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Metamath Proof Explorer

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 ) )