comp — Mathlib

English

The composition of two isometries Q1 →qᵢ Q2 and Q2 →qᵢ Q3 yields a new isometry Q1 →qᵢ Q3.

Русский

Сложение двух изометрий Q1 →qᵢ Q2 и Q2 →qᵢ Q3 даёт новую изометрию Q1 →qᵢ Q3.

LaTeX

$$$\exists f: Q_1 \to qᵢ Q_2, g: Q_2 \to qᵢ Q_3$, then $g \circ f : Q_1 \to qᵢ Q_3$$$

Lean4

/-- The composition of two isometries between quadratic forms. -/
@[simps]
def comp (g : Q₂ →qᵢ Q₃) (f : Q₁ →qᵢ Q₂) : Q₁ →qᵢ Q₃
    where
  toFun x := g (f x)
  map_app' x := by rw [← f.map_app, ← g.map_app]
  __ := (g.toLinearMap : M₂ →ₗ[R] M₃) ∘ₗ (f.toLinearMap : M₁ →ₗ[R] M₂)

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