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Metamath Proof Explorer

Theorem crrei

Description: The real part of a complex number representation. Definition 10-3.1 of Gleason p. 132. (Contributed by NM, 10-May-1999)

Ref Expression
Hypotheses crre.1 ⊢ 𝐴 ∈ ℝ
crre.2 ⊢ 𝐵 ∈ ℝ
Assertion crrei ( ℜ ‘ ( 𝐴 + ( i · 𝐵 ) ) ) = 𝐴

Proof

Step Hyp Ref Expression
1 crre.1 ⊢ 𝐴 ∈ ℝ
2 crre.2 ⊢ 𝐵 ∈ ℝ
3 crre ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ℜ ‘ ( 𝐴 + ( i · 𝐵 ) ) ) = 𝐴 )
4 1 2 3 mp2an ⊢ ( ℜ ‘ ( 𝐴 + ( i · 𝐵 ) ) ) = 𝐴