Theorem resubd

Description: Real part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	recld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	readdd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
Assertion	resubd	⊢ ( 𝜑 → ( ℜ ‘ ( 𝐴 − 𝐵 ) ) = ( ( ℜ ‘ 𝐴 ) − ( ℜ ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		recld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		readdd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		resub	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ℜ ‘ ( 𝐴 − 𝐵 ) ) = ( ( ℜ ‘ 𝐴 ) − ( ℜ ‘ 𝐵 ) ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( ℜ ‘ ( 𝐴 − 𝐵 ) ) = ( ( ℜ ‘ 𝐴 ) − ( ℜ ‘ 𝐵 ) ) )