Basisa linear independent
PureU1.VectorLikeOddPlane.basisa_linear_independent
Project documentation
A point in the span of the whole basis. -/ def Pa' (f : (Fin n) ⊕ (Fin n) → ℚ) : (PureU1 (2 * n + 1)).LinSols := ∑ i, f i • basisa i lemma Pa'_P'_P!' (f : (Fin n) ⊕ (Fin n) → ℚ) : Pa' f = P' (f ∘ Sum.inl) + P!' (f ∘ Sum.inr) := by exact Fintype.sum_sum_type _ /-! ### E.6. The combined basis vectors are linearly independent
Exact Lean statement
theorem basisa_linear_independent : LinearIndependent ℚ (@basisa n.succ)
Formal artifact
Lean source
theorem basisa_linear_independent : LinearIndependent ℚ (@basisa n.succ) := by apply Fintype.linearIndependent_iff.mpr intro f h change Pa' f = 0 at h have h1 : (Pa' f).val = 0 := congrArg ACCSystemLinear.LinSols.val h rw [Pa'_P'_P!'] at h1 change (P' (f ∘ Sum.inl)).val + (P!' (f ∘ Sum.inr)).val = 0 at h1 rw [P!'_val, P'_val] at h1 change Pa (f ∘ Sum.inl) (f ∘ Sum.inr) = 0 at h1 have hf := Pa_zero (f ∘ Sum.inl) (f ∘ Sum.inr) h1 have hg := Pa_zero! (f ∘ Sum.inl) (f ∘ Sum.inr) h1 intro i simp_all only [succ_eq_add_one, Function.comp_apply] cases i <;> simp_all- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/QFT/QED/AnomalyCancellation/Odd/BasisLinear.lean:786-799
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