Theorem readdi

Description: Real part distributes over addition. (Contributed by NM, 28-Jul-1999)

	Ref	Expression
Hypotheses	recl.1	⊢ 𝐴 ∈ ℂ
	readdi.2	⊢ 𝐵 ∈ ℂ
Assertion	readdi	⊢ ( ℜ ‘ ( 𝐴 + 𝐵 ) ) = ( ( ℜ ‘ 𝐴 ) + ( ℜ ‘ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		recl.1	⊢ 𝐴 ∈ ℂ
2		readdi.2	⊢ 𝐵 ∈ ℂ
3		readd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ℜ ‘ ( 𝐴 + 𝐵 ) ) = ( ( ℜ ‘ 𝐴 ) + ( ℜ ‘ 𝐵 ) ) )
4	1 2 3	mp2an	⊢ ( ℜ ‘ ( 𝐴 + 𝐵 ) ) = ( ( ℜ ‘ 𝐴 ) + ( ℜ ‘ 𝐵 ) )