`BookProof.ChapterF4.pseudoInverse_left_inverse` {k m : ℕ} (Φ : Matrix (Fin k) (Fin m) ℝ) [Invertible (Φᵀ * Φ)] : ⅟(Φᵀ * Φ) * Φᵀ * 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 {k m : ℕ} (Φ : Matrix (Fin k) (Fin m) ℝ) [Invertible (Φᵀ * Φ)] : ⅟(Φᵀ * Φ) * Φᵀ * 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]
variable {α κ Ω : Type*} [Fintype α] [DecidableEq α] [Fintype κ] [DecidableEq κ]
  {mΩ : MeasurableSpace Ω}

Formal statement

theorem BookProof.ChapterF4.pseudoInverse_left_inverse {k m : ℕ} (Φ : Matrix (Fin k) (Fin m) ℝ)
    [Invertible (Φᵀ * Φ)] :
    ⅟(Φᵀ * Φ) * Φᵀ * Φ = 1 := by sorry

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