Step |
Hyp |
Ref |
Expression |
1 |
|
itgge0.1 |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝐿1 ) |
2 |
|
itgge0.2 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ ) |
3 |
|
itgge0.3 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 0 ≤ 𝐵 ) |
4 |
|
itgz |
⊢ ∫ 𝐴 0 d 𝑥 = 0 |
5 |
|
fconstmpt |
⊢ ( 𝐴 × { 0 } ) = ( 𝑥 ∈ 𝐴 ↦ 0 ) |
6 |
|
iblmbf |
⊢ ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝐿1 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ MblFn ) |
7 |
1 6
|
syl |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ MblFn ) |
8 |
7 2
|
mbfdm2 |
⊢ ( 𝜑 → 𝐴 ∈ dom vol ) |
9 |
|
ibl0 |
⊢ ( 𝐴 ∈ dom vol → ( 𝐴 × { 0 } ) ∈ 𝐿1 ) |
10 |
8 9
|
syl |
⊢ ( 𝜑 → ( 𝐴 × { 0 } ) ∈ 𝐿1 ) |
11 |
5 10
|
eqeltrrid |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 0 ) ∈ 𝐿1 ) |
12 |
|
0red |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 0 ∈ ℝ ) |
13 |
11 1 12 2 3
|
itgle |
⊢ ( 𝜑 → ∫ 𝐴 0 d 𝑥 ≤ ∫ 𝐴 𝐵 d 𝑥 ) |
14 |
4 13
|
eqbrtrrid |
⊢ ( 𝜑 → 0 ≤ ∫ 𝐴 𝐵 d 𝑥 ) |