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Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Prod eq sum eval

TensorSpecies.Tensorial.prod_eq_sum_eval

physicsquantum field theoryrelativitytensorial

Plain-language statement

Double basis expansion of an element of a tensor product M ⊗[k] M₂ of two Tensorial one-index spaces. Given bases b, b2 of M, M₂ coming from the single-index tensor bases, every x : M ⊗[k] M₂ is the double sum over i, j of the iterated evaluation coefficient toField (evalT 0 j (evalT 0 i (toTensor x))) times b i ⊗ₜ b2 j.

Exact Lean statement

lemma prod_eq_sum_eval {c c2 : C} {M₂ : Type}
    [Tensorial S ![c] M] [AddCommMonoid M₂] [Module k M₂]
    [Tensorial S ![c2] M₂] {b : Module.Basis (basisIdx c) k M}
    {b2 : Module.Basis (basisIdx c2) k M₂}
    (h : b = ((Tensor.basis (S := S) ![c]).map toTensor.symm).reindex ComponentIdx.single)
    (h2 : b2 = ((Tensor.basis (S := S) ![c2]).map toTensor.symm).reindex ComponentIdx.single)
    (x : M ⊗[k] M₂) : x = ∑ i : (basisIdx c), ∑ j : (basisIdx c2),
    toField (evalT 0 (basisIdxCongr (by rfl) j)
      (evalT 0 (basisIdxCongr (by rfl) i) (toTensor x))) • b i ⊗ₜ[k] b2 j

Formal artifact

Lean source

Canonical source
Full Lean sourceLean 4
lemma prod_eq_sum_eval {c c2 : C} {M₂ : Type}    [Tensorial S ![c] M] [AddCommMonoid M₂] [Module k M₂]    [Tensorial S ![c2] M₂] {b : Module.Basis (basisIdx c) k M}    {b2 : Module.Basis (basisIdx c2) k M₂}    (h : b = ((Tensor.basis (S := S) ![c]).map toTensor.symm).reindex ComponentIdx.single)    (h2 : b2 = ((Tensor.basis (S := S) ![c2]).map toTensor.symm).reindex ComponentIdx.single)    (x : M ⊗[k] M₂) : x = ∑ i : (basisIdx c), ∑ j : (basisIdx c2),    toField (evalT 0 (basisIdxCongr (by rfl) j)      (evalT 0 (basisIdxCongr (by rfl) i) (toTensor x))) • b i ⊗ₜ[k] b2 j := by  apply toTensor.injective  conv_lhs => rw [eq_sum_evalT_zero (toTensor x)]  simp only [map_sum, map_smul]  refine Finset.sum_congr rfl fun i _ => ?_  rw [eq_sum_evalT_zero ((evalT 0 i) (toTensor x))]  simp only [map_sum]  refine Finset.sum_congr rfl fun j _ => ?_  simp only [prodT_zero_right, map_smul, permT_basis, prodT_basis']  congr 1  subst h h2  simp only [toTensor_tprod, Function.comp_apply, Module.Basis.coe_reindex, Module.Basis.map_apply,    LinearEquiv.apply_symm_apply, prodT_basis']  congr 1  funext i  fin_cases i <;> rfl
Project
Physlib
License
Apache-2.0
Commit
dd43e9e65791
Source
Physlib/Relativity/Tensors/Tensorial.lean:330-353

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Plain-language statement

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Person-level attribution pending.

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