| import Mathlib | |
| import OAI.Analysis.Conductivity.Model | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory | |
| open scoped ENNReal | |
| namespace ScalarConductivity | |
| theorem det_id_add_rankOne {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| [FiniteDimensional ℝ E] (d : E) (L : E →L[ℝ] ℝ) : | |
| (ContinuousLinearMap.id ℝ E + L.smulRight d).det = 1 + L d := by | |
| let b := Module.finBasis ℝ E | |
| change LinearMap.det (LinearMap.id + L.toLinearMap.smulRight d) = _ | |
| rw [← LinearMap.det_toMatrix b, map_add, LinearMap.toMatrix_id, | |
| LinearMap.toMatrix_smulRight, Matrix.vecMulVec_eq Unit, | |
| Matrix.det_one_add_replicateCol_mul_replicateRow] | |
| congr 1 | |
| conv_rhs => rw [← b.sum_repr d, map_sum] | |
| simp only [dotProduct, Function.comp_apply, map_smul, smul_eq_mul] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| change L (b i) * (b.repr d) i = (b.repr d) i * L (b i) | |
| ring | |
| end ScalarConductivity | |
| end | |
| end OAI | |