conductor_eq_of_eq — Mathlib

English

If the R-subalgebra generated by x maps to the top of S (i.e., is all of S), then the conductor of x is the top ideal.

Русский

Если порождение R<x> равно верхнему пределу (всё S), то кондуктор x — верхний кондуктор.

LaTeX

$$$(R = \\top) \\Rightarrow \\mathrm{conductor}(R,x) = \\top$$$

Lean4

theorem conductor_eq_of_eq {y : S} (h : (R<x> : Set S) = R<y>) : conductor R x = conductor R y :=
  Ideal.ext fun _ => forall_congr' fun _ => Set.ext_iff.mp h _

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