quotKerEquivRange — Mathlib

English

Compatibility of map range and mk' extensionality lemma.

Русский

Совместимость образа карты и экстенсивной леммы mk'.

LaTeX

$$range_mk'_ext$$

Lean4

/-- The first isomorphism theorem for morphisms of Lie algebras. -/
@[simps]
noncomputable def quotKerEquivRange : (L ⧸ f.ker) ≃ₗ⁅R⁆ f.range :=
  {
    (f : L →ₗ[R] L').quotKerEquivRange with
    toFun := (f : L →ₗ[R] L').quotKerEquivRange
    map_lie' := by
      intro x y
      induction x using Submodule.Quotient.induction_on
      induction y using Submodule.Quotient.induction_on
      rw [← SetLike.coe_eq_coe, LieSubalgebra.coe_bracket f.range]
      simp only [← LieSubmodule.Quotient.mk_bracket, LinearMap.quotKerEquivRange_apply_mk, coe_toLinearMap, map_lie] }

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