English
If P and Q are L-projections on X and the action is faithful, then the product P Q is again an L-projection.
Русский
Если P и Q — L-проекции на X и действие сохраняет проклятия, то произведение P Q —Again L-проекция.
LaTeX
$$If $P,Q$ are L-projections on $X$ with faithful scalar action, then $PQ$ is an L-projection.$$
Lean4
theorem mul [FaithfulSMul M X] {P Q : M} (h₁ : IsLprojection X P) (h₂ : IsLprojection X Q) : IsLprojection X (P * Q) :=
by
refine ⟨IsIdempotentElem.mul_of_commute (h₁.commute h₂) h₁.proj h₂.proj, ?_⟩
intro x
refine le_antisymm ?_ ?_
·
calc
‖x‖ = ‖(P * Q) • x + (x - (P * Q) • x)‖ := by rw [add_sub_cancel ((P * Q) • x) x]
_ ≤ ‖(P * Q) • x‖ + ‖x - (P * Q) • x‖ := by apply norm_add_le
_ = ‖(P * Q) • x‖ + ‖(1 - P * Q) • x‖ := by rw [sub_smul, one_smul]
·
calc
‖x‖ = ‖P • Q • x‖ + (‖Q • x - P • Q • x‖ + ‖x - Q • x‖) := by
rw [h₂.Lnorm x, h₁.Lnorm (Q • x), sub_smul, one_smul, sub_smul, one_smul, add_assoc]
_ ≥ ‖P • Q • x‖ + ‖Q • x - P • Q • x + (x - Q • x)‖ :=
((add_le_add_iff_left ‖P • Q • x‖).mpr (norm_add_le (Q • x - P • Q • x) (x - Q • x)))
_ = ‖(P * Q) • x‖ + ‖(1 - P * Q) • x‖ := by rw [sub_add_sub_cancel', sub_smul, one_smul, mul_smul]
