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Mirrors > Home > MPE Home > Th. List > df-aa | Structured version Visualization version GIF version |
Description: Define the set of algebraic numbers. An algebraic number is a root of a nonzero polynomial over the integers. Here we construct it as the union of all kernels (preimages of {0}) of all polynomials in (Poly‘ℤ), except the zero polynomial 0𝑝. (Contributed by Mario Carneiro, 22-Jul-2014.) |
Ref | Expression |
---|---|
df-aa | ⊢ 𝔸 = ∪ 𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(◡𝑓 “ {0}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caa 25690 | . 2 class 𝔸 | |
2 | vf | . . 3 setvar 𝑓 | |
3 | cz 12504 | . . . . 5 class ℤ | |
4 | cply 25561 | . . . . 5 class Poly | |
5 | 3, 4 | cfv 6497 | . . . 4 class (Poly‘ℤ) |
6 | c0p 25049 | . . . . 5 class 0𝑝 | |
7 | 6 | csn 4587 | . . . 4 class {0𝑝} |
8 | 5, 7 | cdif 3908 | . . 3 class ((Poly‘ℤ) ∖ {0𝑝}) |
9 | 2 | cv 1541 | . . . . 5 class 𝑓 |
10 | 9 | ccnv 5633 | . . . 4 class ◡𝑓 |
11 | cc0 11056 | . . . . 5 class 0 | |
12 | 11 | csn 4587 | . . . 4 class {0} |
13 | 10, 12 | cima 5637 | . . 3 class (◡𝑓 “ {0}) |
14 | 2, 8, 13 | ciun 4955 | . 2 class ∪ 𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(◡𝑓 “ {0}) |
15 | 1, 14 | wceq 1542 | 1 wff 𝔸 = ∪ 𝑓 ∈ ((Poly‘ℤ) ∖ {0𝑝})(◡𝑓 “ {0}) |
Colors of variables: wff setvar class |
This definition is referenced by: elaa 25692 |
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