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Metamath Proof Explorer

Theorem reim

Description: The real part of a complex number in terms of the imaginary part function. (Contributed by Mario Carneiro, 31-Mar-2015)

Ref Expression
Assertion reim ( 𝐴 ∈ ℂ → ( ℜ ‘ 𝐴 ) = ( ℑ ‘ ( i · 𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 ax-icn ⊢ i ∈ ℂ
2 mulcl ⊢ ( ( i ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( i · 𝐴 ) ∈ ℂ )
3 1 2 mpan ⊢ ( 𝐴 ∈ ℂ → ( i · 𝐴 ) ∈ ℂ )
4 imval ⊢ ( ( i · 𝐴 ) ∈ ℂ → ( ℑ ‘ ( i · 𝐴 ) ) = ( ℜ ‘ ( ( i · 𝐴 ) / i ) ) )
5 3 4 syl ⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ ( i · 𝐴 ) ) = ( ℜ ‘ ( ( i · 𝐴 ) / i ) ) )
6 ine0 ⊢ i ≠ 0
7 divcan3 ⊢ ( ( 𝐴 ∈ ℂ ∧ i ∈ ℂ ∧ i ≠ 0 ) → ( ( i · 𝐴 ) / i ) = 𝐴 )
8 1 6 7 mp3an23 ⊢ ( 𝐴 ∈ ℂ → ( ( i · 𝐴 ) / i ) = 𝐴 )
9 8 fveq2d ⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ ( ( i · 𝐴 ) / i ) ) = ( ℜ ‘ 𝐴 ) )
10 5 9 eqtr2d ⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ 𝐴 ) = ( ℑ ‘ ( i · 𝐴 ) ) )