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 ) ) )