Theorems · Theorem · ring theory
IsSelfAdjoint.smul_mem_skewAdjoint
∀ {R : Type u_1} {A : Type u_2} [inst : Ring R] [inst_1 : AddCommGroup A] [inst_2 : Module R A]
[inst_3 : StarAddMonoid R] [inst_4 : StarAddMonoid A] [StarModule R A] {r : R},
r ∈ skewAdjoint R → ∀ {a : A}, IsSelfAdjoint a → r • a ∈ skewAdjoint AScalar multiplication of a self-adjoint element by a skew-adjoint element produces a skew-adjoint element.
- Defined in
- Mathlib.Algebra.Star.SelfAdjoint
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 22 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Modulestatement and proof · cited by 20,661
- AddCommGroupstatement and proof · cited by 12,871
- Ringstatement and proof · cited by 7,463
- AddSubgroupstatement · cited by 3,232
- StarModulestatement and proof · cited by 570
- IsSelfAdjointstatement and proof · cited by 545
- neg_smulproof · cited by 306
- StarAddMonoidstatement and proof · cited by 296
- skewAdjointstatement and proof · cited by 34
- StarModule.star_smulproof · cited by 34
Cited by2
Results whose statement or proof uses this declaration.
- IsSelfAdjoint.mem_spectrum_eq_reproof · cited by 3
- IsSelfAdjoint.I_smul_mem_skewAdjointproof · cited by 0