Theorems · Theorem · linear algebra
realPart_add_I_smul_imaginaryPart
∀ {A : Type u_1} [inst : AddCommGroup A] [inst_1 : Module ℂ A] [inst_2 : StarAddMonoid A] [inst_3 : StarModule ℂ A]
(a : A), ↑(realPart a) + Complex.I • ↑(imaginaryPart a) = aThe standard decomposition of ℜ a + Complex.I • ℑ a = a of an element of a star module over
ℂ into a linear combination of self adjoint elements.
- Defined in
- Mathlib.LinearAlgebra.Complex.Module
- Cited by
- 10 results in Mathlib
- Foundations
- Depth 114 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites28
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 and proof · cited by 25,697
- Modulestatement and proof · cited by 20,661
- RingHom.idstatement · cited by 18,349
- AddCommGroupstatement and proof · cited by 12,871
- LinearMapstatement · cited by 10,215
- Complexstatement and proof · cited by 5,565
- AddSubgroupstatement · cited by 3,232
- one_smulproof · cited by 1,374
- Star.starproof · cited by 1,082
- neg_negproof · cited by 960
- Complex.Istatement and proof · cited by 866
Cited by10
Results whose statement or proof uses this declaration.
- nonneg_iff_realPart_imaginaryPartproof · cited by 3
- cfcₙ_im_idproof · cited by 3
- cfc_im_idproof · cited by 3
- CStarAlgebra.linear_combination_nonnegproof · cited by 2
- ComplexStarModule.extproof · cited by 1
- CStarAlgebra.norm_smul_two_inv_smul_add_four_unitaryproof · cited by 1
- star_mul_self_add_self_mul_starproof · cited by 1
- star_mul_self_eq_realPart_sq_add_imaginaryPart_sqproof · cited by 1
- star_mul_self_sub_self_mul_starproof · cited by 1
- span_selfAdjointproof · cited by 0