English
Continuing, the pullback construction yields a Field on K from an injective map into a Field L.
Русский
Продолжая, вытягивание по инъекции даёт поле на K из поля L.
LaTeX
$$$\\text{Field}_{K} \\text{ via } f: K \\hookrightarrow L$$$
Lean4
/-- Pullback a `Field` along an injective function. -/
-- See note [reducible non-instances]
protected abbrev field [Field L] (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ x y, f (x + y) = f x + f y)
(mul : ∀ x y, f (x * y) = f x * f y) (neg : ∀ x, f (-x) = -f x) (sub : ∀ x y, f (x - y) = f x - f y)
(inv : ∀ x, f x⁻¹ = (f x)⁻¹) (div : ∀ x y, f (x / y) = f x / f y) (nsmul : ∀ (n : ℕ) (x), f (n • x) = n • f x)
(zsmul : ∀ (n : ℤ) (x), f (n • x) = n • f x) (nnqsmul : ∀ (q : ℚ≥0) (x), f (q • x) = q • f x)
(qsmul : ∀ (q : ℚ) (x), f (q • x) = q • f x) (npow : ∀ (x) (n : ℕ), f (x ^ n) = f x ^ n)
(zpow : ∀ (x) (n : ℤ), f (x ^ n) = f x ^ n) (natCast : ∀ n : ℕ, f n = n) (intCast : ∀ n : ℤ, f n = n)
(nnratCast : ∀ q : ℚ≥0, f q = q) (ratCast : ∀ q : ℚ, f q = q) : Field K
where
toCommRing := hf.commRing f zero one add mul neg sub nsmul zsmul npow natCast intCast
__ :=
hf.divisionRing f zero one add mul neg sub inv div nsmul zsmul nnqsmul qsmul npow zpow natCast intCast nnratCast
ratCast
