`BookProof.ChapterGravityProjDirectSum.isCompl_range_of_add_eq_id` {R V : Type*} [CommRing R] [AddCommGroup V] [Module R V] (P Q : V →ₗ[R] V) (hsum : P + Q = LinearMap.id) (hPQ : P

Proved

BookProof.ChapterGravityProjDirectSum.isCompl_range_of_add_eq_id

by leonardopedro · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterGravityProjDirectSum.

BookProof.ChapterGravityProjDirectSum.isCompl_range_of_add_eq_id {R V : Type*} [CommRing R] [AddCommGroup V] [Module R V] (P Q : V →ₗ[R] V) (hsum : P + Q = LinearMap.id) (hPQ : P.comp Q = 0) (hQP : Q.comp P = 0) : IsCompl (LinearMap.range P) (LinearMap.range Q)

Formalization note: Lean 4 identifier BookProof.ChapterGravityProjDirectSum.isCompl_range_of_add_eq_id.

Preamble

-- Generated from ChapterGravityProjDirectSum.lean — theorem BookProof.ChapterGravityProjDirectSum.isCompl_range_of_add_eq_id
import Definitions.Def_ChapterGravityProjector
import Definitions.Def_ChapterGravityTimeProj
import Mathlib
import Definitions.Def_ChapterGravityProjDirectSum
open BookProof.ChapterGravityProjDirectSum


open Matrix


open BookProof.ChapterGravityProjector
open BookProof.ChapterGravityTimeProj

Formal statement

theorem BookProof.ChapterGravityProjDirectSum.isCompl_range_of_add_eq_id {R V : Type*} [CommRing R] [AddCommGroup V] [Module R V]
    (P Q : V →ₗ[R] V) (hsum : P + Q = LinearMap.id)
    (hPQ : P.comp Q = 0) (hQP : Q.comp P = 0) :
    IsCompl (LinearMap.range P) (LinearMap.range Q) := by sorry

View graph