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Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Mul Operator comp Restricted eq

QuantumMechanics.SpaceDHilbertSpace.mulOperator_compRestricted_eq

Plain-language statement

(𝓜 μ g).domain = ⊤ is a sufficient condition to ensure equality in mulOperator_compRestricted_ge.

Exact Lean statement

lemma mulOperator_compRestricted_eq
    {μ : Measure (Space d)} (f : Space d → ℂ) {g : Space d → ℂ} (h : (𝓜 μ g).domain = ⊤) :
    𝓜 μ f ∘ᵣ 𝓜 μ g = 𝓜 μ (f • g)

Formal artifact

Lean source

Canonical source
Full Lean sourceLean 4
lemma mulOperator_compRestricted_eq    {μ : Measure (Space d)} (f : Space d  ℂ) {g : Space d  ℂ} (h : (𝓜 μ g).domain = ⊤) :    𝓜 μ f ∘ᵣ 𝓜 μ g = 𝓜 μ (f • g) := by  have hle := mulOperator_compRestricted_le μ f g  refine eq_of_le_of_domain_eq hle ?_  refine eq_of_le_of_ge hle.1 fun ψ hψ  ?_  refine mem_compRestricted_domain_iff.mpr h ▸ Submodule.mem_top, ?_  refine (mem_mulOperator_domain_iff.mp hψ).ae_eq ?_  filter_upwards [mulOperator_apply_ae ψ, h ▸ Submodule.mem_top]  simp_all [mul_assoc]
Project
Physlib
License
Apache-2.0
Commit
dd43e9e65791
Source
Physlib/QuantumMechanics/Operators/Multiplication.lean:485-494

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Related declarations

Project-declaredLean 4.32.0

Adiabatic relation log

adiabatic_relation_log

Plain-language statement

Adiabatic relation in logarithmic form: If S(Ua,Va,N) = S(Ub,Vb,N) with N fixed, then c * log (Ua/Ub) + log (Va/Vb) = 0.

physicsquantum field theoryrelativity

Source project: Physlib

Person-level attribution pending.

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Project-declaredLean 4.32.0

Adiabatic relation Ua Ub Va Vb

adiabatic_relation_UaUbVaVb

Plain-language statement

Adiabatic relation in product form: If S(Ua,Va,N) = S(Ub,Vb,N) with N fixed, then (Ua/Ub)^c * (Va/Vb) = 1.

physicsquantum field theoryrelativity

Source project: Physlib

Person-level attribution pending.

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