apply_ofBijective_symm_apply — Mathlib

English

An illustrative instance of bijective linear map: for a fixed invertible matrix A, the map x ↦ Ax is a bijection with inverse x ↦ A^{-1}x.

Русский

Пусть A обратима; отображение x ↦ Ax биективно, его обратное — x ↦ A^{-1}x.

LaTeX

$$$A \\in \\mathrm{GL}_n(\\mathbb{R}) \\implies x \\mapsto Ax \\text{ is bijective with inverse } x \\mapsto A^{-1}x.$$$

Lean4

@[simp]
theorem apply_ofBijective_symm_apply [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {h} (x : M₂) :
    f ((ofBijective f h).symm x) = x := by
  rw [← ofBijective_apply f (hf := h) ((ofBijective f h).symm x), apply_symm_apply]

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