Metamath Proof Explorer
Description: The real part of a complex number representation. Definition 10-3.1 of
Gleason p. 132. (Contributed by Mario Carneiro, 29-May-2016)
|
|
Ref |
Expression |
|
Hypotheses |
crred.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
|
|
crred.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
|
Assertion |
crred |
⊢ ( 𝜑 → ( ℜ ‘ ( 𝐴 + ( i · 𝐵 ) ) ) = 𝐴 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
crred.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
crred.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
3 |
|
crre |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ℜ ‘ ( 𝐴 + ( i · 𝐵 ) ) ) = 𝐴 ) |
4 |
1 2 3
|
syl2anc |
⊢ ( 𝜑 → ( ℜ ‘ ( 𝐴 + ( i · 𝐵 ) ) ) = 𝐴 ) |