# Metamath Proof Explorer

## Theorem ismbfcn2

Description: A complex function is measurable iff the real and imaginary components of the function are measurable. (Contributed by Mario Carneiro, 13-Aug-2014)

Ref Expression
Hypothesis ismbfcn2.1 ( ( 𝜑𝑥𝐴 ) → 𝐵 ∈ ℂ )
Assertion ismbfcn2 ( 𝜑 → ( ( 𝑥𝐴𝐵 ) ∈ MblFn ↔ ( ( 𝑥𝐴 ↦ ( ℜ ‘ 𝐵 ) ) ∈ MblFn ∧ ( 𝑥𝐴 ↦ ( ℑ ‘ 𝐵 ) ) ∈ MblFn ) ) )

### Proof

Step Hyp Ref Expression
1 ismbfcn2.1 ( ( 𝜑𝑥𝐴 ) → 𝐵 ∈ ℂ )
2 1 fmpttd ( 𝜑 → ( 𝑥𝐴𝐵 ) : 𝐴 ⟶ ℂ )
3 ismbfcn ( ( 𝑥𝐴𝐵 ) : 𝐴 ⟶ ℂ → ( ( 𝑥𝐴𝐵 ) ∈ MblFn ↔ ( ( ℜ ∘ ( 𝑥𝐴𝐵 ) ) ∈ MblFn ∧ ( ℑ ∘ ( 𝑥𝐴𝐵 ) ) ∈ MblFn ) ) )
4 2 3 syl ( 𝜑 → ( ( 𝑥𝐴𝐵 ) ∈ MblFn ↔ ( ( ℜ ∘ ( 𝑥𝐴𝐵 ) ) ∈ MblFn ∧ ( ℑ ∘ ( 𝑥𝐴𝐵 ) ) ∈ MblFn ) ) )
5 ref ℜ : ℂ ⟶ ℝ
6 5 a1i ( 𝜑 → ℜ : ℂ ⟶ ℝ )
7 6 1 cofmpt ( 𝜑 → ( ℜ ∘ ( 𝑥𝐴𝐵 ) ) = ( 𝑥𝐴 ↦ ( ℜ ‘ 𝐵 ) ) )
8 7 eleq1d ( 𝜑 → ( ( ℜ ∘ ( 𝑥𝐴𝐵 ) ) ∈ MblFn ↔ ( 𝑥𝐴 ↦ ( ℜ ‘ 𝐵 ) ) ∈ MblFn ) )
9 imf ℑ : ℂ ⟶ ℝ
10 9 a1i ( 𝜑 → ℑ : ℂ ⟶ ℝ )
11 10 1 cofmpt ( 𝜑 → ( ℑ ∘ ( 𝑥𝐴𝐵 ) ) = ( 𝑥𝐴 ↦ ( ℑ ‘ 𝐵 ) ) )
12 11 eleq1d ( 𝜑 → ( ( ℑ ∘ ( 𝑥𝐴𝐵 ) ) ∈ MblFn ↔ ( 𝑥𝐴 ↦ ( ℑ ‘ 𝐵 ) ) ∈ MblFn ) )
13 8 12 anbi12d ( 𝜑 → ( ( ( ℜ ∘ ( 𝑥𝐴𝐵 ) ) ∈ MblFn ∧ ( ℑ ∘ ( 𝑥𝐴𝐵 ) ) ∈ MblFn ) ↔ ( ( 𝑥𝐴 ↦ ( ℜ ‘ 𝐵 ) ) ∈ MblFn ∧ ( 𝑥𝐴 ↦ ( ℑ ‘ 𝐵 ) ) ∈ MblFn ) ) )
14 4 13 bitrd ( 𝜑 → ( ( 𝑥𝐴𝐵 ) ∈ MblFn ↔ ( ( 𝑥𝐴 ↦ ( ℜ ‘ 𝐵 ) ) ∈ MblFn ∧ ( 𝑥𝐴 ↦ ( ℑ ‘ 𝐵 ) ) ∈ MblFn ) ) )