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Theorems · Theorem · functional analysis

IsStrictlyPositive.of_le

∀ {A : Type u_1} [inst : CStarAlgebra A] [inst_1 : PartialOrder A] [StarOrderedRing A] {a b : A},
  IsStrictlyPositive a → a ≤ b → IsStrictlyPositive b
Defined in
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order
Cited by
4 results in Mathlib
Foundations
Depth 318 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
CStarAlgebraPartialOrderStarOrderedRing
Mathlib docsSourceLoogleLeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

IsStrictlyPositive.add_nonneg · cited by 1IsStrictlyPositive.add_no…CStarAlgebra.convexOn_ringInverse_algebraMap_add · cited by 1CStarAlgebra.convexOn_rin…CStarAlgebra.ringInverse_le_ringInverse · cited by 0CStarAlgebra.ringInverse_…CFC.log_le_log · cited by 0CFC.log_le_logPartialOrder · cited by 6410PartialOrderLE.le.trans · cited by 3151le.transStarOrderedRing · cited by 587StarOrderedRingCStarAlgebra · cited by 123CStarAlgebraIsStrictlyPositive · cited by 75IsStrictlyPositiveIsStrictlyPositive.nonneg · cited by 18IsStrictlyPositive.nonnegCStarAlgebra.isUnit_of_le · cited by 3CStarAlgebra.isUnit_of_leIsStrictlyPositive.of_leCITED BYCITES

Cites7

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Cited by4

Results whose statement or proof uses this declaration.