Theorems · Theorem · linear algebra
nonneg_iff_realPart_imaginaryPart
∀ {A : Type u_1} [inst : NonUnitalRing A] [inst_1 : StarRing A] [inst_2 : PartialOrder A] [StarOrderedRing A]
[inst_4 : Module ℂ A] [inst_5 : StarModule ℂ A] {a : A}, 0 ≤ a ↔ 0 ≤ realPart a ∧ imaginaryPart a = 0- Defined in
- Mathlib.LinearAlgebra.Complex.Module
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 116 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites23
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Realstatement · cited by 25,697
- Modulestatement and proof · cited by 20,661
- RingHom.idstatement · cited by 18,349
- LinearMapstatement · cited by 10,215
- PartialOrderstatement and proof · cited by 6,410
- Complexstatement and proof · cited by 5,565
- AddSubgroupstatement · cited by 3,232
- add_zeroproof · cited by 2,707
- StarRingstatement and proof · cited by 1,686
- Complex.Iproof · cited by 866
- smul_zeroproof · cited by 665
Cited by3
Results whose statement or proof uses this declaration.
- le_iff_realPart_imaginaryPartproof · cited by 2
- nonpos_iff_realPart_imaginaryPartproof · cited by 1
- realPart_nonneg_of_nonnegproof · cited by 0