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159357f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 | 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
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