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Mirrors > Home > ILE Home > Th. List > df-re | GIF version |
Description: Define a function whose value is the real part of a complex number. See reval 10778 for its value, recli 10840 for its closure, and replim 10788 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.) |
Ref | Expression |
---|---|
df-re | ⊢ ℜ = (𝑥 ∈ ℂ ↦ ((𝑥 + (∗‘𝑥)) / 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cre 10769 | . 2 class ℜ | |
2 | vx | . . 3 setvar 𝑥 | |
3 | cc 7743 | . . 3 class ℂ | |
4 | 2 | cv 1341 | . . . . 5 class 𝑥 |
5 | ccj 10768 | . . . . . 6 class ∗ | |
6 | 4, 5 | cfv 5183 | . . . . 5 class (∗‘𝑥) |
7 | caddc 7748 | . . . . 5 class + | |
8 | 4, 6, 7 | co 5837 | . . . 4 class (𝑥 + (∗‘𝑥)) |
9 | c2 8900 | . . . 4 class 2 | |
10 | cdiv 8560 | . . . 4 class / | |
11 | 8, 9, 10 | co 5837 | . . 3 class ((𝑥 + (∗‘𝑥)) / 2) |
12 | 2, 3, 11 | cmpt 4038 | . 2 class (𝑥 ∈ ℂ ↦ ((𝑥 + (∗‘𝑥)) / 2)) |
13 | 1, 12 | wceq 1342 | 1 wff ℜ = (𝑥 ∈ ℂ ↦ ((𝑥 + (∗‘𝑥)) / 2)) |
Colors of variables: wff set class |
This definition is referenced by: reval 10778 ref 10784 |
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