Prove2Me

`BookProof.ChapterA3x.tensorCube_complete_reducibility` : projSym 3 + projAnti 3 + projMixed 3 = 1 ∧ projSym 3 * projSym 3 = projSym 3 ∧ projAnti 3 * projAnti 3 = projAnti 3 ∧ proj

Proved
BookProof.ChapterA3x.tensorCube_complete_reducibility

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterA3x.

BookProof.ChapterA3x.tensorCube_complete_reducibility : projSym 3 + projAnti 3 + projMixed 3 = 1 ∧ projSym 3 * projSym 3 = projSym 3 ∧ projAnti 3 * projAnti 3 = projAnti 3 ∧ projMixed 3 * projMixed 3 = projMixed 3 ∧ projSym 3 * projAnti 3 = 0 ∧ projAnti 3 * projSym 3 = 0 ∧ projSym 3 * projMixed 3 = 0 ∧ projMixed 3 * projSym 3 = 0 ∧ projAnti 3 * projMixed 3 = 0 ∧ projMixed 3 * projAnti 3 = 0 ∧ projMixed 3 ≠ 0

Formalization note: Lean 4 identifier BookProof.ChapterA3x.tensorCube_complete_reducibility.

Preamble
-- Generated from ChapterA3x.lean — theorem BookProof.ChapterA3x.tensorCube_complete_reducibility
import Definitions.Def_ChapterA3
import Definitions.Def_ChapterA3j
import Definitions.Def_ChapterA3p
import Mathlib
import Definitions.Def_ChapterA3x
import Definitions.Def_ChapterA3l
import Definitions.Def_ChapterA3n
import Definitions.Def_ChapterA3o
open BookProof.ChapterA3l
open BookProof.ChapterA3n
open BookProof.ChapterA3o
open BookProof.ChapterA3x


open Matrix
open scoped BigOperators


open BookProof.ChapterA3 BookProof.ChapterA3j BookProof.ChapterA3n BookProof.ChapterA3o
open BookProof.ChapterA3p
Formal statement
theorem BookProof.ChapterA3x.tensorCube_complete_reducibility :
    projSym 3 + projAnti 3 + projMixed 3 = 1 ∧
    projSym 3 * projSym 3 = projSym 3 ∧
    projAnti 3 * projAnti 3 = projAnti 3 ∧
    projMixed 3 * projMixed 3 = projMixed 3 ∧
    projSym 3 * projAnti 3 = 0 ∧ projAnti 3 * projSym 3 = 0 ∧
    projSym 3 * projMixed 3 = 0 ∧ projMixed 3 * projSym 3 = 0 ∧
    projAnti 3 * projMixed 3 = 0 ∧ projMixed 3 * projAnti 3 = 0 ∧
    projMixed 3 ≠ 0 := by sorry

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