Mul Operator add eq
QuantumMechanics.SpaceDHilbertSpace.mulOperator_add_eq
Plain-language statement
(𝓜 μ g).domain = ⊤ is a sufficient condition to ensure equality in mulOperator_add_ge.
Exact Lean statement
@[simp]
lemma mulOperator_add_eq
{μ : Measure (Space d)} (f : Space d → ℂ) {g : Space d → ℂ} (h : (𝓜 μ g).domain = ⊤) :
𝓜 μ (f + g) = 𝓜 μ f + 𝓜 μ gFormal artifact
Lean source
@[simp]lemma mulOperator_add_eq {μ : Measure (Space d)} (f : Space d → ℂ) {g : Space d → ℂ} (h : (𝓜 μ g).domain = ⊤) : 𝓜 μ (f + g) = 𝓜 μ f + 𝓜 μ g := by have hle := mulOperator_add_ge μ f g refine (eq_of_le_of_domain_eq hle ?_).symm refine eq_of_le_of_ge hle.1 fun ψ hψ ↦ ?_ have hg : ψ ∈ (𝓜 μ g).domain := by simp [h] simp only [add_domain, Submodule.mem_inf, mem_mulOperator_domain_iff] at * exact ⟨by simpa [add_mul] using hψ.sub hg, hg⟩- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/QuantumMechanics/Operators/Multiplication.lean:424-433
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