Theorems · Theorem · functional analysis
CFC.sqrt_mul_sqrt_self
∀ {A : Type u_1} [inst : PartialOrder A] [inst_1 : NonUnitalRing A] [inst_2 : TopologicalSpace A] [inst_3 : StarRing A]
[inst_4 : Module ℝ A] [inst_5 : SMulCommClass ℝ A A] [inst_6 : IsScalarTower ℝ A A] [inst_7 : StarOrderedRing A]
[inst_8 : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint] [inst_9 : NonnegSpectrumClass ℝ A]
[IsSemitopologicalRing A] [T2Space A] (a : A),
autoParam (0 ≤ a) CFC.sqrt_mul_sqrt_self._auto_1 → CFC.sqrt a * CFC.sqrt a = a- Cited by
- 11 results in Mathlib
- Foundations
- Depth 212 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites17
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
- TopologicalSpacestatement and proof · cited by 24,529
- Modulestatement and proof · cited by 20,661
- PartialOrderstatement and proof · cited by 6,410
- IsScalarTowerstatement and proof · cited by 3,896
- SMulCommClassstatement and proof · cited by 1,927
- StarRingstatement and proof · cited by 1,686
- T2Spacestatement and proof · cited by 1,351
- StarOrderedRingstatement and proof · cited by 587
- IsSelfAdjointstatement and proof · cited by 545
- NonUnitalRingstatement and proof · cited by 422
- NonnegSpectrumClassstatement and proof · cited by 292
Cited by11
Results whose statement or proof uses this declaration.
- CStarAlgebra.isStrictlyPositive_TFAEproof · cited by 7
- CStarAlgebra.nonneg_TFAEproof · cited by 6
- CFC.abs_mul_absproof · cited by 6
- CFC.norm_star_mul_mul_self_of_nonnegproof · cited by 3
- ProbabilityTheory.covarianceBilin_multivariateGaussianproof · cited by 3
- CFC.conjSqrt_oneproof · cited by 1
- CFC.sq_sqrtproof · cited by 1
- IsStarProjection.mul_right_and_mul_left_of_nonneg_of_leproof · cited by 1
- CFC.mul_self_eqproof · cited by 1
- le_iff_norm_sqrt_mul_rpowproof · cited by 1
- Unitization.sqrt_inrproof · cited by 0