one_apply_posSubmonoid — Mathlib

English

For γ ∈ Units(ValueGroupWithZero R), there exist a,b ∈ posSubmonoid R with valuation a / valuation b = γ.

Русский

Для γ ∈ Units(ValueGroupWithZero R) существуют a,b ∈ posSubmonoid R такие, что valuation a / valuation b = γ.

LaTeX

$$$\\exists a,b \\in \\text{posSubmonoid}(R), \\; \\dfrac{\\mathrm{valuation}(a)}{\\mathrm{valuation}(b)} = \\gamma$$$

Lean4

/-- For any `x ∈ posSubmonoid R`, the trivial valuation `1 : Valuation R Γ` sends `x` to `1`.
In fact, this is true for any `x ≠ 0`. This lemma is a special case useful for shorthand of
`x ∈ posSubmonoid R → x ≠ 0`. -/
theorem one_apply_posSubmonoid [Nontrivial R] [NoZeroDivisors R] [DecidablePred fun x : R ↦ x = 0]
    (x : posSubmonoid R) : (1 : Valuation R Γ) x = 1 :=
  Valuation.one_apply_of_ne_zero (by simp)

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