Prove2Me

`BookProof.ChapterA3q.tensorPow_complete_reducibility` {N : ℕ} (hN : 2 ≤ N) : projSym N + projAnti N + projMixed N = 1 ∧ projSym N * projSym N = projSym N ∧ projAnti N * projAnti N

Proved
BookProof.ChapterA3q.tensorPow_complete_reducibility

by leonardopedro · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterA3q.

BookProof.ChapterA3q.tensorPow_complete_reducibility {N : ℕ} (hN : 2 ≤ N) : projSym N + projAnti N + projMixed N = 1 ∧ projSym N * projSym N = projSym N ∧ projAnti N * projAnti N = projAnti N ∧ projMixed N * projMixed N = projMixed N ∧ projSym N * projAnti N = 0 ∧ projAnti N * projSym N = 0 ∧ projSym N * projMixed N = 0 ∧ projMixed N * projSym N = 0 ∧ projAnti N * projMixed N = 0 ∧ projMixed N * projAnti N = 0

Formalization note: Lean 4 identifier BookProof.ChapterA3q.tensorPow_complete_reducibility.

Preamble
-- Generated from ChapterA3q.lean — theorem BookProof.ChapterA3q.tensorPow_complete_reducibility
import Definitions.Def_ChapterA3
import Definitions.Def_ChapterA3j
import Definitions.Def_ChapterA3p
import Mathlib
import Definitions.Def_ChapterA3q
import Definitions.Def_ChapterA3l
import Definitions.Def_ChapterA3n
import Definitions.Def_ChapterA3o
open BookProof.ChapterA3l
open BookProof.ChapterA3n
open BookProof.ChapterA3o
open BookProof.ChapterA3q


open Matrix
open scoped BigOperators


open BookProof.ChapterA3 BookProof.ChapterA3j BookProof.ChapterA3n BookProof.ChapterA3o
open BookProof.ChapterA3p
Formal statement
theorem BookProof.ChapterA3q.tensorPow_complete_reducibility {N : ℕ} (hN : 2 ≤ N) :
    projSym N + projAnti N + projMixed N = 1 ∧
    projSym N * projSym N = projSym N ∧
    projAnti N * projAnti N = projAnti N ∧
    projMixed N * projMixed N = projMixed N ∧
    projSym N * projAnti N = 0 ∧ projAnti N * projSym N = 0 ∧
    projSym N * projMixed N = 0 ∧ projMixed N * projSym N = 0 ∧
    projAnti N * projMixed N = 0 ∧ projMixed N * projAnti N = 0 := by sorry

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