Plain-language statement
Multiplying an element of SL(2, ℂ) on the left with the metric 𝓔 is equivalent to multiplying the inverse-transpose of that element on the right with the metric.
Exact Lean statement
lemma comm_metricRaw (M : SL(2,ℂ)) : M.1 * metricRaw = metricRaw * (M.1⁻¹)ᵀ
Formal artifact
Lean source
lemma comm_metricRaw (M : SL(2,ℂ)) : M.1 * metricRaw = metricRaw * (M.1⁻¹)ᵀ := by rw [metricRaw] rw [Lorentz.SL2C.inverse_coe, eta_fin_two M.1] rw [SpecialLinearGroup.coe_inv, Matrix.adjugate_fin_two, Matrix.mul_fin_two, eta_fin_two !![M.1 1 1, -M.1 0 1; -M.1 1 0, M.1 0 0]ᵀ] simp only [Fin.isValue, mul_zero, mul_neg, mul_one, zero_add, add_zero, transpose_apply, of_apply, cons_val', cons_val_zero, empty_val', cons_val_fin_one, cons_val_one, cons_mul, Nat.succ_eq_add_one, Nat.reduceAdd, vecMul_cons, head_cons, zero_smul, tail_cons, one_smul, empty_vecMul, neg_smul, neg_cons, neg_neg, neg_empty, empty_mul, Equiv.symm_apply_apply]- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/Relativity/Fermions/Weyl/Metric.lean:35-43
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