Same eq det to Self Adjoint
Lorentz.contrContrContractField.same_eq_det_toSelfAdjoint
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
lemma same_eq_det_toSelfAdjoint (x : ContrMod 3) :
⟪x, x⟫ₘ = det (ContrMod.toSelfAdjoint x).1Formal artifact
Lean source
lemma same_eq_det_toSelfAdjoint (x : ContrMod 3) : ⟪x, x⟫ₘ = det (ContrMod.toSelfAdjoint x).1 := by rw [ContrMod.toSelfAdjoint_apply_coe, as_sum_toSpace, det_fin_two, PauliMatrix.pauliMatrix, PauliMatrix.pauliMatrix, PauliMatrix.pauliMatrix, PauliMatrix.pauliMatrix, ContrMod.toSpace, ContrMod.toFin1dℝ_eq_val] simp only [Fin.isValue, PiLp.inner_apply, Fin.sum_univ_three, ofReal_sub, ofReal_mul, smul_of, smul_cons, smul_zero, real_smul, mul_one, smul_empty, smul_neg, Matrix.sub_apply, Matrix.smul_apply, one_apply_eq, of_apply, cons_val', cons_val_zero, cons_val_fin_one, sub_zero, cons_val_one, sub_neg_eq_add, ne_eq, zero_ne_one, not_false_eq_true, one_apply_ne, zero_sub, one_ne_zero] ring_nf simp only [Fin.isValue, Function.comp_apply, inner_self_eq_norm_sq_to_K, Real.norm_eq_abs, RCLike.ofReal_real_eq_id, id_eq, sq_abs, ofReal_add, ofReal_pow, I_sq, mul_neg, mul_one] ring- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/Relativity/Tensors/RealTensor/Vector/Pre/Contraction.lean:422-436
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