Step | Hyp | Ref
| Expression |
1 | | ffvelrn 6859 |
. . . . . . . . . 10
⊢ ((𝐺:ℝ⟶ℝ ∧
𝑡 ∈ ℝ) →
(𝐺‘𝑡) ∈ ℝ) |
2 | 1 | recnd 10747 |
. . . . . . . . 9
⊢ ((𝐺:ℝ⟶ℝ ∧
𝑡 ∈ ℝ) →
(𝐺‘𝑡) ∈ ℂ) |
3 | | i1ff 24428 |
. . . . . . . . . . 11
⊢ (𝐹 ∈ dom ∫1
→ 𝐹:ℝ⟶ℝ) |
4 | 3 | ffvelrnda 6861 |
. . . . . . . . . 10
⊢ ((𝐹 ∈ dom ∫1
∧ 𝑡 ∈ ℝ)
→ (𝐹‘𝑡) ∈
ℝ) |
5 | 4 | recnd 10747 |
. . . . . . . . 9
⊢ ((𝐹 ∈ dom ∫1
∧ 𝑡 ∈ ℝ)
→ (𝐹‘𝑡) ∈
ℂ) |
6 | | subcl 10963 |
. . . . . . . . 9
⊢ (((𝐺‘𝑡) ∈ ℂ ∧ (𝐹‘𝑡) ∈ ℂ) → ((𝐺‘𝑡) − (𝐹‘𝑡)) ∈ ℂ) |
7 | 2, 5, 6 | syl2anr 600 |
. . . . . . . 8
⊢ (((𝐹 ∈ dom ∫1
∧ 𝑡 ∈ ℝ)
∧ (𝐺:ℝ⟶ℝ ∧ 𝑡 ∈ ℝ)) → ((𝐺‘𝑡) − (𝐹‘𝑡)) ∈ ℂ) |
8 | 7 | anandirs 679 |
. . . . . . 7
⊢ (((𝐹 ∈ dom ∫1
∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → ((𝐺‘𝑡) − (𝐹‘𝑡)) ∈ ℂ) |
9 | 8 | abscld 14886 |
. . . . . 6
⊢ (((𝐹 ∈ dom ∫1
∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) →
(abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) ∈ ℝ) |
10 | 9 | rexrd 10769 |
. . . . 5
⊢ (((𝐹 ∈ dom ∫1
∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) →
(abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) ∈
ℝ*) |
11 | 8 | absge0d 14894 |
. . . . 5
⊢ (((𝐹 ∈ dom ∫1
∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → 0 ≤
(abs‘((𝐺‘𝑡) − (𝐹‘𝑡)))) |
12 | | elxrge0 12931 |
. . . . 5
⊢
((abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) ∈ (0[,]+∞) ↔
((abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) ∈ ℝ* ∧ 0 ≤
(abs‘((𝐺‘𝑡) − (𝐹‘𝑡))))) |
13 | 10, 11, 12 | sylanbrc 586 |
. . . 4
⊢ (((𝐹 ∈ dom ∫1
∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) →
(abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) ∈ (0[,]+∞)) |
14 | 13 | fmpttd 6889 |
. . 3
⊢ ((𝐹 ∈ dom ∫1
∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦
(abs‘((𝐺‘𝑡) − (𝐹‘𝑡)))):ℝ⟶(0[,]+∞)) |
15 | 14 | 3adant2 1132 |
. 2
⊢ ((𝐹 ∈ dom ∫1
∧ 𝐺 ∈
𝐿1 ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦
(abs‘((𝐺‘𝑡) − (𝐹‘𝑡)))):ℝ⟶(0[,]+∞)) |
16 | | reex 10706 |
. . . . . . 7
⊢ ℝ
∈ V |
17 | 16 | a1i 11 |
. . . . . 6
⊢ ((𝐹 ∈ dom ∫1
∧ 𝐺 ∈
𝐿1 ∧ 𝐺:ℝ⟶ℝ) → ℝ
∈ V) |
18 | | fvexd 6689 |
. . . . . 6
⊢ (((𝐹 ∈ dom ∫1
∧ 𝐺 ∈
𝐿1 ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) →
(abs‘(𝐺‘𝑡)) ∈ V) |
19 | | fvexd 6689 |
. . . . . 6
⊢ (((𝐹 ∈ dom ∫1
∧ 𝐺 ∈
𝐿1 ∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) →
(abs‘(𝐹‘𝑡)) ∈ V) |
20 | | eqidd 2739 |
. . . . . 6
⊢ ((𝐹 ∈ dom ∫1
∧ 𝐺 ∈
𝐿1 ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦
(abs‘(𝐺‘𝑡))) = (𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡)))) |
21 | | eqidd 2739 |
. . . . . 6
⊢ ((𝐹 ∈ dom ∫1
∧ 𝐺 ∈
𝐿1 ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦
(abs‘(𝐹‘𝑡))) = (𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡)))) |
22 | 17, 18, 19, 20, 21 | offval2 7444 |
. . . . 5
⊢ ((𝐹 ∈ dom ∫1
∧ 𝐺 ∈
𝐿1 ∧ 𝐺:ℝ⟶ℝ) → ((𝑡 ∈ ℝ ↦
(abs‘(𝐺‘𝑡))) ∘f + (𝑡 ∈ ℝ ↦
(abs‘(𝐹‘𝑡)))) = (𝑡 ∈ ℝ ↦ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))))) |
23 | 22 | fveq2d 6678 |
. . . 4
⊢ ((𝐹 ∈ dom ∫1
∧ 𝐺 ∈
𝐿1 ∧ 𝐺:ℝ⟶ℝ) →
(∫2‘((𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡))) ∘f + (𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡))))) = (∫2‘(𝑡 ∈ ℝ ↦
((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡)))))) |
24 | | id 22 |
. . . . . . . . . 10
⊢ (𝐺:ℝ⟶ℝ →
𝐺:ℝ⟶ℝ) |
25 | 24 | feqmptd 6737 |
. . . . . . . . 9
⊢ (𝐺:ℝ⟶ℝ →
𝐺 = (𝑡 ∈ ℝ ↦ (𝐺‘𝑡))) |
26 | | absf 14787 |
. . . . . . . . . . 11
⊢
abs:ℂ⟶ℝ |
27 | 26 | a1i 11 |
. . . . . . . . . 10
⊢ (𝐺:ℝ⟶ℝ →
abs:ℂ⟶ℝ) |
28 | 27 | feqmptd 6737 |
. . . . . . . . 9
⊢ (𝐺:ℝ⟶ℝ →
abs = (𝑥 ∈ ℂ
↦ (abs‘𝑥))) |
29 | | fveq2 6674 |
. . . . . . . . 9
⊢ (𝑥 = (𝐺‘𝑡) → (abs‘𝑥) = (abs‘(𝐺‘𝑡))) |
30 | 2, 25, 28, 29 | fmptco 6901 |
. . . . . . . 8
⊢ (𝐺:ℝ⟶ℝ →
(abs ∘ 𝐺) = (𝑡 ∈ ℝ ↦
(abs‘(𝐺‘𝑡)))) |
31 | 30 | adantl 485 |
. . . . . . 7
⊢ ((𝐺 ∈ 𝐿1
∧ 𝐺:ℝ⟶ℝ) → (abs ∘
𝐺) = (𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡)))) |
32 | | iblmbf 24520 |
. . . . . . . . 9
⊢ (𝐺 ∈ 𝐿1
→ 𝐺 ∈
MblFn) |
33 | | ftc1anclem1 35473 |
. . . . . . . . 9
⊢ ((𝐺:ℝ⟶ℝ ∧
𝐺 ∈ MblFn) → (abs
∘ 𝐺) ∈
MblFn) |
34 | 32, 33 | sylan2 596 |
. . . . . . . 8
⊢ ((𝐺:ℝ⟶ℝ ∧
𝐺 ∈
𝐿1) → (abs ∘ 𝐺) ∈ MblFn) |
35 | 34 | ancoms 462 |
. . . . . . 7
⊢ ((𝐺 ∈ 𝐿1
∧ 𝐺:ℝ⟶ℝ) → (abs ∘
𝐺) ∈
MblFn) |
36 | 31, 35 | eqeltrrd 2834 |
. . . . . 6
⊢ ((𝐺 ∈ 𝐿1
∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦
(abs‘(𝐺‘𝑡))) ∈
MblFn) |
37 | 36 | 3adant1 1131 |
. . . . 5
⊢ ((𝐹 ∈ dom ∫1
∧ 𝐺 ∈
𝐿1 ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦
(abs‘(𝐺‘𝑡))) ∈
MblFn) |
38 | 2 | abscld 14886 |
. . . . . . . 8
⊢ ((𝐺:ℝ⟶ℝ ∧
𝑡 ∈ ℝ) →
(abs‘(𝐺‘𝑡)) ∈
ℝ) |
39 | 2 | absge0d 14894 |
. . . . . . . 8
⊢ ((𝐺:ℝ⟶ℝ ∧
𝑡 ∈ ℝ) → 0
≤ (abs‘(𝐺‘𝑡))) |
40 | | elrege0 12928 |
. . . . . . . 8
⊢
((abs‘(𝐺‘𝑡)) ∈ (0[,)+∞) ↔
((abs‘(𝐺‘𝑡)) ∈ ℝ ∧ 0 ≤
(abs‘(𝐺‘𝑡)))) |
41 | 38, 39, 40 | sylanbrc 586 |
. . . . . . 7
⊢ ((𝐺:ℝ⟶ℝ ∧
𝑡 ∈ ℝ) →
(abs‘(𝐺‘𝑡)) ∈
(0[,)+∞)) |
42 | 41 | fmpttd 6889 |
. . . . . 6
⊢ (𝐺:ℝ⟶ℝ →
(𝑡 ∈ ℝ ↦
(abs‘(𝐺‘𝑡))):ℝ⟶(0[,)+∞)) |
43 | 42 | 3ad2ant3 1136 |
. . . . 5
⊢ ((𝐹 ∈ dom ∫1
∧ 𝐺 ∈
𝐿1 ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦
(abs‘(𝐺‘𝑡))):ℝ⟶(0[,)+∞)) |
44 | | iftrue 4420 |
. . . . . . . . 9
⊢ (𝑡 ∈ ℝ → if(𝑡 ∈ ℝ,
(abs‘(𝐺‘𝑡)), 0) = (abs‘(𝐺‘𝑡))) |
45 | 44 | mpteq2ia 5121 |
. . . . . . . 8
⊢ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ ℝ,
(abs‘(𝐺‘𝑡)), 0)) = (𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡))) |
46 | 45 | fveq2i 6677 |
. . . . . . 7
⊢
(∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ℝ, (abs‘(𝐺‘𝑡)), 0))) = (∫2‘(𝑡 ∈ ℝ ↦
(abs‘(𝐺‘𝑡)))) |
47 | 1 | adantll 714 |
. . . . . . . . . 10
⊢ (((𝐺 ∈ 𝐿1
∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → (𝐺‘𝑡) ∈ ℝ) |
48 | | simpr 488 |
. . . . . . . . . . . 12
⊢ ((𝐺 ∈ 𝐿1
∧ 𝐺:ℝ⟶ℝ) → 𝐺:ℝ⟶ℝ) |
49 | 48 | feqmptd 6737 |
. . . . . . . . . . 11
⊢ ((𝐺 ∈ 𝐿1
∧ 𝐺:ℝ⟶ℝ) → 𝐺 = (𝑡 ∈ ℝ ↦ (𝐺‘𝑡))) |
50 | | simpl 486 |
. . . . . . . . . . 11
⊢ ((𝐺 ∈ 𝐿1
∧ 𝐺:ℝ⟶ℝ) → 𝐺 ∈
𝐿1) |
51 | 49, 50 | eqeltrrd 2834 |
. . . . . . . . . 10
⊢ ((𝐺 ∈ 𝐿1
∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦ (𝐺‘𝑡)) ∈
𝐿1) |
52 | 47, 51, 36 | iblabsnc 35464 |
. . . . . . . . 9
⊢ ((𝐺 ∈ 𝐿1
∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦
(abs‘(𝐺‘𝑡))) ∈
𝐿1) |
53 | 38 | adantll 714 |
. . . . . . . . . 10
⊢ (((𝐺 ∈ 𝐿1
∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) →
(abs‘(𝐺‘𝑡)) ∈
ℝ) |
54 | 39 | adantll 714 |
. . . . . . . . . 10
⊢ (((𝐺 ∈ 𝐿1
∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → 0 ≤
(abs‘(𝐺‘𝑡))) |
55 | 53, 54 | iblpos 24545 |
. . . . . . . . 9
⊢ ((𝐺 ∈ 𝐿1
∧ 𝐺:ℝ⟶ℝ) → ((𝑡 ∈ ℝ ↦
(abs‘(𝐺‘𝑡))) ∈ 𝐿1
↔ ((𝑡 ∈ ℝ
↦ (abs‘(𝐺‘𝑡))) ∈ MblFn ∧
(∫2‘(𝑡
∈ ℝ ↦ if(𝑡
∈ ℝ, (abs‘(𝐺‘𝑡)), 0))) ∈ ℝ))) |
56 | 52, 55 | mpbid 235 |
. . . . . . . 8
⊢ ((𝐺 ∈ 𝐿1
∧ 𝐺:ℝ⟶ℝ) → ((𝑡 ∈ ℝ ↦
(abs‘(𝐺‘𝑡))) ∈ MblFn ∧
(∫2‘(𝑡
∈ ℝ ↦ if(𝑡
∈ ℝ, (abs‘(𝐺‘𝑡)), 0))) ∈ ℝ)) |
57 | 56 | simprd 499 |
. . . . . . 7
⊢ ((𝐺 ∈ 𝐿1
∧ 𝐺:ℝ⟶ℝ) →
(∫2‘(𝑡
∈ ℝ ↦ if(𝑡
∈ ℝ, (abs‘(𝐺‘𝑡)), 0))) ∈ ℝ) |
58 | 46, 57 | eqeltrrid 2838 |
. . . . . 6
⊢ ((𝐺 ∈ 𝐿1
∧ 𝐺:ℝ⟶ℝ) →
(∫2‘(𝑡
∈ ℝ ↦ (abs‘(𝐺‘𝑡)))) ∈ ℝ) |
59 | 58 | 3adant1 1131 |
. . . . 5
⊢ ((𝐹 ∈ dom ∫1
∧ 𝐺 ∈
𝐿1 ∧ 𝐺:ℝ⟶ℝ) →
(∫2‘(𝑡
∈ ℝ ↦ (abs‘(𝐺‘𝑡)))) ∈ ℝ) |
60 | 5 | abscld 14886 |
. . . . . . . 8
⊢ ((𝐹 ∈ dom ∫1
∧ 𝑡 ∈ ℝ)
→ (abs‘(𝐹‘𝑡)) ∈ ℝ) |
61 | 5 | absge0d 14894 |
. . . . . . . 8
⊢ ((𝐹 ∈ dom ∫1
∧ 𝑡 ∈ ℝ)
→ 0 ≤ (abs‘(𝐹‘𝑡))) |
62 | | elrege0 12928 |
. . . . . . . 8
⊢
((abs‘(𝐹‘𝑡)) ∈ (0[,)+∞) ↔
((abs‘(𝐹‘𝑡)) ∈ ℝ ∧ 0 ≤
(abs‘(𝐹‘𝑡)))) |
63 | 60, 61, 62 | sylanbrc 586 |
. . . . . . 7
⊢ ((𝐹 ∈ dom ∫1
∧ 𝑡 ∈ ℝ)
→ (abs‘(𝐹‘𝑡)) ∈ (0[,)+∞)) |
64 | 63 | fmpttd 6889 |
. . . . . 6
⊢ (𝐹 ∈ dom ∫1
→ (𝑡 ∈ ℝ
↦ (abs‘(𝐹‘𝑡))):ℝ⟶(0[,)+∞)) |
65 | 64 | 3ad2ant1 1134 |
. . . . 5
⊢ ((𝐹 ∈ dom ∫1
∧ 𝐺 ∈
𝐿1 ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦
(abs‘(𝐹‘𝑡))):ℝ⟶(0[,)+∞)) |
66 | | iftrue 4420 |
. . . . . . . . 9
⊢ (𝑡 ∈ ℝ → if(𝑡 ∈ ℝ,
(abs‘(𝐹‘𝑡)), 0) = (abs‘(𝐹‘𝑡))) |
67 | 66 | mpteq2ia 5121 |
. . . . . . . 8
⊢ (𝑡 ∈ ℝ ↦ if(𝑡 ∈ ℝ,
(abs‘(𝐹‘𝑡)), 0)) = (𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡))) |
68 | 67 | fveq2i 6677 |
. . . . . . 7
⊢
(∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ℝ, (abs‘(𝐹‘𝑡)), 0))) = (∫2‘(𝑡 ∈ ℝ ↦
(abs‘(𝐹‘𝑡)))) |
69 | 3 | feqmptd 6737 |
. . . . . . . . . . 11
⊢ (𝐹 ∈ dom ∫1
→ 𝐹 = (𝑡 ∈ ℝ ↦ (𝐹‘𝑡))) |
70 | | i1fibl 24560 |
. . . . . . . . . . 11
⊢ (𝐹 ∈ dom ∫1
→ 𝐹 ∈
𝐿1) |
71 | 69, 70 | eqeltrrd 2834 |
. . . . . . . . . 10
⊢ (𝐹 ∈ dom ∫1
→ (𝑡 ∈ ℝ
↦ (𝐹‘𝑡)) ∈
𝐿1) |
72 | 26 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝐹 ∈ dom ∫1
→ abs:ℂ⟶ℝ) |
73 | 72 | feqmptd 6737 |
. . . . . . . . . . . 12
⊢ (𝐹 ∈ dom ∫1
→ abs = (𝑥 ∈
ℂ ↦ (abs‘𝑥))) |
74 | | fveq2 6674 |
. . . . . . . . . . . 12
⊢ (𝑥 = (𝐹‘𝑡) → (abs‘𝑥) = (abs‘(𝐹‘𝑡))) |
75 | 5, 69, 73, 74 | fmptco 6901 |
. . . . . . . . . . 11
⊢ (𝐹 ∈ dom ∫1
→ (abs ∘ 𝐹) =
(𝑡 ∈ ℝ ↦
(abs‘(𝐹‘𝑡)))) |
76 | | i1fmbf 24427 |
. . . . . . . . . . . 12
⊢ (𝐹 ∈ dom ∫1
→ 𝐹 ∈
MblFn) |
77 | | ftc1anclem1 35473 |
. . . . . . . . . . . 12
⊢ ((𝐹:ℝ⟶ℝ ∧
𝐹 ∈ MblFn) → (abs
∘ 𝐹) ∈
MblFn) |
78 | 3, 76, 77 | syl2anc 587 |
. . . . . . . . . . 11
⊢ (𝐹 ∈ dom ∫1
→ (abs ∘ 𝐹)
∈ MblFn) |
79 | 75, 78 | eqeltrrd 2834 |
. . . . . . . . . 10
⊢ (𝐹 ∈ dom ∫1
→ (𝑡 ∈ ℝ
↦ (abs‘(𝐹‘𝑡))) ∈ MblFn) |
80 | 4, 71, 79 | iblabsnc 35464 |
. . . . . . . . 9
⊢ (𝐹 ∈ dom ∫1
→ (𝑡 ∈ ℝ
↦ (abs‘(𝐹‘𝑡))) ∈
𝐿1) |
81 | 60, 61 | iblpos 24545 |
. . . . . . . . 9
⊢ (𝐹 ∈ dom ∫1
→ ((𝑡 ∈ ℝ
↦ (abs‘(𝐹‘𝑡))) ∈ 𝐿1 ↔
((𝑡 ∈ ℝ ↦
(abs‘(𝐹‘𝑡))) ∈ MblFn ∧
(∫2‘(𝑡
∈ ℝ ↦ if(𝑡
∈ ℝ, (abs‘(𝐹‘𝑡)), 0))) ∈ ℝ))) |
82 | 80, 81 | mpbid 235 |
. . . . . . . 8
⊢ (𝐹 ∈ dom ∫1
→ ((𝑡 ∈ ℝ
↦ (abs‘(𝐹‘𝑡))) ∈ MblFn ∧
(∫2‘(𝑡
∈ ℝ ↦ if(𝑡
∈ ℝ, (abs‘(𝐹‘𝑡)), 0))) ∈ ℝ)) |
83 | 82 | simprd 499 |
. . . . . . 7
⊢ (𝐹 ∈ dom ∫1
→ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ ℝ, (abs‘(𝐹‘𝑡)), 0))) ∈ ℝ) |
84 | 68, 83 | eqeltrrid 2838 |
. . . . . 6
⊢ (𝐹 ∈ dom ∫1
→ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡)))) ∈ ℝ) |
85 | 84 | 3ad2ant1 1134 |
. . . . 5
⊢ ((𝐹 ∈ dom ∫1
∧ 𝐺 ∈
𝐿1 ∧ 𝐺:ℝ⟶ℝ) →
(∫2‘(𝑡
∈ ℝ ↦ (abs‘(𝐹‘𝑡)))) ∈ ℝ) |
86 | 37, 43, 59, 65, 85 | itg2addnc 35454 |
. . . 4
⊢ ((𝐹 ∈ dom ∫1
∧ 𝐺 ∈
𝐿1 ∧ 𝐺:ℝ⟶ℝ) →
(∫2‘((𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡))) ∘f + (𝑡 ∈ ℝ ↦ (abs‘(𝐹‘𝑡))))) = ((∫2‘(𝑡 ∈ ℝ ↦
(abs‘(𝐺‘𝑡)))) +
(∫2‘(𝑡
∈ ℝ ↦ (abs‘(𝐹‘𝑡)))))) |
87 | 23, 86 | eqtr3d 2775 |
. . 3
⊢ ((𝐹 ∈ dom ∫1
∧ 𝐺 ∈
𝐿1 ∧ 𝐺:ℝ⟶ℝ) →
(∫2‘(𝑡
∈ ℝ ↦ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))))) = ((∫2‘(𝑡 ∈ ℝ ↦
(abs‘(𝐺‘𝑡)))) +
(∫2‘(𝑡
∈ ℝ ↦ (abs‘(𝐹‘𝑡)))))) |
88 | 59, 85 | readdcld 10748 |
. . 3
⊢ ((𝐹 ∈ dom ∫1
∧ 𝐺 ∈
𝐿1 ∧ 𝐺:ℝ⟶ℝ) →
((∫2‘(𝑡 ∈ ℝ ↦ (abs‘(𝐺‘𝑡)))) + (∫2‘(𝑡 ∈ ℝ ↦
(abs‘(𝐹‘𝑡))))) ∈
ℝ) |
89 | 87, 88 | eqeltrd 2833 |
. 2
⊢ ((𝐹 ∈ dom ∫1
∧ 𝐺 ∈
𝐿1 ∧ 𝐺:ℝ⟶ℝ) →
(∫2‘(𝑡
∈ ℝ ↦ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))))) ∈ ℝ) |
90 | | readdcl 10698 |
. . . . . . . . 9
⊢
(((abs‘(𝐺‘𝑡)) ∈ ℝ ∧ (abs‘(𝐹‘𝑡)) ∈ ℝ) → ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))) ∈ ℝ) |
91 | 38, 60, 90 | syl2anr 600 |
. . . . . . . 8
⊢ (((𝐹 ∈ dom ∫1
∧ 𝑡 ∈ ℝ)
∧ (𝐺:ℝ⟶ℝ ∧ 𝑡 ∈ ℝ)) →
((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))) ∈ ℝ) |
92 | 91 | anandirs 679 |
. . . . . . 7
⊢ (((𝐹 ∈ dom ∫1
∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) →
((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))) ∈ ℝ) |
93 | 92 | rexrd 10769 |
. . . . . 6
⊢ (((𝐹 ∈ dom ∫1
∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) →
((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))) ∈
ℝ*) |
94 | 38 | adantll 714 |
. . . . . . 7
⊢ (((𝐹 ∈ dom ∫1
∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) →
(abs‘(𝐺‘𝑡)) ∈
ℝ) |
95 | 60 | adantlr 715 |
. . . . . . 7
⊢ (((𝐹 ∈ dom ∫1
∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) →
(abs‘(𝐹‘𝑡)) ∈
ℝ) |
96 | 39 | adantll 714 |
. . . . . . 7
⊢ (((𝐹 ∈ dom ∫1
∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → 0 ≤
(abs‘(𝐺‘𝑡))) |
97 | 61 | adantlr 715 |
. . . . . . 7
⊢ (((𝐹 ∈ dom ∫1
∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → 0 ≤
(abs‘(𝐹‘𝑡))) |
98 | 94, 95, 96, 97 | addge0d 11294 |
. . . . . 6
⊢ (((𝐹 ∈ dom ∫1
∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) → 0 ≤
((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡)))) |
99 | | elxrge0 12931 |
. . . . . 6
⊢
(((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))) ∈ (0[,]+∞) ↔
(((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))) ∈ ℝ* ∧ 0 ≤
((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))))) |
100 | 93, 98, 99 | sylanbrc 586 |
. . . . 5
⊢ (((𝐹 ∈ dom ∫1
∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) →
((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))) ∈ (0[,]+∞)) |
101 | 100 | fmpttd 6889 |
. . . 4
⊢ ((𝐹 ∈ dom ∫1
∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦
((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡)))):ℝ⟶(0[,]+∞)) |
102 | 101 | 3adant2 1132 |
. . 3
⊢ ((𝐹 ∈ dom ∫1
∧ 𝐺 ∈
𝐿1 ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦
((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡)))):ℝ⟶(0[,]+∞)) |
103 | | abs2dif2 14783 |
. . . . . . . 8
⊢ (((𝐺‘𝑡) ∈ ℂ ∧ (𝐹‘𝑡) ∈ ℂ) → (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) ≤ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡)))) |
104 | 2, 5, 103 | syl2anr 600 |
. . . . . . 7
⊢ (((𝐹 ∈ dom ∫1
∧ 𝑡 ∈ ℝ)
∧ (𝐺:ℝ⟶ℝ ∧ 𝑡 ∈ ℝ)) →
(abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) ≤ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡)))) |
105 | 104 | anandirs 679 |
. . . . . 6
⊢ (((𝐹 ∈ dom ∫1
∧ 𝐺:ℝ⟶ℝ) ∧ 𝑡 ∈ ℝ) →
(abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) ≤ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡)))) |
106 | 105 | ralrimiva 3096 |
. . . . 5
⊢ ((𝐹 ∈ dom ∫1
∧ 𝐺:ℝ⟶ℝ) →
∀𝑡 ∈ ℝ
(abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) ≤ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡)))) |
107 | 16 | a1i 11 |
. . . . . 6
⊢ ((𝐹 ∈ dom ∫1
∧ 𝐺:ℝ⟶ℝ) → ℝ
∈ V) |
108 | | eqidd 2739 |
. . . . . 6
⊢ ((𝐹 ∈ dom ∫1
∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦
(abs‘((𝐺‘𝑡) − (𝐹‘𝑡)))) = (𝑡 ∈ ℝ ↦ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))))) |
109 | | eqidd 2739 |
. . . . . 6
⊢ ((𝐹 ∈ dom ∫1
∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦
((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡)))) = (𝑡 ∈ ℝ ↦ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))))) |
110 | 107, 9, 92, 108, 109 | ofrfval2 7445 |
. . . . 5
⊢ ((𝐹 ∈ dom ∫1
∧ 𝐺:ℝ⟶ℝ) → ((𝑡 ∈ ℝ ↦
(abs‘((𝐺‘𝑡) − (𝐹‘𝑡)))) ∘r ≤ (𝑡 ∈ ℝ ↦
((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡)))) ↔ ∀𝑡 ∈ ℝ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))) ≤ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))))) |
111 | 106, 110 | mpbird 260 |
. . . 4
⊢ ((𝐹 ∈ dom ∫1
∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦
(abs‘((𝐺‘𝑡) − (𝐹‘𝑡)))) ∘r ≤ (𝑡 ∈ ℝ ↦
((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))))) |
112 | 111 | 3adant2 1132 |
. . 3
⊢ ((𝐹 ∈ dom ∫1
∧ 𝐺 ∈
𝐿1 ∧ 𝐺:ℝ⟶ℝ) → (𝑡 ∈ ℝ ↦
(abs‘((𝐺‘𝑡) − (𝐹‘𝑡)))) ∘r ≤ (𝑡 ∈ ℝ ↦
((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))))) |
113 | | itg2le 24492 |
. . 3
⊢ (((𝑡 ∈ ℝ ↦
(abs‘((𝐺‘𝑡) − (𝐹‘𝑡)))):ℝ⟶(0[,]+∞) ∧
(𝑡 ∈ ℝ ↦
((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡)))):ℝ⟶(0[,]+∞) ∧
(𝑡 ∈ ℝ ↦
(abs‘((𝐺‘𝑡) − (𝐹‘𝑡)))) ∘r ≤ (𝑡 ∈ ℝ ↦
((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))))) → (∫2‘(𝑡 ∈ ℝ ↦
(abs‘((𝐺‘𝑡) − (𝐹‘𝑡))))) ≤ (∫2‘(𝑡 ∈ ℝ ↦
((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡)))))) |
114 | 15, 102, 112, 113 | syl3anc 1372 |
. 2
⊢ ((𝐹 ∈ dom ∫1
∧ 𝐺 ∈
𝐿1 ∧ 𝐺:ℝ⟶ℝ) →
(∫2‘(𝑡
∈ ℝ ↦ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))))) ≤ (∫2‘(𝑡 ∈ ℝ ↦
((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡)))))) |
115 | | itg2lecl 24491 |
. 2
⊢ (((𝑡 ∈ ℝ ↦
(abs‘((𝐺‘𝑡) − (𝐹‘𝑡)))):ℝ⟶(0[,]+∞) ∧
(∫2‘(𝑡
∈ ℝ ↦ ((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡))))) ∈ ℝ ∧
(∫2‘(𝑡
∈ ℝ ↦ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))))) ≤ (∫2‘(𝑡 ∈ ℝ ↦
((abs‘(𝐺‘𝑡)) + (abs‘(𝐹‘𝑡)))))) → (∫2‘(𝑡 ∈ ℝ ↦
(abs‘((𝐺‘𝑡) − (𝐹‘𝑡))))) ∈ ℝ) |
116 | 15, 89, 114, 115 | syl3anc 1372 |
1
⊢ ((𝐹 ∈ dom ∫1
∧ 𝐺 ∈
𝐿1 ∧ 𝐺:ℝ⟶ℝ) →
(∫2‘(𝑡
∈ ℝ ↦ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))))) ∈ ℝ) |