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Metamath Proof Explorer

Theorem remul2d

Description: Real part of a product. (Contributed by Mario Carneiro, 29-May-2016)

Ref Expression
Hypotheses crred.1 ⊢ φ → A ∈ ℝ
remul2d.2 ⊢ φ → B ∈ ℂ
Assertion remul2d ⊢ φ → ℜ ⁡ A ⁢ B = A ⁢ ℜ ⁡ B

Proof

Step Hyp Ref Expression
1 crred.1 ⊢ φ → A ∈ ℝ
2 remul2d.2 ⊢ φ → B ∈ ℂ
3 remul2 ⊢ A ∈ ℝ ∧ B ∈ ℂ → ℜ ⁡ A ⁢ B = A ⁢ ℜ ⁡ B
4 1 2 3 syl2anc ⊢ φ → ℜ ⁡ A ⁢ B = A ⁢ ℜ ⁡ B