Theorems · Theorem · functional analysis

le_iff_norm_sqrt_mul_rpow

∀ {A : Type u_1} [inst : CStarAlgebra A] [inst_1 : PartialOrder A] [inst_2 : StarOrderedRing A] (a b : A),
  autoParam (0 ≤ a) le_iff_norm_sqrt_mul_rpow._auto_1 →
    autoParam (IsStrictlyPositive b) le_iff_norm_sqrt_mul_rpow._auto_3 → (a ≤ b ↔ ‖CFC.sqrt a * b ^ (-(1 / 2))‖ ≤ 1)

Defined in: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order
Cited by: 1 results in Mathlib
Foundations: Depth 321 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CStarAlgebra PartialOrder StarOrderedRing

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

Realstatement and proof · cited by 25,697
TopologicalSpaceproof · cited by 24,529
Algebraproof · cited by 11,388
Ringproof · cited by 7,463
PartialOrderstatement and proof · cited by 6,410
Complexstatement · cited by 5,565
Norm.normstatement and proof · cited by 5,413
mul_oneproof · cited by 3,885
LE.le.transproof · cited by 3,151
one_mulproof · cited by 2,841
Unitsproof · cited by 2,804
Nat.cast_oneproof · cited by 2,501

Cited by1

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le_iff_norm_sqrt_mul_sqrt_invproof · cited by 1