Proof of Theorem itgle
Step | Hyp | Ref
| Expression |
1 | | itgle.1 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈
𝐿1) |
2 | | itgle.3 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
3 | 2 | iblrelem 23557 |
. . . . 5
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ↔
((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) ∈ ℝ ∧
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))) ∈ ℝ))) |
4 | 1, 3 | mpbid 222 |
. . . 4
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) ∈ ℝ ∧
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))) ∈ ℝ)) |
5 | 4 | simp2d 1074 |
. . 3
⊢ (𝜑 →
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) ∈ ℝ) |
6 | | itgle.2 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈
𝐿1) |
7 | | itgle.4 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ) |
8 | 7 | iblrelem 23557 |
. . . . 5
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1 ↔
((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn ∧
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) ∈ ℝ ∧
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0))) ∈ ℝ))) |
9 | 6, 8 | mpbid 222 |
. . . 4
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn ∧
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) ∈ ℝ ∧
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0))) ∈ ℝ)) |
10 | 9 | simp3d 1075 |
. . 3
⊢ (𝜑 →
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0))) ∈ ℝ) |
11 | 9 | simp2d 1074 |
. . 3
⊢ (𝜑 →
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) ∈ ℝ) |
12 | 4 | simp3d 1075 |
. . 3
⊢ (𝜑 →
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))) ∈ ℝ) |
13 | 2 | ad2ant2r 783 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵)) → 𝐵 ∈ ℝ) |
14 | 13 | rexrd 10089 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵)) → 𝐵 ∈
ℝ*) |
15 | | simprr 796 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ 𝐵) |
16 | | elxrge0 12281 |
. . . . . . 7
⊢ (𝐵 ∈ (0[,]+∞) ↔
(𝐵 ∈
ℝ* ∧ 0 ≤ 𝐵)) |
17 | 14, 15, 16 | sylanbrc 698 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵)) → 𝐵 ∈ (0[,]+∞)) |
18 | | 0e0iccpnf 12283 |
. . . . . . 7
⊢ 0 ∈
(0[,]+∞) |
19 | 18 | a1i 11 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ∈
(0[,]+∞)) |
20 | 17, 19 | ifclda 4120 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ (0[,]+∞)) |
21 | | eqid 2622 |
. . . . 5
⊢ (𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) |
22 | 20, 21 | fmptd 6385 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵,
0)):ℝ⟶(0[,]+∞)) |
23 | 7 | ad2ant2r 783 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶)) → 𝐶 ∈ ℝ) |
24 | 23 | rexrd 10089 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶)) → 𝐶 ∈
ℝ*) |
25 | | simprr 796 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶)) → 0 ≤ 𝐶) |
26 | | elxrge0 12281 |
. . . . . . 7
⊢ (𝐶 ∈ (0[,]+∞) ↔
(𝐶 ∈
ℝ* ∧ 0 ≤ 𝐶)) |
27 | 24, 25, 26 | sylanbrc 698 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶)) → 𝐶 ∈ (0[,]+∞)) |
28 | 18 | a1i 11 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶)) → 0 ∈
(0[,]+∞)) |
29 | 27, 28 | ifclda 4120 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ (0[,]+∞)) |
30 | | eqid 2622 |
. . . . 5
⊢ (𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) |
31 | 29, 30 | fmptd 6385 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶,
0)):ℝ⟶(0[,]+∞)) |
32 | | 0re 10040 |
. . . . . . . . . . . 12
⊢ 0 ∈
ℝ |
33 | | max1 12016 |
. . . . . . . . . . . 12
⊢ ((0
∈ ℝ ∧ 𝐶
∈ ℝ) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0)) |
34 | 32, 7, 33 | sylancr 695 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0)) |
35 | | ifcl 4130 |
. . . . . . . . . . . . 13
⊢ ((𝐶 ∈ ℝ ∧ 0 ∈
ℝ) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ) |
36 | 7, 32, 35 | sylancl 694 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ) |
37 | | itgle.5 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝐶) |
38 | | max2 12018 |
. . . . . . . . . . . . 13
⊢ ((0
∈ ℝ ∧ 𝐶
∈ ℝ) → 𝐶
≤ if(0 ≤ 𝐶, 𝐶, 0)) |
39 | 32, 7, 38 | sylancr 695 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ≤ if(0 ≤ 𝐶, 𝐶, 0)) |
40 | 2, 7, 36, 37, 39 | letrd 10194 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ if(0 ≤ 𝐶, 𝐶, 0)) |
41 | 32 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ∈ ℝ) |
42 | | maxle 12022 |
. . . . . . . . . . . 12
⊢ ((0
∈ ℝ ∧ 𝐵
∈ ℝ ∧ if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ) → (if(0 ≤ 𝐵, 𝐵, 0) ≤ if(0 ≤ 𝐶, 𝐶, 0) ↔ (0 ≤ if(0 ≤ 𝐶, 𝐶, 0) ∧ 𝐵 ≤ if(0 ≤ 𝐶, 𝐶, 0)))) |
43 | 41, 2, 36, 42 | syl3anc 1326 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) ≤ if(0 ≤ 𝐶, 𝐶, 0) ↔ (0 ≤ if(0 ≤ 𝐶, 𝐶, 0) ∧ 𝐵 ≤ if(0 ≤ 𝐶, 𝐶, 0)))) |
44 | 34, 40, 43 | mpbir2and 957 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ≤ if(0 ≤ 𝐶, 𝐶, 0)) |
45 | | iftrue 4092 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) = if(0 ≤ 𝐵, 𝐵, 0)) |
46 | 45 | adantl 482 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) = if(0 ≤ 𝐵, 𝐵, 0)) |
47 | | iftrue 4092 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) = if(0 ≤ 𝐶, 𝐶, 0)) |
48 | 47 | adantl 482 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) = if(0 ≤ 𝐶, 𝐶, 0)) |
49 | 44, 46, 48 | 3brtr4d 4685 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ≤ if(𝑥 ∈ 𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0)) |
50 | 49 | ex 450 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ≤ if(𝑥 ∈ 𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0))) |
51 | | 0le0 11110 |
. . . . . . . . . 10
⊢ 0 ≤
0 |
52 | 51 | a1i 11 |
. . . . . . . . 9
⊢ (¬
𝑥 ∈ 𝐴 → 0 ≤ 0) |
53 | | iffalse 4095 |
. . . . . . . . 9
⊢ (¬
𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) = 0) |
54 | | iffalse 4095 |
. . . . . . . . 9
⊢ (¬
𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) = 0) |
55 | 52, 53, 54 | 3brtr4d 4685 |
. . . . . . . 8
⊢ (¬
𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ≤ if(𝑥 ∈ 𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0)) |
56 | 50, 55 | pm2.61d1 171 |
. . . . . . 7
⊢ (𝜑 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ≤ if(𝑥 ∈ 𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0)) |
57 | | ifan 4134 |
. . . . . . 7
⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = if(𝑥 ∈ 𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) |
58 | | ifan 4134 |
. . . . . . 7
⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) = if(𝑥 ∈ 𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) |
59 | 56, 57, 58 | 3brtr4g 4687 |
. . . . . 6
⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ≤ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) |
60 | 59 | ralrimivw 2967 |
. . . . 5
⊢ (𝜑 → ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ≤ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) |
61 | | reex 10027 |
. . . . . . 7
⊢ ℝ
∈ V |
62 | 61 | a1i 11 |
. . . . . 6
⊢ (𝜑 → ℝ ∈
V) |
63 | | eqidd 2623 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) |
64 | | eqidd 2623 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) |
65 | 62, 20, 29, 63, 64 | ofrfval2 6915 |
. . . . 5
⊢ (𝜑 → ((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∘𝑟 ≤
(𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) ↔ ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ≤ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) |
66 | 60, 65 | mpbird 247 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∘𝑟 ≤
(𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) |
67 | | itg2le 23506 |
. . . 4
⊢ (((𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)):ℝ⟶(0[,]+∞) ∧
(𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)):ℝ⟶(0[,]+∞) ∧
(𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∘𝑟 ≤
(𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) →
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) ≤ (∫2‘(𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))) |
68 | 22, 31, 66, 67 | syl3anc 1326 |
. . 3
⊢ (𝜑 →
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) ≤ (∫2‘(𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))) |
69 | 7 | renegcld 10457 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐶 ∈ ℝ) |
70 | 69 | ad2ant2r 783 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶)) → -𝐶 ∈ ℝ) |
71 | 70 | rexrd 10089 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶)) → -𝐶 ∈
ℝ*) |
72 | | simprr 796 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶)) → 0 ≤ -𝐶) |
73 | | elxrge0 12281 |
. . . . . . 7
⊢ (-𝐶 ∈ (0[,]+∞) ↔
(-𝐶 ∈
ℝ* ∧ 0 ≤ -𝐶)) |
74 | 71, 72, 73 | sylanbrc 698 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶)) → -𝐶 ∈ (0[,]+∞)) |
75 | 18 | a1i 11 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶)) → 0 ∈
(0[,]+∞)) |
76 | 74, 75 | ifclda 4120 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) ∈ (0[,]+∞)) |
77 | | eqid 2622 |
. . . . 5
⊢ (𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)) |
78 | 76, 77 | fmptd 6385 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶,
0)):ℝ⟶(0[,]+∞)) |
79 | 2 | renegcld 10457 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℝ) |
80 | 79 | ad2ant2r 783 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵)) → -𝐵 ∈ ℝ) |
81 | 80 | rexrd 10089 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵)) → -𝐵 ∈
ℝ*) |
82 | | simprr 796 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵)) → 0 ≤ -𝐵) |
83 | | elxrge0 12281 |
. . . . . . 7
⊢ (-𝐵 ∈ (0[,]+∞) ↔
(-𝐵 ∈
ℝ* ∧ 0 ≤ -𝐵)) |
84 | 81, 82, 83 | sylanbrc 698 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵)) → -𝐵 ∈ (0[,]+∞)) |
85 | 18 | a1i 11 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ¬ (𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵)) → 0 ∈
(0[,]+∞)) |
86 | 84, 85 | ifclda 4120 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0) ∈ (0[,]+∞)) |
87 | | eqid 2622 |
. . . . 5
⊢ (𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)) |
88 | 86, 87 | fmptd 6385 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵,
0)):ℝ⟶(0[,]+∞)) |
89 | | max1 12016 |
. . . . . . . . . . . 12
⊢ ((0
∈ ℝ ∧ -𝐵
∈ ℝ) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0)) |
90 | 32, 79, 89 | sylancr 695 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0)) |
91 | | ifcl 4130 |
. . . . . . . . . . . . 13
⊢ ((-𝐵 ∈ ℝ ∧ 0 ∈
ℝ) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ) |
92 | 79, 32, 91 | sylancl 694 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ) |
93 | 2, 7 | lenegd 10606 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 ≤ 𝐶 ↔ -𝐶 ≤ -𝐵)) |
94 | 37, 93 | mpbid 222 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐶 ≤ -𝐵) |
95 | | max2 12018 |
. . . . . . . . . . . . 13
⊢ ((0
∈ ℝ ∧ -𝐵
∈ ℝ) → -𝐵
≤ if(0 ≤ -𝐵, -𝐵, 0)) |
96 | 32, 79, 95 | sylancr 695 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ≤ if(0 ≤ -𝐵, -𝐵, 0)) |
97 | 69, 79, 92, 94, 96 | letrd 10194 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐶 ≤ if(0 ≤ -𝐵, -𝐵, 0)) |
98 | | maxle 12022 |
. . . . . . . . . . . 12
⊢ ((0
∈ ℝ ∧ -𝐶
∈ ℝ ∧ if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ) → (if(0 ≤
-𝐶, -𝐶, 0) ≤ if(0 ≤ -𝐵, -𝐵, 0) ↔ (0 ≤ if(0 ≤ -𝐵, -𝐵, 0) ∧ -𝐶 ≤ if(0 ≤ -𝐵, -𝐵, 0)))) |
99 | 41, 69, 92, 98 | syl3anc 1326 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) ≤ if(0 ≤ -𝐵, -𝐵, 0) ↔ (0 ≤ if(0 ≤ -𝐵, -𝐵, 0) ∧ -𝐶 ≤ if(0 ≤ -𝐵, -𝐵, 0)))) |
100 | 90, 97, 99 | mpbir2and 957 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐶, -𝐶, 0) ≤ if(0 ≤ -𝐵, -𝐵, 0)) |
101 | | iftrue 4092 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) = if(0 ≤ -𝐶, -𝐶, 0)) |
102 | 101 | adantl 482 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) = if(0 ≤ -𝐶, -𝐶, 0)) |
103 | | iftrue 4092 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0) = if(0 ≤ -𝐵, -𝐵, 0)) |
104 | 103 | adantl 482 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0) = if(0 ≤ -𝐵, -𝐵, 0)) |
105 | 100, 102,
104 | 3brtr4d 4685 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) ≤ if(𝑥 ∈ 𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0)) |
106 | 105 | ex 450 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) ≤ if(𝑥 ∈ 𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0))) |
107 | | iffalse 4095 |
. . . . . . . . 9
⊢ (¬
𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) = 0) |
108 | | iffalse 4095 |
. . . . . . . . 9
⊢ (¬
𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0) = 0) |
109 | 52, 107, 108 | 3brtr4d 4685 |
. . . . . . . 8
⊢ (¬
𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) ≤ if(𝑥 ∈ 𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0)) |
110 | 106, 109 | pm2.61d1 171 |
. . . . . . 7
⊢ (𝜑 → if(𝑥 ∈ 𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) ≤ if(𝑥 ∈ 𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0)) |
111 | | ifan 4134 |
. . . . . . 7
⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) = if(𝑥 ∈ 𝐴, if(0 ≤ -𝐶, -𝐶, 0), 0) |
112 | | ifan 4134 |
. . . . . . 7
⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0) = if(𝑥 ∈ 𝐴, if(0 ≤ -𝐵, -𝐵, 0), 0) |
113 | 110, 111,
112 | 3brtr4g 4687 |
. . . . . 6
⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) ≤ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)) |
114 | 113 | ralrimivw 2967 |
. . . . 5
⊢ (𝜑 → ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) ≤ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)) |
115 | | eqidd 2623 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0))) |
116 | | eqidd 2623 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))) |
117 | 62, 76, 86, 115, 116 | ofrfval2 6915 |
. . . . 5
⊢ (𝜑 → ((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)) ∘𝑟 ≤
(𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)) ↔ ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0) ≤ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))) |
118 | 114, 117 | mpbird 247 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)) ∘𝑟 ≤
(𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))) |
119 | | itg2le 23506 |
. . . 4
⊢ (((𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)):ℝ⟶(0[,]+∞) ∧
(𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)):ℝ⟶(0[,]+∞) ∧
(𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0)) ∘𝑟 ≤
(𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))) →
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0))) ≤ (∫2‘(𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)))) |
120 | 78, 88, 118, 119 | syl3anc 1326 |
. . 3
⊢ (𝜑 →
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0))) ≤ (∫2‘(𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)))) |
121 | 5, 10, 11, 12, 68, 120 | le2subd 10647 |
. 2
⊢ (𝜑 →
((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) −
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)))) ≤
((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) −
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0))))) |
122 | 2, 1 | itgrevallem1 23561 |
. 2
⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = ((∫2‘(𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) −
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))))) |
123 | 7, 6 | itgrevallem1 23561 |
. 2
⊢ (𝜑 → ∫𝐴𝐶 d𝑥 = ((∫2‘(𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) −
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐶), -𝐶, 0))))) |
124 | 121, 122,
123 | 3brtr4d 4685 |
1
⊢ (𝜑 → ∫𝐴𝐵 d𝑥 ≤ ∫𝐴𝐶 d𝑥) |