Basis left
Lorentz.contrContrContractField.basis_left
Project documentation
The metric tensor is non-degenerate. -/ lemma nondegenerate : (∀ (x : ContrMod d), ⟪x, y⟫ₘ = 0) ↔ y = 0 := by refine Iff.intro (fun h => ?) (fun h => ?) · exact (self_parity_eq_zero_iff _).mp ((symm _ _).trans $ h _) · simp [h] set_option backward.isDefEq.respectTransparency false in lemma matrix_apply_eq_iff_sub : ⟪x, Λ *ᵥ y⟫ₘ = ⟪x, Λ' *ᵥ y⟫ₘ ↔ ⟪x, (Λ...
Exact Lean statement
@[simp]
lemma basis_left {v : ContrMod d} (μ : Fin 1 ⊕ Fin d) :
⟪ ContrMod.stdBasis μ, v⟫ₘ = η μ μ * v.toFin1dℝ μFormal artifact
Lean source
@[simp]lemma basis_left {v : ContrMod d} (μ : Fin 1 ⊕ Fin d) : ⟪ ContrMod.stdBasis μ, v⟫ₘ = η μ μ * v.toFin1dℝ μ := by rw [as_sum] rcases μ with μ | μ · fin_cases μ simp only [Fin.zero_eta, Fin.isValue, ContrMod.stdBasis_apply_same, one_mul, ContrMod.stdBasis_inl_apply_inr, zero_mul, Finset.sum_const_zero, sub_zero, minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_apply_eq, Sum.elim_inl] rfl · simp only [Fin.isValue, ContrMod.stdBasis_apply, reduceCtorEq, ↓reduceIte, zero_mul, Sum.inr.injEq, ite_mul, one_mul, Finset.sum_ite_eq, Finset.mem_univ, zero_sub, minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_apply_eq, Sum.elim_inr, neg_mul, neg_inj] rfl- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/Relativity/Tensors/RealTensor/Vector/Pre/Contraction.lean:382-395
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