powQuotSuccInclusion_injective — Mathlib

English

For a fixed i, the map quotientToQuotientRangePowQuotSucc p P i is injective when certain ramification/prime conditions hold, ensuring distinct residue-classes map to distinct graded quotients.

Русский

При выполнении условий разветвления отображение quotientToQuotientRangePowQuotSucc инъективно на соответствующих степенях.

LaTeX

$$$(quotientToQuotientRangePowQuotSucc p P i)$ is injective for appropriate i$$

Lean4

theorem powQuotSuccInclusion_injective (i : ℕ) : Function.Injective (powQuotSuccInclusion p P i) :=
  by
  rw [← LinearMap.ker_eq_bot, LinearMap.ker_eq_bot']
  rintro ⟨x, hx⟩ hx0
  rw [Subtype.ext_iff] at hx0 ⊢
  rwa [powQuotSuccInclusion_apply_coe] at hx0

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