Documentation

Mathlib.Data.Int.WithZero

WithZero #

In this file we provide some basic API lemmas for the WithZero construction and we define the morphism WithZeroMultInt.toNNReal.

Main Definitions #

Main Results #

Tags #

WithZero, multiplicative, nnreal

Given a nonzero e : ℝ≥0, this is the map ℤₘ₀ → ℝ≥0 sending 0 ↦ 0 and x ↦ e^(WithZero.unzero hx).toAdd when x ≠ 0 as a MonoidWithZeroHom.

Equations
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Instances For
    theorem WithZeroMulInt.toNNReal_ne_zero {e : NNReal} {m : WithZero (Multiplicative )} (he : e 0) (hm : m 0) :
    (toNNReal he) m 0

    toNNReal sends nonzero elements to nonzero elements.

    theorem WithZeroMulInt.toNNReal_pos {e : NNReal} {m : WithZero (Multiplicative )} (he : e 0) (hm : m 0) :
    0 < (toNNReal he) m

    toNNReal sends nonzero elements to positive elements.

    The map toNNReal is strictly monotone whenever 1 < e.