Std Basis decomp
Lorentz.ContrMod.stdBasis_decomp
Plain-language statement
Decomposition of a contravariant Lorentz vector into the standard basis.
Exact Lean statement
lemma stdBasis_decomp (v : ContrMod d) : v = ∑ i, v.toFin1dℝ i • stdBasis i
Formal artifact
Lean source
lemma stdBasis_decomp (v : ContrMod d) : v = ∑ i, v.toFin1dℝ i • stdBasis i := by apply toFin1dℝEquiv.injective simp only [map_sum, _root_.map_smul] funext μ rw [Fintype.sum_apply μ fun c => toFin1dℝEquiv v c • toFin1dℝEquiv (stdBasis c)] change _ = ∑ x : Fin 1 ⊕ Fin d, toFin1dℝEquiv v x • (toFin1dℝEquiv (stdBasis x) μ) rw [Finset.sum_eq_single_of_mem μ (Finset.mem_univ μ)] · simp only [stdBasis_toFin1dℝEquiv_apply_same, smul_eq_mul, mul_one] · intro b _ hbμ rw [stdBasis_toFin1dℝEquiv_apply_ne hbμ] simp only [smul_eq_mul, mul_zero]- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/Relativity/Tensors/RealTensor/Vector/Pre/Modules.lean:117-127
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