . . . . . . . 20 abs ∈ (ℂ–cn→ℝ)
6563, 64sselii 3580 . . . . . . . . . . . . . . . . . . 19 abs ∈ (ℂ–cn→ℂ)
6665a1i 11 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → abs ∈ (ℂ–cn→ℂ))
6756reseq1d 5355 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((ℝ D 𝐹) ↾ ((1st𝑥)(,)(2nd𝑥))) = ((𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ↾ ((1st𝑥)(,)(2nd𝑥))))
6867adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((ℝ D 𝐹) ↾ ((1st𝑥)(,)(2nd𝑥))) = ((𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ↾ ((1st𝑥)(,)(2nd𝑥))))
6947resmptd 5411 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ↾ ((1st𝑥)(,)(2nd𝑥))) = (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ ((ℝ D 𝐹)‘𝑡)))
7068, 69eqtrd 2655 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((ℝ D 𝐹) ↾ ((1st𝑥)(,)(2nd𝑥))) = (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ ((ℝ D 𝐹)‘𝑡)))
7120adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (ℝ D 𝐹) ∈ ((𝐴(,)𝐵)–cn→ℂ))
72 rescncf 22608 . . . . . . . . . . . . . . . . . . . 20 (((1st𝑥)(,)(2nd𝑥)) ⊆ (𝐴(,)𝐵) → ((ℝ D 𝐹) ∈ ((𝐴(,)𝐵)–cn→ℂ) → ((ℝ D 𝐹) ↾ ((1st𝑥)(,)(2nd𝑥))) ∈ (((1st𝑥)(,)(2nd𝑥))–cn→ℂ)))
7347, 71, 72sylc 65 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((ℝ D 𝐹) ↾ ((1st𝑥)(,)(2nd𝑥))) ∈ (((1st𝑥)(,)(2nd𝑥))–cn→ℂ))
7470, 73eqeltrrd 2699 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ (((1st𝑥)(,)(2nd𝑥))–cn→ℂ))
7566, 74cncfmpt1f 22624 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ (abs‘((ℝ D 𝐹)‘𝑡))) ∈ (((1st𝑥)(,)(2nd𝑥))–cn→ℂ))
76 cnmbf 23332 . . . . . . . . . . . . . . . . 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)
7752, 75, 76sylancr 694 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ (abs‘((ℝ D 𝐹)‘𝑡))) ∈ MblFn)
7851, 59itgcl 23456 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ)
7978cjcld 13870 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (∗‘∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ∈ ℂ)
80 ioossre 12177 . . . . . . . . . . . . . . . . . . . . 21 ((1st𝑥)(,)(2nd𝑥)) ⊆ ℝ
8180, 60sstri 3592 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑥)(,)(2nd𝑥)) ⊆ ℂ
82 cncfmptc 22622 . . . . . . . . . . . . . . . . . . . 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))
8381, 61, 82mp3an23 1413 . . . . . . . . . . . . . . . . . . 19 ((∗‘∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ∈ ℂ → (𝑠 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ (∗‘∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡)) ∈ (((1st𝑥)(,)(2nd𝑥))–cn→ℂ))
8479, 83syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑠 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ (∗‘∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡)) ∈ (((1st𝑥)(,)(2nd𝑥))–cn→ℂ))
85 nfcv 2761 . . . . . . . . . . . . . . . . . . . 20 𝑠((ℝ D 𝐹)‘𝑡)
86 nfcsb1v 3530 . . . . . . . . . . . . . . . . . . . 20 𝑡𝑠 / 𝑡((ℝ D 𝐹)‘𝑡)
87 csbeq1a 3523 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = 𝑠 → ((ℝ D 𝐹)‘𝑡) = 𝑠 / 𝑡((ℝ D 𝐹)‘𝑡))
8885, 86, 87cbvmpt 4709 . . . . . . . . . . . . . . . . . . 19 (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ ((ℝ D 𝐹)‘𝑡)) = (𝑠 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ 𝑠 / 𝑡((ℝ D 𝐹)‘𝑡))
8988, 74syl5eqelr 2703 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑠 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ 𝑠 / 𝑡((ℝ D 𝐹)‘𝑡)) ∈ (((1st𝑥)(,)(2nd𝑥))–cn→ℂ))
9084, 89mulcncf 23123 . . . . . . . . . . . . . . . . 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))
91 cnmbf 23332 . . . . . . . . . . . . . . . . 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)
9252, 90, 91sylancr 694 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑠 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ ((∗‘∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) · 𝑠 / 𝑡((ℝ D 𝐹)‘𝑡))) ∈ MblFn)
9351, 59, 77, 92itgabsnc 33111 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (abs‘∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ≤ ∫((1st𝑥)(,)(2nd𝑥))(abs‘((ℝ D 𝐹)‘𝑡)) d𝑡)
9451abscld 14109 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st𝑥)(,)(2nd𝑥))) → (abs‘((ℝ D 𝐹)‘𝑡)) ∈ ℝ)
9554a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st𝑥)(,)(2nd𝑥))) → ((ℝ D 𝐹)‘𝑡) ∈ V)
9695, 59, 77iblabsnc 33106 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↦ (abs‘((ℝ D 𝐹)‘𝑡))) ∈ 𝐿1)
9751absge0d 14117 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st𝑥)(,)(2nd𝑥))) → 0 ≤ (abs‘((ℝ D 𝐹)‘𝑡)))
9894, 96, 97itgposval 23468 . . . . . . . . . . . . . . 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))))
9993, 98breqtrd 4639 . . . . . . . . . . . . . 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))))
100 itgeq1 23445 . . . . . . . . . . . . . . . 16 (((1st𝑥)(,)(2nd𝑥)) = 𝑠 → ∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡 = ∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡)
101100fveq2d 6152 . . . . . . . . . . . . . . 15 (((1st𝑥)(,)(2nd𝑥)) = 𝑠 → (abs‘∫((1st𝑥)(,)(2nd𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) = (abs‘∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡))
102 eleq2 2687 . . . . . . . . . . . . . . . . . 18 (((1st𝑥)(,)(2nd𝑥)) = 𝑠 → (𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)) ↔ 𝑡𝑠))
103102ifbid 4080 . . . . . . . . . . . . . . . . 17 (((1st𝑥)(,)(2nd𝑥)) = 𝑠 → if(𝑡 ∈ ((1st𝑥)(,)(2nd𝑥)), (abs‘((ℝ D 𝐹)‘𝑡)), 0) = if(𝑡𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0))
104103mpteq2dv 4705 . . . . . . . . . . . . . . . 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)))
105104fveq2d 6152 . . . . . . . . . . . . . . 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))))
106101, 105breq12d 4626 . . . . . . . . . . . . . 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)))))
10799, 106syl5ibcom 235 . . . . . . . . . . . . 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)))))
10833, 107sylbid 230 . . . . . . . . . . . 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)))))
109108rexlimdva 3024 . . . . . . . . . . 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)))))
11027, 109syl5 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)))))
111110ralrimiv 2959 . . . . . . . . 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))))
11214, 1, 3, 5, 16, 18, 19, 22, 111ftc1anc 33125 . . . . . . . 8 (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡) ∈ ((𝐴[,]𝐵)–cn→ℂ))
113 ftc2nc.f . . . . . . . . . . 11 (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))
114 cncff 22604 . . . . . . . . . . 11 (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
115113, 114syl 17 . . . . . . . . . 10 (𝜑𝐹:(𝐴[,]𝐵)⟶ℂ)
116115feqmptd 6206 . . . . . . . . 9 (𝜑𝐹 = (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹𝑥)))
117116, 113eqeltrrd 2699 . . . . . . . 8 (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
11811, 13, 112, 117cncfmpt2f 22625 . . . . . . 7 (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥))) ∈ ((𝐴[,]𝐵)–cn→ℂ))
11960a1i 11 . . . . . . . . . 10 (𝜑 → ℝ ⊆ ℂ)
120 iccssre 12197 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
1211, 3, 120syl2anc 692 . . . . . . . . . 10 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
12254a1i 11 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ 𝑡 ∈ (𝐴(,)𝑥)) → ((ℝ D 𝐹)‘𝑡) ∈ V)
1233adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ)
124123rexrd 10033 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ*)
125 elicc2 12180 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴𝑥𝑥𝐵)))
1261, 3, 125syl2anc 692 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴𝑥𝑥𝐵)))
127126biimpa 501 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝐴𝑥𝑥𝐵))
128127simp3d 1073 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → 𝑥𝐵)
129 iooss2 12153 . . . . . . . . . . . . . 14 ((𝐵 ∈ ℝ*𝑥𝐵) → (𝐴(,)𝑥) ⊆ (𝐴(,)𝐵))
130124, 128, 129syl2anc 692 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐴(,)𝑥) ⊆ (𝐴(,)𝐵))
131 ioombl 23240 . . . . . . . . . . . . . 14 (𝐴(,)𝑥) ∈ dom vol
132131a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐴(,)𝑥) ∈ dom vol)
13354a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝐴[,]𝐵)) ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ V)
13457adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿1)
135130, 132, 133, 134iblss 23477 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ (𝐴(,)𝑥) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ 𝐿1)
136122, 135itgcl 23456 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ)
137115ffvelrnda 6315 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (𝐹𝑥) ∈ ℂ)
138136, 137subcld 10336 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴[,]𝐵)) → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)) ∈ ℂ)
13911tgioo2 22514 . . . . . . . . . 10 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
140 iccntr 22532 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
1411, 3, 140syl2anc 692 . . . . . . . . . 10 (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
142119, 121, 138, 139, 11, 141dvmptntr 23640 . . . . . . . . 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)))))
143 reelprrecn 9972 . . . . . . . . . . 11 ℝ ∈ {ℝ, ℂ}
144143a1i 11 . . . . . . . . . 10 (𝜑 → ℝ ∈ {ℝ, ℂ})
145 ioossicc 12201 . . . . . . . . . . . 12 (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
146145sseli 3579 . . . . . . . . . . 11 (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ (𝐴[,]𝐵))
147146, 136sylan2 491 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ)
14822ffvelrnda 6315 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
14914, 1, 3, 5, 20, 19ftc1cnnc 33116 . . . . . . . . . . 11 (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡)) = (ℝ D 𝐹))
150119, 121, 136, 139, 11, 141dvmptntr 23640 . . . . . . . . . . 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)))
15122feqmptd 6206 . . . . . . . . . . 11 (𝜑 → (ℝ D 𝐹) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥)))
152149, 150, 1513eqtr3d 2663 . . . . . . . . . 10 (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥)))
153146, 137sylan2 491 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → (𝐹𝑥) ∈ ℂ)
154116oveq2d 6620 . . . . . . . . . . 11 (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹𝑥))))
155119, 121, 137, 139, 11, 141dvmptntr 23640 . . . . . . . . . . 11 (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹𝑥))) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹𝑥))))
156154, 151, 1553eqtr3rd 2664 . . . . . . . . . 10 (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥)))
157144, 147, 148, 152, 153, 148, 156dvmptsub 23636 . . . . . . . . 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))))
158148subidd 10324 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐹)‘𝑥) − ((ℝ D 𝐹)‘𝑥)) = 0)
159158mpteq2dva 4704 . . . . . . . . 9 (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑥) − ((ℝ D 𝐹)‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0))
160142, 157, 1593eqtrd 2659 . . . . . . . 8 (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0))
161 fconstmpt 5123 . . . . . . . 8 ((𝐴(,)𝐵) × {0}) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0)
162160, 161syl6eqr 2673 . . . . . . 7 (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))) = ((𝐴(,)𝐵) × {0}))
1631, 3, 118, 162dveq0 23667 . . . . . 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)}))
164163fveq1d 6150 . . . . 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))
165 oveq2 6612 . . . . . . . . 9 (𝑥 = 𝐵 → (𝐴(,)𝑥) = (𝐴(,)𝐵))
166 itgeq1 23445 . . . . . . . . 9 ((𝐴(,)𝑥) = (𝐴(,)𝐵) → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡)
167165, 166syl 17 . . . . . . . 8 (𝑥 = 𝐵 → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡)
168 fveq2 6148 . . . . . . . 8 (𝑥 = 𝐵 → (𝐹𝑥) = (𝐹𝐵))
169167, 168oveq12d 6622 . . . . . . 7 (𝑥 = 𝐵 → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)) = (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝐵)))
170 eqid 2621 . . . . . . 7 (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥))) = (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))
171 ovex 6632 . . . . . . 7 (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝐵)) ∈ V
172169, 170, 171fvmpt 6239 . . . . . 6 (𝐵 ∈ (𝐴[,]𝐵) → ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))‘𝐵) = (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝐵)))
1737, 172syl 17 . . . . 5 (𝜑 → ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))‘𝐵) = (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝐵)))
174164, 173eqtr3d 2657 . . . 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)))
175 lbicc2 12230 . . . . . 6 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
1762, 4, 5, 175syl3anc 1323 . . . . 5 (𝜑𝐴 ∈ (𝐴[,]𝐵))
177 oveq2 6612 . . . . . . . . . . 11 (𝑥 = 𝐴 → (𝐴(,)𝑥) = (𝐴(,)𝐴))
178 iooid 12145 . . . . . . . . . . 11 (𝐴(,)𝐴) = ∅
179177, 178syl6eq 2671 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝐴(,)𝑥) = ∅)
180 itgeq1 23445 . . . . . . . . . 10 ((𝐴(,)𝑥) = ∅ → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = ∫∅((ℝ D 𝐹)‘𝑡) d𝑡)
181179, 180syl 17 . . . . . . . . 9 (𝑥 = 𝐴 → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = ∫∅((ℝ D 𝐹)‘𝑡) d𝑡)
182 itg0 23452 . . . . . . . . 9 ∫∅((ℝ D 𝐹)‘𝑡) d𝑡 = 0
183181, 182syl6eq 2671 . . . . . . . 8 (𝑥 = 𝐴 → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = 0)
184 fveq2 6148 . . . . . . . 8 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
185183, 184oveq12d 6622 . . . . . . 7 (𝑥 = 𝐴 → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)) = (0 − (𝐹𝐴)))
186 df-neg 10213 . . . . . . 7 -(𝐹𝐴) = (0 − (𝐹𝐴))
187185, 186syl6eqr 2673 . . . . . 6 (𝑥 = 𝐴 → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)) = -(𝐹𝐴))
188 negex 10223 . . . . . 6 -(𝐹𝐴) ∈ V
189187, 170, 188fvmpt 6239 . . . . 5 (𝐴 ∈ (𝐴[,]𝐵) → ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))‘𝐴) = -(𝐹𝐴))
190176, 189syl 17 . . . 4 (𝜑 → ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝑥)))‘𝐴) = -(𝐹𝐴))
19110, 174, 1903eqtr3d 2663 . . 3 (𝜑 → (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝐵)) = -(𝐹𝐴))
192191oveq2d 6620 . 2 (𝜑 → ((𝐹𝐵) + (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝐵))) = ((𝐹𝐵) + -(𝐹𝐴)))
193115, 7ffvelrnd 6316 . . 3 (𝜑 → (𝐹𝐵) ∈ ℂ)
19454a1i 11 . . . 4 ((𝜑𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ V)
195194, 57itgcl 23456 . . 3 (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ)
196193, 195pncan3d 10339 . 2 (𝜑 → ((𝐹𝐵) + (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹𝐵))) = ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡)
197115, 176ffvelrnd 6316 . . 3 (𝜑 → (𝐹𝐴) ∈ ℂ)
198193, 197negsubd 10342 . 2 (𝜑 → ((𝐹𝐵) + -(𝐹𝐴)) = ((𝐹𝐵) − (𝐹𝐴)))
199192, 196, 1983eqtr3d 2663 1 (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹𝐵) − (𝐹𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wrex 2908  Vcvv 3186  csb 3514  wss 3555  c0 3891  ifcif 4058  𝒫 cpw 4130  {csn 4148  {cpr 4150  cop 4154   class class class wbr 4613  cmpt 4673   × cxp 5072  dom cdm 5074  ran crn 5075  cres 5076  cima 5077  Fun wfun 5841  wf 5843  cfv 5847  (class class class)co 6604  1st c1st 7111  2nd c2nd 7112  cc 9878  cr 9879  0cc0 9880   + caddc 9883   · cmul 9885  *cxr 10017  cle 10019  cmin 10210  -cneg 10211  (,)cioo 12117  [,]cicc 12120  ccj 13770  abscabs 13908  TopOpenctopn 16003  topGenctg 16019  fldccnfld 19665  intcnt 20731   Cn ccn 20938   ×t ctx 21273  cnccncf 22587  volcvol 23139  MblFncmbf 23289  2citg2 23291  𝐿1cibl 23292  citg 23293   D cdv 23533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958  ax-addf 9959  ax-mulf 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-disj 4584  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-ofr 6851  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-omul 7510  df-er 7687  df-map 7804  df-pm 7805  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-fi 8261  df-sup 8292  df-inf 8293  df-oi 8359  df-card 8709  df-acn 8712  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-q 11733  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-ioo 12121  df-ico 12123  df-icc 12124  df-fz 12269  df-fzo 12407  df-fl 12533  df-mod 12609  df-seq 12742  df-exp 12801  df-hash 13058  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-clim 14153  df-rlim 14154  df-sum 14351  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-starv 15877  df-sca 15878  df-vsca 15879  df-ip 15880  df-tset 15881  df-ple 15882  df-ds 15885  df-unif 15886  df-hom 15887  df-cco 15888  df-rest 16004  df-topn 16005  df-0g 16023  df-gsum 16024  df-topgen 16025  df-pt 16026  df-prds 16029  df-xrs 16083  df-qtop 16088  df-imas 16089  df-xps 16091  df-mre 16167  df-mrc 16168  df-acs 16170  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-submnd 17257  df-mulg 17462  df-cntz 17671  df-cmn 18116  df-psmet 19657  df-xmet 19658  df-met 19659  df-bl 19660  df-mopn 19661  df-fbas 19662  df-fg 19663  df-cnfld 19666  df-top 20621  df-bases 20622  df-topon 20623  df-topsp 20624  df-cld 20733  df-ntr 20734  df-cls 20735  df-nei 20812  df-lp 20850  df-perf 20851  df-cn 20941  df-cnp 20942  df-haus 21029  df-cmp 21100  df-tx 21275  df-hmeo 21468  df-fil 21560  df-fm 21652  df-flim 21653  df-flf 21654  df-xms 22035  df-ms 22036  df-tms 22037  df-cncf 22589  df-ovol 23140  df-vol 23141  df-mbf 23294  df-itg1 23295  df-itg2 23296  df-ibl 23297  df-itg 23298  df-0p 23343  df-limc 23536  df-dv 23537
This theorem is referenced by:  areacirc  33137
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