subtype_injective — Mathlib

English

The subtype construction is injective: if P.subtype hP = Q.subtype hQ, then P = Q.

Русский

Конструкция subtype инъективна: если P.subtype hP = Q.subtype hQ, то P = Q.

LaTeX

$$$ (P.subtype\\, hP) = (Q.subtype\\, hQ) \\Rightarrow P = Q $$$

Lean4

theorem subtype_injective {P Q : Sylow p G} {hP : P ≤ N} {hQ : Q ≤ N} (h : P.subtype hP = Q.subtype hQ) : P = Q :=
  by
  rw [SetLike.ext_iff] at h ⊢
  exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩

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