`BookProof.ChapterA3x.projSym_apply` {N : ℕ} (a b : Idx N) : projSym N a b = (Nat.factorial N : ℂ)⁻¹ * ∑ σ : Equiv.Perm (Fin N), (if b = a ∘ σ then (1 : ℂ) else 0)

Proved

BookProof.ChapterA3x.projSym_apply

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterA3x.

BookProof.ChapterA3x.projSym_apply {N : ℕ} (a b : Idx N) : projSym N a b = (Nat.factorial N : ℂ)⁻¹ * ∑ σ : Equiv.Perm (Fin N), (if b = a ∘ σ then (1 : ℂ) else 0)

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

Preamble

-- Generated from ChapterA3x.lean — theorem BookProof.ChapterA3x.projSym_apply
import Definitions.Def_ChapterA3
import Definitions.Def_ChapterA3j
import Definitions.Def_ChapterA3o
import Definitions.Def_ChapterA3p
import Mathlib
import Definitions.Def_ChapterA3x
import Definitions.Def_ChapterA3l
import Definitions.Def_ChapterA3n
open BookProof.ChapterA3l
open BookProof.ChapterA3n
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.projSym_apply {N : ℕ} (a b : Idx N) :
    projSym N a b = (Nat.factorial N : ℂ)⁻¹ * ∑ σ : Equiv.Perm (Fin N),
      (if b = a ∘ σ then (1 : ℂ) else 0) := by sorry

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