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Metamath Proof Explorer

Theorem redivd

Description: Real part of a division. Related to remul2 . (Contributed by Mario Carneiro, 29-May-2016)

Ref Expression
Hypotheses crred.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
remul2d.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
redivd.2 ⊢ ( 𝜑 → 𝐴 ≠ 0 )
Assertion redivd ( 𝜑 → ( ℜ ‘ ( 𝐵 / 𝐴 ) ) = ( ( ℜ ‘ 𝐵 ) / 𝐴 ) )

Proof

Step Hyp Ref Expression
1 crred.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 remul2d.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3 redivd.2 ⊢ ( 𝜑 → 𝐴 ≠ 0 )
4 rediv ⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ) → ( ℜ ‘ ( 𝐵 / 𝐴 ) ) = ( ( ℜ ‘ 𝐵 ) / 𝐴 ) )
5 2 1 3 4 syl3anc ⊢ ( 𝜑 → ( ℜ ‘ ( 𝐵 / 𝐴 ) ) = ( ( ℜ ‘ 𝐵 ) / 𝐴 ) )