fieldEquivOfAlgEquiv_algebraMap — Mathlib

English

This states that the fieldEquivOfAlgEquiv respects algebra maps on B, i.e., the algebraMap from B to L followed by the field equivalence equals the algebraMap from A to L after applying f.

Русский

Это утверждение, что fieldEquivOfAlgEquiv сохраняет algebraMap на B, то есть композиция дает совпадающий результат.

LaTeX

$$$\text{fieldEquivOfAlgEquiv } FA FB FC f (\operatorname{algebraMap} B FB b) = \operatorname{algebraMap} C FC (f b)$$$

Lean4

/-- This says that `fieldEquivOfAlgEquiv f` is an extension of `f` (i.e., it agrees with `f` on
`B`). Whereas `(fieldEquivOfAlgEquiv f).commutes` says that `fieldEquivOfAlgEquiv f` fixes `K`. -/
@[simp]
theorem fieldEquivOfAlgEquiv_algebraMap (f : B ≃ₐ[A] C) (b : B) :
    fieldEquivOfAlgEquiv FA FB FC f (algebraMap B FB b) = algebraMap C FC (f b) :=
  ringEquivOfRingEquiv_algebraMap f.toRingEquiv b

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