Step |
Hyp |
Ref |
Expression |
1 |
|
iccmbl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 [,] 𝐵 ) ∈ dom vol ) |
2 |
|
cnmbf |
⊢ ( ( ( 𝐴 [,] 𝐵 ) ∈ dom vol ∧ 𝐹 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) → 𝐹 ∈ MblFn ) |
3 |
1 2
|
stoic3 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐹 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) → 𝐹 ∈ MblFn ) |
4 |
|
simp3 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐹 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) → 𝐹 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) |
5 |
|
cncff |
⊢ ( 𝐹 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ℂ ) |
6 |
|
fdm |
⊢ ( 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ℂ → dom 𝐹 = ( 𝐴 [,] 𝐵 ) ) |
7 |
4 5 6
|
3syl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐹 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) → dom 𝐹 = ( 𝐴 [,] 𝐵 ) ) |
8 |
7
|
fveq2d |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐹 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) → ( vol ‘ dom 𝐹 ) = ( vol ‘ ( 𝐴 [,] 𝐵 ) ) ) |
9 |
|
iccvolcl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( vol ‘ ( 𝐴 [,] 𝐵 ) ) ∈ ℝ ) |
10 |
9
|
3adant3 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐹 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) → ( vol ‘ ( 𝐴 [,] 𝐵 ) ) ∈ ℝ ) |
11 |
8 10
|
eqeltrd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐹 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) → ( vol ‘ dom 𝐹 ) ∈ ℝ ) |
12 |
|
cniccbdd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐹 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) → ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ( abs ‘ ( 𝐹 ‘ 𝑦 ) ) ≤ 𝑥 ) |
13 |
7
|
raleqdv |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐹 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) → ( ∀ 𝑦 ∈ dom 𝐹 ( abs ‘ ( 𝐹 ‘ 𝑦 ) ) ≤ 𝑥 ↔ ∀ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ( abs ‘ ( 𝐹 ‘ 𝑦 ) ) ≤ 𝑥 ) ) |
14 |
13
|
rexbidv |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐹 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) → ( ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ dom 𝐹 ( abs ‘ ( 𝐹 ‘ 𝑦 ) ) ≤ 𝑥 ↔ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ( abs ‘ ( 𝐹 ‘ 𝑦 ) ) ≤ 𝑥 ) ) |
15 |
12 14
|
mpbird |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐹 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) → ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ dom 𝐹 ( abs ‘ ( 𝐹 ‘ 𝑦 ) ) ≤ 𝑥 ) |
16 |
|
bddiblnc |
⊢ ( ( 𝐹 ∈ MblFn ∧ ( vol ‘ dom 𝐹 ) ∈ ℝ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ dom 𝐹 ( abs ‘ ( 𝐹 ‘ 𝑦 ) ) ≤ 𝑥 ) → 𝐹 ∈ 𝐿1 ) |
17 |
3 11 15 16
|
syl3anc |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐹 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) → 𝐹 ∈ 𝐿1 ) |