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Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Mul Operator smul ge

QuantumMechanics.SpaceDHilbertSpace.mulOperator_smul_ge

Plain-language statement

Scalar multiplication and mulOperator commute except possibly for c = 0 where the domains of 0 • 𝓜 μ f and 𝓜 μ 0 = 0 may not agree. See mulOperator_smul_eq for equality when c ≠ 0.

Exact Lean statement

lemma mulOperator_smul_ge (μ : Measure (Space d)) (c : ℂ) (f : Space d → ℂ) :
    c • 𝓜 μ f ≤ 𝓜 μ (c • f)

Formal artifact

Lean source

Canonical source
Full Lean sourceLean 4
lemma mulOperator_smul_ge (μ : Measure (Space d)) (c : ℂ) (f : Space d  ℂ) :    c • 𝓜 μ f  𝓜 μ (c • f) := by  refine le_of_le_graph fun u h  ?_  rw [mem_graph_iff] at *  obtain ⟨⟨v, hv, hvu, hvu' := h  have hv' : v  (𝓜 μ (c • f)).domain := by    rw [smul_domain, mem_mulOperator_domain_iff] at *    simpa using hv.const_smul c  refine ⟨⟨v, hv', hvu, ?_  rw [ hvu', ext_iff]  filter_upwards [mulOperator_apply_ae v, hv, mulOperator_apply_ae v, hv',    coeFn_smul c (𝓜 μ f v, hv)]  simp_all [mul_assoc]
Project
Physlib
License
Apache-2.0
Commit
dd43e9e65791
Source
Physlib/QuantumMechanics/Operators/Multiplication.lean:370-382

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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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