Metamath Proof Explorer


Theorem ftc2nc

Description: Choice-free proof of ftc2 . (Contributed by Brendan Leahy, 19-Jun-2018)

Ref Expression
Hypotheses 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 ftc2nc ( 𝜑 → ∫ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 = ( ( 𝐹𝐵 ) − ( 𝐹𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 ftc2nc.a ( 𝜑𝐴 ∈ ℝ )
2 ftc2nc.b ( 𝜑𝐵 ∈ ℝ )
3 ftc2nc.le ( 𝜑𝐴𝐵 )
4 ftc2nc.c ( 𝜑 → ( ℝ D 𝐹 ) ∈ ( ( 𝐴 (,) 𝐵 ) –cn→ ℂ ) )
5 ftc2nc.i ( 𝜑 → ( ℝ D 𝐹 ) ∈ 𝐿1 )
6 ftc2nc.f ( 𝜑𝐹 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) )
7 1 rexrd ( 𝜑𝐴 ∈ ℝ* )
8 2 rexrd ( 𝜑𝐵 ∈ ℝ* )
9 ubicc2 ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵 ) → 𝐵 ∈ ( 𝐴 [,] 𝐵 ) )
10 7 8 3 9 syl3anc ( 𝜑𝐵 ∈ ( 𝐴 [,] 𝐵 ) )
11 fvex ( ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹𝑥 ) ) ) ‘ 𝐴 ) ∈ V
12 11 fvconst2 ( 𝐵 ∈ ( 𝐴 [,] 𝐵 ) → ( ( ( 𝐴 [,] 𝐵 ) × { ( ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹𝑥 ) ) ) ‘ 𝐴 ) } ) ‘ 𝐵 ) = ( ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹𝑥 ) ) ) ‘ 𝐴 ) )
13 10 12 syl ( 𝜑 → ( ( ( 𝐴 [,] 𝐵 ) × { ( ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹𝑥 ) ) ) ‘ 𝐴 ) } ) ‘ 𝐵 ) = ( ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹𝑥 ) ) ) ‘ 𝐴 ) )
14 eqid ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld )
15 14 subcn − ∈ ( ( ( TopOpen ‘ ℂfld ) ×t ( TopOpen ‘ ℂfld ) ) Cn ( TopOpen ‘ ℂfld ) )
16 15 a1i ( 𝜑 → − ∈ ( ( ( TopOpen ‘ ℂfld ) ×t ( TopOpen ‘ ℂfld ) ) Cn ( TopOpen ‘ ℂfld ) ) )
17 eqid ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) = ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 )
18 ssidd ( 𝜑 → ( 𝐴 (,) 𝐵 ) ⊆ ( 𝐴 (,) 𝐵 ) )
19 ioossre ( 𝐴 (,) 𝐵 ) ⊆ ℝ
20 19 a1i ( 𝜑 → ( 𝐴 (,) 𝐵 ) ⊆ ℝ )
21 cncff ( ( ℝ D 𝐹 ) ∈ ( ( 𝐴 (,) 𝐵 ) –cn→ ℂ ) → ( ℝ D 𝐹 ) : ( 𝐴 (,) 𝐵 ) ⟶ ℂ )
22 4 21 syl ( 𝜑 → ( ℝ D 𝐹 ) : ( 𝐴 (,) 𝐵 ) ⟶ ℂ )
23 ioof (,) : ( ℝ* × ℝ* ) ⟶ 𝒫 ℝ
24 ffun ( (,) : ( ℝ* × ℝ* ) ⟶ 𝒫 ℝ → Fun (,) )
25 23 24 ax-mp Fun (,)
26 fvelima ( ( Fun (,) ∧ 𝑠 ∈ ( (,) “ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) ) → ∃ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ( (,) ‘ 𝑥 ) = 𝑠 )
27 25 26 mpan ( 𝑠 ∈ ( (,) “ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ∃ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ( (,) ‘ 𝑥 ) = 𝑠 )
28 1st2nd2 ( 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) → 𝑥 = ⟨ ( 1st𝑥 ) , ( 2nd𝑥 ) ⟩ )
29 28 fveq2d ( 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) → ( (,) ‘ 𝑥 ) = ( (,) ‘ ⟨ ( 1st𝑥 ) , ( 2nd𝑥 ) ⟩ ) )
30 df-ov ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) = ( (,) ‘ ⟨ ( 1st𝑥 ) , ( 2nd𝑥 ) ⟩ )
31 29 30 eqtr4di ( 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) → ( (,) ‘ 𝑥 ) = ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) )
32 31 eqeq1d ( 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) → ( ( (,) ‘ 𝑥 ) = 𝑠 ↔ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) = 𝑠 ) )
33 32 adantl ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( ( (,) ‘ 𝑥 ) = 𝑠 ↔ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) = 𝑠 ) )
34 7 8 jca ( 𝜑 → ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) )
35 34 adantr ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) )
36 xp1st ( 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) → ( 1st𝑥 ) ∈ ( 𝐴 [,] 𝐵 ) )
37 elicc1 ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( 1st𝑥 ) ∈ ( 𝐴 [,] 𝐵 ) ↔ ( ( 1st𝑥 ) ∈ ℝ*𝐴 ≤ ( 1st𝑥 ) ∧ ( 1st𝑥 ) ≤ 𝐵 ) ) )
38 7 8 37 syl2anc ( 𝜑 → ( ( 1st𝑥 ) ∈ ( 𝐴 [,] 𝐵 ) ↔ ( ( 1st𝑥 ) ∈ ℝ*𝐴 ≤ ( 1st𝑥 ) ∧ ( 1st𝑥 ) ≤ 𝐵 ) ) )
39 38 biimpa ( ( 𝜑 ∧ ( 1st𝑥 ) ∈ ( 𝐴 [,] 𝐵 ) ) → ( ( 1st𝑥 ) ∈ ℝ*𝐴 ≤ ( 1st𝑥 ) ∧ ( 1st𝑥 ) ≤ 𝐵 ) )
40 39 simp2d ( ( 𝜑 ∧ ( 1st𝑥 ) ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐴 ≤ ( 1st𝑥 ) )
41 36 40 sylan2 ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → 𝐴 ≤ ( 1st𝑥 ) )
42 xp2nd ( 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) → ( 2nd𝑥 ) ∈ ( 𝐴 [,] 𝐵 ) )
43 iccleub ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ∧ ( 2nd𝑥 ) ∈ ( 𝐴 [,] 𝐵 ) ) → ( 2nd𝑥 ) ≤ 𝐵 )
44 43 3expa ( ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) ∧ ( 2nd𝑥 ) ∈ ( 𝐴 [,] 𝐵 ) ) → ( 2nd𝑥 ) ≤ 𝐵 )
45 34 42 44 syl2an ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( 2nd𝑥 ) ≤ 𝐵 )
46 ioossioo ( ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) ∧ ( 𝐴 ≤ ( 1st𝑥 ) ∧ ( 2nd𝑥 ) ≤ 𝐵 ) ) → ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ⊆ ( 𝐴 (,) 𝐵 ) )
47 35 41 45 46 syl12anc ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ⊆ ( 𝐴 (,) 𝐵 ) )
48 47 sselda ( ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑡 ∈ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ) → 𝑡 ∈ ( 𝐴 (,) 𝐵 ) )
49 22 ffvelrnda ( ( 𝜑𝑡 ∈ ( 𝐴 (,) 𝐵 ) ) → ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ∈ ℂ )
50 49 adantlr ( ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑡 ∈ ( 𝐴 (,) 𝐵 ) ) → ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ∈ ℂ )
51 48 50 syldan ( ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑡 ∈ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ) → ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ∈ ℂ )
52 ioombl ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ∈ dom vol
53 52 a1i ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ∈ dom vol )
54 fvexd ( ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑡 ∈ ( 𝐴 (,) 𝐵 ) ) → ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ∈ V )
55 22 feqmptd ( 𝜑 → ( ℝ D 𝐹 ) = ( 𝑡 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) )
56 55 5 eqeltrrd ( 𝜑 → ( 𝑡 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ∈ 𝐿1 )
57 56 adantr ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( 𝑡 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ∈ 𝐿1 )
58 47 53 54 57 iblss ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( 𝑡 ∈ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ↦ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ∈ 𝐿1 )
59 ax-resscn ℝ ⊆ ℂ
60 ssid ℂ ⊆ ℂ
61 cncfss ( ( ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ ) → ( ℂ –cn→ ℝ ) ⊆ ( ℂ –cn→ ℂ ) )
62 59 60 61 mp2an ( ℂ –cn→ ℝ ) ⊆ ( ℂ –cn→ ℂ )
63 abscncf abs ∈ ( ℂ –cn→ ℝ )
64 62 63 sselii abs ∈ ( ℂ –cn→ ℂ )
65 64 a1i ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → abs ∈ ( ℂ –cn→ ℂ ) )
66 55 reseq1d ( 𝜑 → ( ( ℝ D 𝐹 ) ↾ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ) = ( ( 𝑡 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ↾ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ) )
67 66 adantr ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( ( ℝ D 𝐹 ) ↾ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ) = ( ( 𝑡 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ↾ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ) )
68 47 resmptd ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( ( 𝑡 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ↾ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ) = ( 𝑡 ∈ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ↦ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) )
69 67 68 eqtrd ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( ( ℝ D 𝐹 ) ↾ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ) = ( 𝑡 ∈ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ↦ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) )
70 4 adantr ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( ℝ D 𝐹 ) ∈ ( ( 𝐴 (,) 𝐵 ) –cn→ ℂ ) )
71 rescncf ( ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ⊆ ( 𝐴 (,) 𝐵 ) → ( ( ℝ D 𝐹 ) ∈ ( ( 𝐴 (,) 𝐵 ) –cn→ ℂ ) → ( ( ℝ D 𝐹 ) ↾ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ) ∈ ( ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) –cn→ ℂ ) ) )
72 47 70 71 sylc ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( ( ℝ D 𝐹 ) ↾ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ) ∈ ( ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) –cn→ ℂ ) )
73 69 72 eqeltrrd ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( 𝑡 ∈ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ↦ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ∈ ( ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) –cn→ ℂ ) )
74 65 73 cncfmpt1f ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( 𝑡 ∈ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ↦ ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ) ∈ ( ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) –cn→ ℂ ) )
75 cnmbf ( ( ( ( 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 )
76 52 74 75 sylancr ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( 𝑡 ∈ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ↦ ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ) ∈ MblFn )
77 51 58 itgcl ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ∫ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ∈ ℂ )
78 77 cjcld ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( ∗ ‘ ∫ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ∈ ℂ )
79 ioossre ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ⊆ ℝ
80 79 59 sstri ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ⊆ ℂ
81 cncfmptc ( ( ( ∗ ‘ ∫ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ∈ ℂ ∧ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ⊆ ℂ ∧ ℂ ⊆ ℂ ) → ( 𝑠 ∈ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ↦ ( ∗ ‘ ∫ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ) ∈ ( ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) –cn→ ℂ ) )
82 80 60 81 mp3an23 ( ( ∗ ‘ ∫ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ∈ ℂ → ( 𝑠 ∈ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ↦ ( ∗ ‘ ∫ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ) ∈ ( ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) –cn→ ℂ ) )
83 78 82 syl ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( 𝑠 ∈ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ↦ ( ∗ ‘ ∫ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ) ∈ ( ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) –cn→ ℂ ) )
84 nfcv 𝑠 ( ( ℝ D 𝐹 ) ‘ 𝑡 )
85 nfcsb1v 𝑡 𝑠 / 𝑡 ( ( ℝ D 𝐹 ) ‘ 𝑡 )
86 csbeq1a ( 𝑡 = 𝑠 → ( ( ℝ D 𝐹 ) ‘ 𝑡 ) = 𝑠 / 𝑡 ( ( ℝ D 𝐹 ) ‘ 𝑡 ) )
87 84 85 86 cbvmpt ( 𝑡 ∈ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ↦ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) = ( 𝑠 ∈ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ↦ 𝑠 / 𝑡 ( ( ℝ D 𝐹 ) ‘ 𝑡 ) )
88 87 73 eqeltrrid ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( 𝑠 ∈ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ↦ 𝑠 / 𝑡 ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ∈ ( ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) –cn→ ℂ ) )
89 83 88 mulcncf ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( 𝑠 ∈ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ↦ ( ( ∗ ‘ ∫ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) · 𝑠 / 𝑡 ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ) ∈ ( ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) –cn→ ℂ ) )
90 cnmbf ( ( ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ∈ dom vol ∧ ( 𝑠 ∈ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ↦ ( ( ∗ ‘ ∫ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) · 𝑠 / 𝑡 ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ) ∈ ( ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) –cn→ ℂ ) ) → ( 𝑠 ∈ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ↦ ( ( ∗ ‘ ∫ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) · 𝑠 / 𝑡 ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ) ∈ MblFn )
91 52 89 90 sylancr ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( 𝑠 ∈ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ↦ ( ( ∗ ‘ ∫ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) · 𝑠 / 𝑡 ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ) ∈ MblFn )
92 51 58 76 91 itgabsnc ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( abs ‘ ∫ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ≤ ∫ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) d 𝑡 )
93 51 abscld ( ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑡 ∈ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ) → ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ∈ ℝ )
94 fvexd ( ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑡 ∈ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ) → ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ∈ V )
95 94 58 76 iblabsnc ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( 𝑡 ∈ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ↦ ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ) ∈ 𝐿1 )
96 51 absge0d ( ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑡 ∈ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ) → 0 ≤ ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) )
97 93 95 96 itgposval ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ∫ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) d 𝑡 = ( ∫2 ‘ ( 𝑡 ∈ ℝ ↦ if ( 𝑡 ∈ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) , ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) , 0 ) ) ) )
98 92 97 breqtrd ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( abs ‘ ∫ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ≤ ( ∫2 ‘ ( 𝑡 ∈ ℝ ↦ if ( 𝑡 ∈ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) , ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) , 0 ) ) ) )
99 itgeq1 ( ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) = 𝑠 → ∫ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 = ∫ 𝑠 ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 )
100 99 fveq2d ( ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) = 𝑠 → ( abs ‘ ∫ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) = ( abs ‘ ∫ 𝑠 ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) )
101 eleq2 ( ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) = 𝑠 → ( 𝑡 ∈ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ↔ 𝑡𝑠 ) )
102 101 ifbid ( ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) = 𝑠 → if ( 𝑡 ∈ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) , ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) , 0 ) = if ( 𝑡𝑠 , ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) , 0 ) )
103 102 mpteq2dv ( ( ( 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 ) ) )
104 103 fveq2d ( ( ( 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 ) ) ) )
105 100 104 breq12d ( ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) = 𝑠 → ( ( abs ‘ ∫ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ≤ ( ∫2 ‘ ( 𝑡 ∈ ℝ ↦ if ( 𝑡 ∈ ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) , ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) , 0 ) ) ) ↔ ( abs ‘ ∫ 𝑠 ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ≤ ( ∫2 ‘ ( 𝑡 ∈ ℝ ↦ if ( 𝑡𝑠 , ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) , 0 ) ) ) ) )
106 98 105 syl5ibcom ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( ( ( 1st𝑥 ) (,) ( 2nd𝑥 ) ) = 𝑠 → ( abs ‘ ∫ 𝑠 ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ≤ ( ∫2 ‘ ( 𝑡 ∈ ℝ ↦ if ( 𝑡𝑠 , ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) , 0 ) ) ) ) )
107 33 106 sylbid ( ( 𝜑𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( ( (,) ‘ 𝑥 ) = 𝑠 → ( abs ‘ ∫ 𝑠 ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ≤ ( ∫2 ‘ ( 𝑡 ∈ ℝ ↦ if ( 𝑡𝑠 , ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) , 0 ) ) ) ) )
108 107 rexlimdva ( 𝜑 → ( ∃ 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ( (,) ‘ 𝑥 ) = 𝑠 → ( abs ‘ ∫ 𝑠 ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ≤ ( ∫2 ‘ ( 𝑡 ∈ ℝ ↦ if ( 𝑡𝑠 , ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) , 0 ) ) ) ) )
109 27 108 syl5 ( 𝜑 → ( 𝑠 ∈ ( (,) “ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) → ( abs ‘ ∫ 𝑠 ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ≤ ( ∫2 ‘ ( 𝑡 ∈ ℝ ↦ if ( 𝑡𝑠 , ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) , 0 ) ) ) ) )
110 109 ralrimiv ( 𝜑 → ∀ 𝑠 ∈ ( (,) “ ( ( 𝐴 [,] 𝐵 ) × ( 𝐴 [,] 𝐵 ) ) ) ( abs ‘ ∫ 𝑠 ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ≤ ( ∫2 ‘ ( 𝑡 ∈ ℝ ↦ if ( 𝑡𝑠 , ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) , 0 ) ) ) )
111 17 1 2 3 18 20 5 22 110 ftc1anc ( 𝜑 → ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) )
112 cncff ( 𝐹 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ℂ )
113 6 112 syl ( 𝜑𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ℂ )
114 113 feqmptd ( 𝜑𝐹 = ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( 𝐹𝑥 ) ) )
115 114 6 eqeltrrd ( 𝜑 → ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( 𝐹𝑥 ) ) ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) )
116 14 16 111 115 cncfmpt2f ( 𝜑 → ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹𝑥 ) ) ) ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) )
117 59 a1i ( 𝜑 → ℝ ⊆ ℂ )
118 iccssre ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 [,] 𝐵 ) ⊆ ℝ )
119 1 2 118 syl2anc ( 𝜑 → ( 𝐴 [,] 𝐵 ) ⊆ ℝ )
120 fvexd ( ( ( 𝜑𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑡 ∈ ( 𝐴 (,) 𝑥 ) ) → ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ∈ V )
121 2 adantr ( ( 𝜑𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐵 ∈ ℝ )
122 121 rexrd ( ( 𝜑𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐵 ∈ ℝ* )
123 elicc2 ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑥 ∈ ℝ ∧ 𝐴𝑥𝑥𝐵 ) ) )
124 1 2 123 syl2anc ( 𝜑 → ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑥 ∈ ℝ ∧ 𝐴𝑥𝑥𝐵 ) ) )
125 124 biimpa ( ( 𝜑𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝑥 ∈ ℝ ∧ 𝐴𝑥𝑥𝐵 ) )
126 125 simp3d ( ( 𝜑𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑥𝐵 )
127 iooss2 ( ( 𝐵 ∈ ℝ*𝑥𝐵 ) → ( 𝐴 (,) 𝑥 ) ⊆ ( 𝐴 (,) 𝐵 ) )
128 122 126 127 syl2anc ( ( 𝜑𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐴 (,) 𝑥 ) ⊆ ( 𝐴 (,) 𝐵 ) )
129 ioombl ( 𝐴 (,) 𝑥 ) ∈ dom vol
130 129 a1i ( ( 𝜑𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐴 (,) 𝑥 ) ∈ dom vol )
131 fvexd ( ( ( 𝜑𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑡 ∈ ( 𝐴 (,) 𝐵 ) ) → ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ∈ V )
132 56 adantr ( ( 𝜑𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝑡 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ∈ 𝐿1 )
133 128 130 131 132 iblss ( ( 𝜑𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝑡 ∈ ( 𝐴 (,) 𝑥 ) ↦ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ) ∈ 𝐿1 )
134 120 133 itgcl ( ( 𝜑𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ∈ ℂ )
135 113 ffvelrnda ( ( 𝜑𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹𝑥 ) ∈ ℂ )
136 134 135 subcld ( ( 𝜑𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹𝑥 ) ) ∈ ℂ )
137 14 tgioo2 ( topGen ‘ ran (,) ) = ( ( TopOpen ‘ ℂfld ) ↾t ℝ )
138 iccntr ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝐴 [,] 𝐵 ) ) = ( 𝐴 (,) 𝐵 ) )
139 1 2 138 syl2anc ( 𝜑 → ( ( int ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝐴 [,] 𝐵 ) ) = ( 𝐴 (,) 𝐵 ) )
140 117 119 136 137 14 139 dvmptntr ( 𝜑 → ( ℝ D ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹𝑥 ) ) ) ) = ( ℝ D ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹𝑥 ) ) ) ) )
141 reelprrecn ℝ ∈ { ℝ , ℂ }
142 141 a1i ( 𝜑 → ℝ ∈ { ℝ , ℂ } )
143 ioossicc ( 𝐴 (,) 𝐵 ) ⊆ ( 𝐴 [,] 𝐵 )
144 143 sseli ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) → 𝑥 ∈ ( 𝐴 [,] 𝐵 ) )
145 144 134 sylan2 ( ( 𝜑𝑥 ∈ ( 𝐴 (,) 𝐵 ) ) → ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ∈ ℂ )
146 22 ffvelrnda ( ( 𝜑𝑥 ∈ ( 𝐴 (,) 𝐵 ) ) → ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ∈ ℂ )
147 17 1 2 3 4 5 ftc1cnnc ( 𝜑 → ( ℝ D ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ) = ( ℝ D 𝐹 ) )
148 117 119 134 137 14 139 dvmptntr ( 𝜑 → ( ℝ D ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ) = ( ℝ D ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ) )
149 22 feqmptd ( 𝜑 → ( ℝ D 𝐹 ) = ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ) )
150 147 148 149 3eqtr3d ( 𝜑 → ( ℝ D ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ) ) = ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ) )
151 144 135 sylan2 ( ( 𝜑𝑥 ∈ ( 𝐴 (,) 𝐵 ) ) → ( 𝐹𝑥 ) ∈ ℂ )
152 114 oveq2d ( 𝜑 → ( ℝ D 𝐹 ) = ( ℝ D ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( 𝐹𝑥 ) ) ) )
153 117 119 135 137 14 139 dvmptntr ( 𝜑 → ( ℝ D ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( 𝐹𝑥 ) ) ) = ( ℝ D ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( 𝐹𝑥 ) ) ) )
154 152 149 153 3eqtr3rd ( 𝜑 → ( ℝ D ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( 𝐹𝑥 ) ) ) = ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ) )
155 142 145 146 150 151 146 154 dvmptsub ( 𝜑 → ( ℝ D ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹𝑥 ) ) ) ) = ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( ( ( ℝ D 𝐹 ) ‘ 𝑥 ) − ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ) ) )
156 146 subidd ( ( 𝜑𝑥 ∈ ( 𝐴 (,) 𝐵 ) ) → ( ( ( ℝ D 𝐹 ) ‘ 𝑥 ) − ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ) = 0 )
157 156 mpteq2dva ( 𝜑 → ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( ( ( ℝ D 𝐹 ) ‘ 𝑥 ) − ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ 0 ) )
158 140 155 157 3eqtrd ( 𝜑 → ( ℝ D ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹𝑥 ) ) ) ) = ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ 0 ) )
159 fconstmpt ( ( 𝐴 (,) 𝐵 ) × { 0 } ) = ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ 0 )
160 158 159 eqtr4di ( 𝜑 → ( ℝ D ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹𝑥 ) ) ) ) = ( ( 𝐴 (,) 𝐵 ) × { 0 } ) )
161 1 2 116 160 dveq0 ( 𝜑 → ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹𝑥 ) ) ) = ( ( 𝐴 [,] 𝐵 ) × { ( ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹𝑥 ) ) ) ‘ 𝐴 ) } ) )
162 161 fveq1d ( 𝜑 → ( ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹𝑥 ) ) ) ‘ 𝐵 ) = ( ( ( 𝐴 [,] 𝐵 ) × { ( ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹𝑥 ) ) ) ‘ 𝐴 ) } ) ‘ 𝐵 ) )
163 oveq2 ( 𝑥 = 𝐵 → ( 𝐴 (,) 𝑥 ) = ( 𝐴 (,) 𝐵 ) )
164 itgeq1 ( ( 𝐴 (,) 𝑥 ) = ( 𝐴 (,) 𝐵 ) → ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 = ∫ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 )
165 163 164 syl ( 𝑥 = 𝐵 → ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 = ∫ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 )
166 fveq2 ( 𝑥 = 𝐵 → ( 𝐹𝑥 ) = ( 𝐹𝐵 ) )
167 165 166 oveq12d ( 𝑥 = 𝐵 → ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹𝑥 ) ) = ( ∫ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹𝐵 ) ) )
168 eqid ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹𝑥 ) ) ) = ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹𝑥 ) ) )
169 ovex ( ∫ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹𝐵 ) ) ∈ V
170 167 168 169 fvmpt ( 𝐵 ∈ ( 𝐴 [,] 𝐵 ) → ( ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹𝑥 ) ) ) ‘ 𝐵 ) = ( ∫ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹𝐵 ) ) )
171 10 170 syl ( 𝜑 → ( ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹𝑥 ) ) ) ‘ 𝐵 ) = ( ∫ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹𝐵 ) ) )
172 162 171 eqtr3d ( 𝜑 → ( ( ( 𝐴 [,] 𝐵 ) × { ( ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹𝑥 ) ) ) ‘ 𝐴 ) } ) ‘ 𝐵 ) = ( ∫ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹𝐵 ) ) )
173 lbicc2 ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵 ) → 𝐴 ∈ ( 𝐴 [,] 𝐵 ) )
174 7 8 3 173 syl3anc ( 𝜑𝐴 ∈ ( 𝐴 [,] 𝐵 ) )
175 oveq2 ( 𝑥 = 𝐴 → ( 𝐴 (,) 𝑥 ) = ( 𝐴 (,) 𝐴 ) )
176 iooid ( 𝐴 (,) 𝐴 ) = ∅
177 175 176 eqtrdi ( 𝑥 = 𝐴 → ( 𝐴 (,) 𝑥 ) = ∅ )
178 itgeq1 ( ( 𝐴 (,) 𝑥 ) = ∅ → ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 = ∫ ∅ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 )
179 177 178 syl ( 𝑥 = 𝐴 → ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 = ∫ ∅ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 )
180 itg0 ∫ ∅ ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 = 0
181 179 180 eqtrdi ( 𝑥 = 𝐴 → ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 = 0 )
182 fveq2 ( 𝑥 = 𝐴 → ( 𝐹𝑥 ) = ( 𝐹𝐴 ) )
183 181 182 oveq12d ( 𝑥 = 𝐴 → ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹𝑥 ) ) = ( 0 − ( 𝐹𝐴 ) ) )
184 df-neg - ( 𝐹𝐴 ) = ( 0 − ( 𝐹𝐴 ) )
185 183 184 eqtr4di ( 𝑥 = 𝐴 → ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹𝑥 ) ) = - ( 𝐹𝐴 ) )
186 negex - ( 𝐹𝐴 ) ∈ V
187 185 168 186 fvmpt ( 𝐴 ∈ ( 𝐴 [,] 𝐵 ) → ( ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹𝑥 ) ) ) ‘ 𝐴 ) = - ( 𝐹𝐴 ) )
188 174 187 syl ( 𝜑 → ( ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ∫ ( 𝐴 (,) 𝑥 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹𝑥 ) ) ) ‘ 𝐴 ) = - ( 𝐹𝐴 ) )
189 13 172 188 3eqtr3d ( 𝜑 → ( ∫ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹𝐵 ) ) = - ( 𝐹𝐴 ) )
190 189 oveq2d ( 𝜑 → ( ( 𝐹𝐵 ) + ( ∫ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹𝐵 ) ) ) = ( ( 𝐹𝐵 ) + - ( 𝐹𝐴 ) ) )
191 113 10 ffvelrnd ( 𝜑 → ( 𝐹𝐵 ) ∈ ℂ )
192 fvexd ( ( 𝜑𝑡 ∈ ( 𝐴 (,) 𝐵 ) ) → ( ( ℝ D 𝐹 ) ‘ 𝑡 ) ∈ V )
193 192 56 itgcl ( 𝜑 → ∫ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 ∈ ℂ )
194 191 193 pncan3d ( 𝜑 → ( ( 𝐹𝐵 ) + ( ∫ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 − ( 𝐹𝐵 ) ) ) = ∫ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 )
195 113 174 ffvelrnd ( 𝜑 → ( 𝐹𝐴 ) ∈ ℂ )
196 191 195 negsubd ( 𝜑 → ( ( 𝐹𝐵 ) + - ( 𝐹𝐴 ) ) = ( ( 𝐹𝐵 ) − ( 𝐹𝐴 ) ) )
197 190 194 196 3eqtr3d ( 𝜑 → ∫ ( 𝐴 (,) 𝐵 ) ( ( ℝ D 𝐹 ) ‘ 𝑡 ) d 𝑡 = ( ( 𝐹𝐵 ) − ( 𝐹𝐴 ) ) )