Theorem remimd

Description: Value of the conjugate of a complex number. The value is the real part minus _i times the imaginary part. Definition 10-3.2 of Gleason p. 132. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypothesis	recld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
Assertion	remimd	⊢ ( 𝜑 → ( ∗ ‘ 𝐴 ) = ( ( ℜ ‘ 𝐴 ) − ( i · ( ℑ ‘ 𝐴 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		recld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		remim	⊢ ( 𝐴 ∈ ℂ → ( ∗ ‘ 𝐴 ) = ( ( ℜ ‘ 𝐴 ) − ( i · ( ℑ ‘ 𝐴 ) ) ) )
3	1 2	syl	⊢ ( 𝜑 → ( ∗ ‘ 𝐴 ) = ( ( ℜ ‘ 𝐴 ) − ( i · ( ℑ ‘ 𝐴 ) ) ) )