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Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Levi Civita basis repr eq zero of eq

realLorentzTensor.leviCivita_basis_repr_eq_zero_of_eq

Plain-language statement

The Levi-Civita tensor vanishes on any multi-index with a repeated value: if two distinct index positions i ≠ j carry the same basis index, the component is zero.

Exact Lean statement

lemma leviCivita_basis_repr_eq_zero_of_eq
    {b : ComponentIdx (S := realLorentzTensor 3) ![Color.up, Color.up, Color.up, Color.up]}
    {i j : Fin 4} (hij : i ≠ j) (h : b i = b j) :
    (Tensor.basis _).repr ε4 b = 0

Formal artifact

Lean source

Canonical source
Full Lean sourceLean 4
lemma leviCivita_basis_repr_eq_zero_of_eq    {b : ComponentIdx (S := realLorentzTensor 3) ![Color.up, Color.up, Color.up, Color.up]}    {i j : Fin 4} (hij : i  j) (h : b i = b j) :    (Tensor.basis _).repr ε4 b = 0 := by  rw [leviCivita_basis_repr_apply]  have hdet : generalizedKroneckerDelta (fun i => finSumFinEquiv (b i))      (id : Fin 4  Fin 4) = 0 := by    rw [show generalizedKroneckerDelta (fun i => finSumFinEquiv (b i)) (id : Fin 4  Fin 4)          = Matrix.det (fun a c => ((kroneckerDelta (finSumFinEquiv (b a)) (id c) : ) : ))          from rfl]    refine Matrix.det_zero_of_row_eq hij (funext fun c => ?_)    rw [congrArg (⇑finSumFinEquiv) h]  rw [hdet, Int.cast_zero]
Project
Physlib
License
Apache-2.0
Commit
dd43e9e65791
Source
Physlib/Relativity/Tensors/LeviCivita/Basic.lean:101-113

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Plain-language statement

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physicsquantum field theoryrelativity

Source project: Physlib

Person-level attribution pending.

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Project-declaredLean 4.32.0

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Plain-language statement

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Source project: Physlib

Person-level attribution pending.

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