`BookProof.ChapterF4.pseudoinverse_left_inverse` {m n : ℕ} (Φ : Matrix (Fin m) (Fin n) ℂ) (h : IsUnit (Φᴴ * Φ).det) : ((Φᴴ * Φ)⁻¹ * Φᴴ) * Φ = 1

Proved

BookProof.ChapterF4.pseudoinverse_left_inverse

by leonardopedro · Oct 8, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterF4.

BookProof.ChapterF4.pseudoinverse_left_inverse {m n : ℕ} (Φ : Matrix (Fin m) (Fin n) ℂ) (h : IsUnit (Φᴴ * Φ).det) : ((Φᴴ * Φ)⁻¹ * Φᴴ) * Φ = 1

Formalization note: Lean 4 identifier BookProof.ChapterF4.pseudoinverse_left_inverse.

Preamble

-- Generated from ChapterF4.lean — theorem BookProof.ChapterF4.pseudoinverse_left_inverse
import Mathlib
import Definitions.Def_ChapterF4
open BookProof.ChapterF4


open scoped BigOperators Matrix

variable {d k : ℕ}
variable {A : Type*} [NormedRing A] [NormedAlgebra ℂ A] [StarRing A] [ContinuousStar A]
  [CompleteSpace A] [StarModule ℂ A]

Formal statement

theorem BookProof.ChapterF4.pseudoinverse_left_inverse {m n : ℕ} (Φ : Matrix (Fin m) (Fin n) ℂ)
    (h : IsUnit (Φᴴ * Φ).det) :
    ((Φᴴ * Φ)⁻¹ * Φᴴ) * Φ = 1 := by sorry

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