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A point in the span of the whole basis. -/ def Pa' (f : (Fin n.succ) ⊕ (Fin n) → ℚ) : (PureU1 (2 * n.succ)).LinSols := ∑ i, f i • basisa i lemma Pa'_P'_P!' (f : (Fin n.succ) ⊕ (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 -/ theorem basisa_lin...
Exact Lean statement
lemma Pa'_eq (f f' : (Fin n.succ) ⊕ (Fin n) → ℚ) : Pa' f = Pa' f' ↔ f = f'
Formal artifact
Lean source
lemma Pa'_eq (f f' : (Fin n.succ) ⊕ (Fin n) → ℚ) : Pa' f = Pa' f' ↔ f = f' := by refine Iff.intro (fun h => (funext (fun i => ?_))) (fun h => ?_) · rw [Pa', Pa'] at h have h1 : ∑ i : Fin (succ n) ⊕ Fin n, (f i + (- f' i)) • basisa i = 0 := by simp only [add_smul, neg_smul] rw [Finset.sum_add_distrib] rw [h] rw [← Finset.sum_add_distrib] simp have h2 := Fintype.linearIndependent_iff.mp basisa_linear_independent _ h1 linarith [h2 i] · rw [h]- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/QFT/QED/AnomalyCancellation/Even/BasisLinear.lean:847-858
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