Theorem reim0

Description: The imaginary part of a real number is 0. (Contributed by NM, 18-Mar-2005) (Revised by Mario Carneiro, 7-Nov-2013)

	Ref	Expression
Assertion	reim0	⊢ ( 𝐴 ∈ ℝ → ( ℑ ‘ 𝐴 ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		recn	⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
2		it0e0	⊢ ( i · 0 ) = 0
3	2	oveq2i	⊢ ( 𝐴 + ( i · 0 ) ) = ( 𝐴 + 0 )
4		addid1	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 + 0 ) = 𝐴 )
5	3 4	syl5eq	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 + ( i · 0 ) ) = 𝐴 )
6	1 5	syl	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 + ( i · 0 ) ) = 𝐴 )
7	6	fveq2d	⊢ ( 𝐴 ∈ ℝ → ( ℑ ‘ ( 𝐴 + ( i · 0 ) ) ) = ( ℑ ‘ 𝐴 ) )
8		0re	⊢ 0 ∈ ℝ
9		crim	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ∈ ℝ ) → ( ℑ ‘ ( 𝐴 + ( i · 0 ) ) ) = 0 )
10	8 9	mpan2	⊢ ( 𝐴 ∈ ℝ → ( ℑ ‘ ( 𝐴 + ( i · 0 ) ) ) = 0 )
11	7 10	eqtr3d	⊢ ( 𝐴 ∈ ℝ → ( ℑ ‘ 𝐴 ) = 0 )