| Step | Hyp | Ref
| Expression |
| 1 | | ftc2nc.a |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ ℝ) |
| 2 | 1 | rexrd 11311 |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
| 3 | | ftc2nc.b |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ ℝ) |
| 4 | 3 | rexrd 11311 |
. . . . . 6
⊢ (𝜑 → 𝐵 ∈
ℝ*) |
| 5 | | ftc2nc.le |
. . . . . 6
⊢ (𝜑 → 𝐴 ≤ 𝐵) |
| 6 | | ubicc2 13505 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ 𝐴
≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵)) |
| 7 | 2, 4, 5, 6 | syl3anc 1373 |
. . . . 5
⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐵)) |
| 8 | | fvex 6919 |
. . . . . 6
⊢ ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴) ∈ V |
| 9 | 8 | fvconst2 7224 |
. . . . 5
⊢ (𝐵 ∈ (𝐴[,]𝐵) → (((𝐴[,]𝐵) × {((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴)})‘𝐵) = ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴)) |
| 10 | 7, 9 | syl 17 |
. . . 4
⊢ (𝜑 → (((𝐴[,]𝐵) × {((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴)})‘𝐵) = ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴)) |
| 11 | | eqid 2737 |
. . . . . . . 8
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) |
| 12 | 11 | subcn 24888 |
. . . . . . . . 9
⊢ −
∈ (((TopOpen‘ℂfld) ×t
(TopOpen‘ℂfld)) Cn
(TopOpen‘ℂfld)) |
| 13 | 12 | a1i 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 |
| 17 | 16 | a1i 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) |
| 21 | 19, 20 | syl 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 (,)) |
| 24 | 22, 23 | ax-mp 5 |
. . . . . . . . . . . 12
⊢ Fun
(,) |
| 25 | | fvelima 6974 |
. . . . . . . . . . . 12
⊢ ((Fun (,)
∧ 𝑠 ∈ ((,) “
((𝐴[,]𝐵) × (𝐴[,]𝐵)))) → ∃𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))((,)‘𝑥) = 𝑠) |
| 26 | 24, 25 | mpan 690 |
. . . . . . . . . . 11
⊢ (𝑠 ∈ ((,) “ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ∃𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))((,)‘𝑥) = 𝑠) |
| 27 | | 1st2nd2 8053 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → 𝑥 = 〈(1st ‘𝑥), (2nd ‘𝑥)〉) |
| 28 | 27 | fveq2d 6910 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → ((,)‘𝑥) = ((,)‘〈(1st
‘𝑥), (2nd
‘𝑥)〉)) |
| 29 | | df-ov 7434 |
. . . . . . . . . . . . . . . 16
⊢
((1st ‘𝑥)(,)(2nd ‘𝑥)) = ((,)‘〈(1st
‘𝑥), (2nd
‘𝑥)〉) |
| 30 | 28, 29 | eqtr4di 2795 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → ((,)‘𝑥) = ((1st ‘𝑥)(,)(2nd ‘𝑥))) |
| 31 | 30 | eqeq1d 2739 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → (((,)‘𝑥) = 𝑠 ↔ ((1st ‘𝑥)(,)(2nd ‘𝑥)) = 𝑠)) |
| 32 | 31 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (((,)‘𝑥) = 𝑠 ↔ ((1st ‘𝑥)(,)(2nd ‘𝑥)) = 𝑠)) |
| 33 | 2, 4 | jca 511 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 𝐵 ∈
ℝ*)) |
| 34 | 33 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈
ℝ*)) |
| 35 | | xp1st 8046 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → (1st ‘𝑥) ∈ (𝐴[,]𝐵)) |
| 36 | | elicc1 13431 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → ((1st ‘𝑥) ∈ (𝐴[,]𝐵) ↔ ((1st ‘𝑥) ∈ ℝ*
∧ 𝐴 ≤
(1st ‘𝑥)
∧ (1st ‘𝑥) ≤ 𝐵))) |
| 37 | 2, 4, 36 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((1st
‘𝑥) ∈ (𝐴[,]𝐵) ↔ ((1st ‘𝑥) ∈ ℝ*
∧ 𝐴 ≤
(1st ‘𝑥)
∧ (1st ‘𝑥) ≤ 𝐵))) |
| 38 | 37 | biimpa 476 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ (1st
‘𝑥) ∈ (𝐴[,]𝐵)) → ((1st ‘𝑥) ∈ ℝ*
∧ 𝐴 ≤
(1st ‘𝑥)
∧ (1st ‘𝑥) ≤ 𝐵)) |
| 39 | 38 | simp2d 1144 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (1st
‘𝑥) ∈ (𝐴[,]𝐵)) → 𝐴 ≤ (1st ‘𝑥)) |
| 40 | 35, 39 | sylan2 593 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → 𝐴 ≤ (1st ‘𝑥)) |
| 41 | | xp2nd 8047 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵)) → (2nd ‘𝑥) ∈ (𝐴[,]𝐵)) |
| 42 | | iccleub 13442 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ (2nd ‘𝑥) ∈ (𝐴[,]𝐵)) → (2nd ‘𝑥) ≤ 𝐵) |
| 43 | 42 | 3expa 1119 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) ∧ (2nd ‘𝑥) ∈ (𝐴[,]𝐵)) → (2nd ‘𝑥) ≤ 𝐵) |
| 44 | 33, 41, 43 | syl2an 596 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (2nd ‘𝑥) ≤ 𝐵) |
| 45 | | ioossioo 13481 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) ∧ (𝐴 ≤ (1st ‘𝑥) ∧ (2nd
‘𝑥) ≤ 𝐵)) → ((1st
‘𝑥)(,)(2nd
‘𝑥)) ⊆ (𝐴(,)𝐵)) |
| 46 | 34, 40, 44, 45 | syl12anc 837 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((1st ‘𝑥)(,)(2nd ‘𝑥)) ⊆ (𝐴(,)𝐵)) |
| 47 | 46 | sselda 3983 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥))) → 𝑡 ∈ (𝐴(,)𝐵)) |
| 48 | 21 | ffvelcdmda 7104 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ ℂ) |
| 49 | 48 | adantlr 715 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ ℂ) |
| 50 | 47, 49 | syldan 591 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥))) → ((ℝ D 𝐹)‘𝑡) ∈ ℂ) |
| 51 | | ioombl 25600 |
. . . . . . . . . . . . . . . . . 18
⊢
((1st ‘𝑥)(,)(2nd ‘𝑥)) ∈ dom vol |
| 52 | 51 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((1st ‘𝑥)(,)(2nd ‘𝑥)) ∈ dom
vol) |
| 53 | | fvexd 6921 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ V) |
| 54 | 21 | feqmptd 6977 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (ℝ D 𝐹) = (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡))) |
| 55 | 54, 18 | eqeltrrd 2842 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ∈
𝐿1) |
| 56 | 55 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ∈
𝐿1) |
| 57 | 46, 52, 53, 56 | iblss 25840 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ ((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)) |
| 61 | 58, 59, 60 | mp2an 692 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(ℂ–cn→ℝ)
⊆ (ℂ–cn→ℂ) |
| 62 | | abscncf 24927 |
. . . . . . . . . . . . . . . . . . . 20
⊢ abs
∈ (ℂ–cn→ℝ) |
| 63 | 61, 62 | sselii 3980 |
. . . . . . . . . . . . . . . . . . 19
⊢ abs
∈ (ℂ–cn→ℂ) |
| 64 | 63 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → abs ∈ (ℂ–cn→ℂ)) |
| 65 | 54 | reseq1d 5996 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((ℝ D 𝐹) ↾ ((1st
‘𝑥)(,)(2nd
‘𝑥))) = ((𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ↾ ((1st ‘𝑥)(,)(2nd ‘𝑥)))) |
| 66 | 65 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((ℝ D 𝐹) ↾ ((1st ‘𝑥)(,)(2nd ‘𝑥))) = ((𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ↾ ((1st ‘𝑥)(,)(2nd ‘𝑥)))) |
| 67 | 46 | resmptd 6058 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ↾ ((1st ‘𝑥)(,)(2nd ‘𝑥))) = (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ ((ℝ D 𝐹)‘𝑡))) |
| 68 | 66, 67 | eqtrd 2777 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((ℝ D 𝐹) ↾ ((1st ‘𝑥)(,)(2nd ‘𝑥))) = (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ ((ℝ D 𝐹)‘𝑡))) |
| 69 | 19 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (ℝ D 𝐹) ∈ ((𝐴(,)𝐵)–cn→ℂ)) |
| 70 | | rescncf 24923 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((1st ‘𝑥)(,)(2nd ‘𝑥)) ⊆ (𝐴(,)𝐵) → ((ℝ D 𝐹) ∈ ((𝐴(,)𝐵)–cn→ℂ) → ((ℝ D 𝐹) ↾ ((1st ‘𝑥)(,)(2nd ‘𝑥))) ∈ (((1st
‘𝑥)(,)(2nd
‘𝑥))–cn→ℂ))) |
| 71 | 46, 69, 70 | sylc 65 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ((ℝ D 𝐹) ↾ ((1st ‘𝑥)(,)(2nd ‘𝑥))) ∈ (((1st
‘𝑥)(,)(2nd
‘𝑥))–cn→ℂ)) |
| 72 | 68, 71 | eqeltrrd 2842 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ ((ℝ D 𝐹)‘𝑡)) ∈ (((1st ‘𝑥)(,)(2nd ‘𝑥))–cn→ℂ)) |
| 73 | 64, 72 | cncfmpt1f 24940 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ (abs‘((ℝ
D 𝐹)‘𝑡))) ∈ (((1st
‘𝑥)(,)(2nd
‘𝑥))–cn→ℂ)) |
| 74 | | cnmbf 25694 |
. . . . . . . . . . . . . . . . 17
⊢
((((1st ‘𝑥)(,)(2nd ‘𝑥)) ∈ dom vol ∧ (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ (abs‘((ℝ
D 𝐹)‘𝑡))) ∈ (((1st
‘𝑥)(,)(2nd
‘𝑥))–cn→ℂ)) → (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ (abs‘((ℝ
D 𝐹)‘𝑡))) ∈
MblFn) |
| 75 | 51, 73, 74 | sylancr 587 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ (abs‘((ℝ
D 𝐹)‘𝑡))) ∈
MblFn) |
| 76 | 50, 57 | itgcl 25819 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ∫((1st
‘𝑥)(,)(2nd
‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ) |
| 77 | 76 | cjcld 15235 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) →
(∗‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ∈ ℂ) |
| 78 | | ioossre 13448 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((1st ‘𝑥)(,)(2nd ‘𝑥)) ⊆ ℝ |
| 79 | 78, 58 | sstri 3993 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((1st ‘𝑥)(,)(2nd ‘𝑥)) ⊆ ℂ |
| 80 | | cncfmptc 24938 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((∗‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ∈ ℂ ∧ ((1st
‘𝑥)(,)(2nd
‘𝑥)) ⊆ ℂ
∧ ℂ ⊆ ℂ) → (𝑠 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦
(∗‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡)) ∈ (((1st ‘𝑥)(,)(2nd ‘𝑥))–cn→ℂ)) |
| 81 | 79, 59, 80 | mp3an23 1455 |
. . . . . . . . . . . . . . . . . . 19
⊢
((∗‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ∈ ℂ → (𝑠 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦
(∗‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡)) ∈ (((1st ‘𝑥)(,)(2nd ‘𝑥))–cn→ℂ)) |
| 82 | 77, 81 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑠 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦
(∗‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡)) ∈ (((1st ‘𝑥)(,)(2nd ‘𝑥))–cn→ℂ)) |
| 83 | | nfcv 2905 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑠((ℝ D 𝐹)‘𝑡) |
| 84 | | nfcsb1v 3923 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑡⦋𝑠 / 𝑡⦌((ℝ D 𝐹)‘𝑡) |
| 85 | | csbeq1a 3913 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑠 → ((ℝ D 𝐹)‘𝑡) = ⦋𝑠 / 𝑡⦌((ℝ D 𝐹)‘𝑡)) |
| 86 | 83, 84, 85 | cbvmpt 5253 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 ∈ ((1st
‘𝑥)(,)(2nd
‘𝑥)) ↦
((ℝ D 𝐹)‘𝑡)) = (𝑠 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ ⦋𝑠 / 𝑡⦌((ℝ D 𝐹)‘𝑡)) |
| 87 | 86, 72 | eqeltrrid 2846 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑠 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ ⦋𝑠 / 𝑡⦌((ℝ D 𝐹)‘𝑡)) ∈ (((1st ‘𝑥)(,)(2nd ‘𝑥))–cn→ℂ)) |
| 88 | 82, 87 | mulcncf 25480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑠 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦
((∗‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) · ⦋𝑠 / 𝑡⦌((ℝ D 𝐹)‘𝑡))) ∈ (((1st ‘𝑥)(,)(2nd ‘𝑥))–cn→ℂ)) |
| 89 | | cnmbf 25694 |
. . . . . . . . . . . . . . . . 17
⊢
((((1st ‘𝑥)(,)(2nd ‘𝑥)) ∈ dom vol ∧ (𝑠 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦
((∗‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) · ⦋𝑠 / 𝑡⦌((ℝ D 𝐹)‘𝑡))) ∈ (((1st ‘𝑥)(,)(2nd ‘𝑥))–cn→ℂ)) → (𝑠 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦
((∗‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) · ⦋𝑠 / 𝑡⦌((ℝ D 𝐹)‘𝑡))) ∈ MblFn) |
| 90 | 51, 88, 89 | sylancr 587 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑠 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦
((∗‘∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) · ⦋𝑠 / 𝑡⦌((ℝ D 𝐹)‘𝑡))) ∈ MblFn) |
| 91 | 50, 57, 75, 90 | itgabsnc 37696 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (abs‘∫((1st
‘𝑥)(,)(2nd
‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ≤ ∫((1st ‘𝑥)(,)(2nd ‘𝑥))(abs‘((ℝ D 𝐹)‘𝑡)) d𝑡) |
| 92 | 50 | abscld 15475 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥))) → (abs‘((ℝ
D 𝐹)‘𝑡)) ∈
ℝ) |
| 93 | | fvexd 6921 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥))) → ((ℝ D 𝐹)‘𝑡) ∈ V) |
| 94 | 93, 57, 75 | iblabsnc 37691 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↦ (abs‘((ℝ
D 𝐹)‘𝑡))) ∈
𝐿1) |
| 95 | 50 | absge0d 15483 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) ∧ 𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥))) → 0 ≤
(abs‘((ℝ D 𝐹)‘𝑡))) |
| 96 | 92, 94, 95 | itgposval 25831 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → ∫((1st
‘𝑥)(,)(2nd
‘𝑥))(abs‘((ℝ D 𝐹)‘𝑡)) d𝑡 = (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ((1st
‘𝑥)(,)(2nd
‘𝑥)),
(abs‘((ℝ D 𝐹)‘𝑡)), 0)))) |
| 97 | 91, 96 | breqtrd 5169 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (abs‘∫((1st
‘𝑥)(,)(2nd
‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ((1st
‘𝑥)(,)(2nd
‘𝑥)),
(abs‘((ℝ D 𝐹)‘𝑡)), 0)))) |
| 98 | | itgeq1 25808 |
. . . . . . . . . . . . . . . 16
⊢
(((1st ‘𝑥)(,)(2nd ‘𝑥)) = 𝑠 → ∫((1st ‘𝑥)(,)(2nd ‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡 = ∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡) |
| 99 | 98 | fveq2d 6910 |
. . . . . . . . . . . . . . 15
⊢
(((1st ‘𝑥)(,)(2nd ‘𝑥)) = 𝑠 → (abs‘∫((1st
‘𝑥)(,)(2nd
‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) = (abs‘∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡)) |
| 100 | | eleq2 2830 |
. . . . . . . . . . . . . . . . . 18
⊢
(((1st ‘𝑥)(,)(2nd ‘𝑥)) = 𝑠 → (𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)) ↔ 𝑡 ∈ 𝑠)) |
| 101 | 100 | ifbid 4549 |
. . . . . . . . . . . . . . . . 17
⊢
(((1st ‘𝑥)(,)(2nd ‘𝑥)) = 𝑠 → if(𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)), (abs‘((ℝ D 𝐹)‘𝑡)), 0) = if(𝑡 ∈ 𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0)) |
| 102 | 101 | mpteq2dv 5244 |
. . . . . . . . . . . . . . . 16
⊢
(((1st ‘𝑥)(,)(2nd ‘𝑥)) = 𝑠 → (𝑡 ∈ ℝ ↦ if(𝑡 ∈ ((1st ‘𝑥)(,)(2nd ‘𝑥)), (abs‘((ℝ D 𝐹)‘𝑡)), 0)) = (𝑡 ∈ ℝ ↦ if(𝑡 ∈ 𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0))) |
| 103 | 102 | fveq2d 6910 |
. . . . . . . . . . . . . . 15
⊢
(((1st ‘𝑥)(,)(2nd ‘𝑥)) = 𝑠 → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ((1st
‘𝑥)(,)(2nd
‘𝑥)),
(abs‘((ℝ D 𝐹)‘𝑡)), 0))) = (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ 𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0)))) |
| 104 | 99, 103 | breq12d 5156 |
. . . . . . . . . . . . . 14
⊢
(((1st ‘𝑥)(,)(2nd ‘𝑥)) = 𝑠 → ((abs‘∫((1st
‘𝑥)(,)(2nd
‘𝑥))((ℝ D 𝐹)‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ((1st
‘𝑥)(,)(2nd
‘𝑥)),
(abs‘((ℝ D 𝐹)‘𝑡)), 0))) ↔ (abs‘∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ 𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0))))) |
| 105 | 97, 104 | syl5ibcom 245 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (((1st ‘𝑥)(,)(2nd ‘𝑥)) = 𝑠 → (abs‘∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ 𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0))))) |
| 106 | 32, 105 | sylbid 240 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (((,)‘𝑥) = 𝑠 → (abs‘∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ 𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0))))) |
| 107 | 106 | rexlimdva 3155 |
. . . . . . . . . . 11
⊢ (𝜑 → (∃𝑥 ∈ ((𝐴[,]𝐵) × (𝐴[,]𝐵))((,)‘𝑥) = 𝑠 → (abs‘∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ 𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0))))) |
| 108 | 26, 107 | syl5 34 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑠 ∈ ((,) “ ((𝐴[,]𝐵) × (𝐴[,]𝐵))) → (abs‘∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ 𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0))))) |
| 109 | 108 | ralrimiv 3145 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑠 ∈ ((,) “ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))(abs‘∫𝑠((ℝ D 𝐹)‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ 𝑠, (abs‘((ℝ D 𝐹)‘𝑡)), 0)))) |
| 110 | 14, 1, 3, 5, 15, 17, 18, 21, 109 | ftc1anc 37708 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡) ∈ ((𝐴[,]𝐵)–cn→ℂ)) |
| 111 | | ftc2nc.f |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ)) |
| 112 | | cncff 24919 |
. . . . . . . . . . 11
⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ) → 𝐹:(𝐴[,]𝐵)⟶ℂ) |
| 113 | 111, 112 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ) |
| 114 | 113 | feqmptd 6977 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥))) |
| 115 | 114, 111 | eqeltrrd 2842 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℂ)) |
| 116 | 11, 13, 110, 115 | cncfmpt2f 24941 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥))) ∈ ((𝐴[,]𝐵)–cn→ℂ)) |
| 117 | 58 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → ℝ ⊆
ℂ) |
| 118 | | iccssre 13469 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) |
| 119 | 1, 3, 118 | syl2anc 584 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
| 120 | | fvexd 6921 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ 𝑡 ∈ (𝐴(,)𝑥)) → ((ℝ D 𝐹)‘𝑡) ∈ V) |
| 121 | 3 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ) |
| 122 | 121 | rexrd 11311 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈
ℝ*) |
| 123 | | elicc2 13452 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵))) |
| 124 | 1, 3, 123 | syl2anc 584 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵))) |
| 125 | 124 | biimpa 476 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)) |
| 126 | 125 | simp3d 1145 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ 𝐵) |
| 127 | | iooss2 13423 |
. . . . . . . . . . . . . 14
⊢ ((𝐵 ∈ ℝ*
∧ 𝑥 ≤ 𝐵) → (𝐴(,)𝑥) ⊆ (𝐴(,)𝐵)) |
| 128 | 122, 126,
127 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐴(,)𝑥) ⊆ (𝐴(,)𝐵)) |
| 129 | | ioombl 25600 |
. . . . . . . . . . . . . 14
⊢ (𝐴(,)𝑥) ∈ dom vol |
| 130 | 129 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐴(,)𝑥) ∈ dom vol) |
| 131 | | fvexd 6921 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ V) |
| 132 | 55 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑡)) ∈
𝐿1) |
| 133 | 128, 130,
131, 132 | iblss 25840 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ (𝐴(,)𝑥) ↦ ((ℝ D 𝐹)‘𝑡)) ∈
𝐿1) |
| 134 | 120, 133 | itgcl 25819 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ) |
| 135 | 113 | ffvelcdmda 7104 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℂ) |
| 136 | 134, 135 | subcld 11620 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)) ∈ 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)) |
| 139 | 1, 3, 138 | syl2anc 584 |
. . . . . . . . . 10
⊢ (𝜑 →
((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵)) |
| 140 | 117, 119,
136, 137, 11, 139 | dvmptntr 26009 |
. . . . . . . . 9
⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥))))) |
| 141 | | reelprrecn 11247 |
. . . . . . . . . . 11
⊢ ℝ
∈ {ℝ, ℂ} |
| 142 | 141 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → ℝ ∈ {ℝ,
ℂ}) |
| 143 | | ioossicc 13473 |
. . . . . . . . . . . 12
⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) |
| 144 | 143 | sseli 3979 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ (𝐴[,]𝐵)) |
| 145 | 144, 134 | sylan2 593 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ) |
| 146 | 21 | ffvelcdmda 7104 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ) |
| 147 | 14, 1, 3, 5, 19, 18 | ftc1cnnc 37699 |
. . . . . . . . . . 11
⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡)) = (ℝ D 𝐹)) |
| 148 | 117, 119,
134, 137, 11, 139 | dvmptntr 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))) |
| 149 | 21 | feqmptd 6977 |
. . . . . . . . . . 11
⊢ (𝜑 → (ℝ D 𝐹) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥))) |
| 150 | 147, 148,
149 | 3eqtr3d 2785 |
. . . . . . . . . 10
⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥))) |
| 151 | 144, 135 | sylan2 593 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑥) ∈ ℂ) |
| 152 | 114 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥)))) |
| 153 | 117, 119,
135, 137, 11, 139 | dvmptntr 26009 |
. . . . . . . . . . 11
⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐹‘𝑥))) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥)))) |
| 154 | 152, 149,
153 | 3eqtr3rd 2786 |
. . . . . . . . . 10
⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥))) |
| 155 | 142, 145,
146, 150, 151, 146, 154 | dvmptsub 26005 |
. . . . . . . . 9
⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑥) − ((ℝ D 𝐹)‘𝑥)))) |
| 156 | 146 | subidd 11608 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐹)‘𝑥) − ((ℝ D 𝐹)‘𝑥)) = 0) |
| 157 | 156 | mpteq2dva 5242 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑥) − ((ℝ D 𝐹)‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0)) |
| 158 | 140, 155,
157 | 3eqtrd 2781 |
. . . . . . . 8
⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0)) |
| 159 | | fconstmpt 5747 |
. . . . . . . 8
⊢ ((𝐴(,)𝐵) × {0}) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 0) |
| 160 | 158, 159 | eqtr4di 2795 |
. . . . . . 7
⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))) = ((𝐴(,)𝐵) × {0})) |
| 161 | 1, 3, 116, 160 | dveq0 26039 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥))) = ((𝐴[,]𝐵) × {((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴)})) |
| 162 | 161 | fveq1d 6908 |
. . . . 5
⊢ (𝜑 → ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐵) = (((𝐴[,]𝐵) × {((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘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) |
| 165 | 163, 164 | syl 17 |
. . . . . . . 8
⊢ (𝑥 = 𝐵 → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡) |
| 166 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵)) |
| 167 | 165, 166 | oveq12d 7449 |
. . . . . . 7
⊢ (𝑥 = 𝐵 → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)) = (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵))) |
| 168 | | eqid 2737 |
. . . . . . 7
⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥))) = (𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥))) |
| 169 | | ovex 7464 |
. . . . . . 7
⊢
(∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵)) ∈ V |
| 170 | 167, 168,
169 | fvmpt 7016 |
. . . . . 6
⊢ (𝐵 ∈ (𝐴[,]𝐵) → ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐵) = (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵))) |
| 171 | 7, 170 | syl 17 |
. . . . 5
⊢ (𝜑 → ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐵) = (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵))) |
| 172 | 162, 171 | eqtr3d 2779 |
. . . 4
⊢ (𝜑 → (((𝐴[,]𝐵) × {((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴)})‘𝐵) = (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵))) |
| 173 | | lbicc2 13504 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ 𝐴
≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) |
| 174 | 2, 4, 5, 173 | syl3anc 1373 |
. . . . 5
⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵)) |
| 175 | | oveq2 7439 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝐴 → (𝐴(,)𝑥) = (𝐴(,)𝐴)) |
| 176 | | iooid 13415 |
. . . . . . . . . . 11
⊢ (𝐴(,)𝐴) = ∅ |
| 177 | 175, 176 | eqtrdi 2793 |
. . . . . . . . . 10
⊢ (𝑥 = 𝐴 → (𝐴(,)𝑥) = ∅) |
| 178 | | itgeq1 25808 |
. . . . . . . . . 10
⊢ ((𝐴(,)𝑥) = ∅ → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = ∫∅((ℝ D 𝐹)‘𝑡) d𝑡) |
| 179 | 177, 178 | syl 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 |
| 181 | 179, 180 | eqtrdi 2793 |
. . . . . . . 8
⊢ (𝑥 = 𝐴 → ∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 = 0) |
| 182 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴)) |
| 183 | 181, 182 | oveq12d 7449 |
. . . . . . 7
⊢ (𝑥 = 𝐴 → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)) = (0 − (𝐹‘𝐴))) |
| 184 | | df-neg 11495 |
. . . . . . 7
⊢ -(𝐹‘𝐴) = (0 − (𝐹‘𝐴)) |
| 185 | 183, 184 | eqtr4di 2795 |
. . . . . 6
⊢ (𝑥 = 𝐴 → (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)) = -(𝐹‘𝐴)) |
| 186 | | negex 11506 |
. . . . . 6
⊢ -(𝐹‘𝐴) ∈ V |
| 187 | 185, 168,
186 | fvmpt 7016 |
. . . . 5
⊢ (𝐴 ∈ (𝐴[,]𝐵) → ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴) = -(𝐹‘𝐴)) |
| 188 | 174, 187 | syl 17 |
. . . 4
⊢ (𝜑 → ((𝑥 ∈ (𝐴[,]𝐵) ↦ (∫(𝐴(,)𝑥)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝑥)))‘𝐴) = -(𝐹‘𝐴)) |
| 189 | 10, 172, 188 | 3eqtr3d 2785 |
. . 3
⊢ (𝜑 → (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵)) = -(𝐹‘𝐴)) |
| 190 | 189 | oveq2d 7447 |
. 2
⊢ (𝜑 → ((𝐹‘𝐵) + (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵))) = ((𝐹‘𝐵) + -(𝐹‘𝐴))) |
| 191 | 113, 7 | ffvelcdmd 7105 |
. . 3
⊢ (𝜑 → (𝐹‘𝐵) ∈ ℂ) |
| 192 | | fvexd 6921 |
. . . 4
⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑡) ∈ V) |
| 193 | 192, 55 | itgcl 25819 |
. . 3
⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 ∈ ℂ) |
| 194 | 191, 193 | pncan3d 11623 |
. 2
⊢ (𝜑 → ((𝐹‘𝐵) + (∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 − (𝐹‘𝐵))) = ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡) |
| 195 | 113, 174 | ffvelcdmd 7105 |
. . 3
⊢ (𝜑 → (𝐹‘𝐴) ∈ ℂ) |
| 196 | 191, 195 | negsubd 11626 |
. 2
⊢ (𝜑 → ((𝐹‘𝐵) + -(𝐹‘𝐴)) = ((𝐹‘𝐵) − (𝐹‘𝐴))) |
| 197 | 190, 194,
196 | 3eqtr3d 2785 |
1
⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴))) |