Theorems · Theorem · group theory
Matrix.GeneralLinearGroup.mem_center_iff_val_mem_range_scalar
∀ {R : Type u_1} {n : Type u_2} [inst : Fintype n] [inst_1 : DecidableEq n] [inst_2 : CommRing R] {g : GL n R},
g ∈ Subgroup.center (GL n R) ↔ ↑g ∈ Set.range ⇑(Matrix.scalar n)The center of GL n R consists of scalar matrices.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 84 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- FintypeDecidableEqCommRing
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites21
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
- Setstatement · cited by 53,352
- CommRingstatement and proof · cited by 17,173
- RingHomstatement · cited by 10,189
- Fintypestatement and proof · cited by 7,736
- Set.rangestatement and proof · cited by 4,705
- Matrixstatement and proof · cited by 4,303
- Subgroupstatement · cited by 3,593
- Units.valstatement and proof · cited by 1,966
- Matrix.GeneralLinearGroupstatement and proof · cited by 556
- Subgroup.centerstatement and proof · cited by 121
- Commute.allproof · cited by 119
Cited by2
Results whose statement or proof uses this declaration.
- Matrix.GeneralLinearGroup.center_eq_range_scalarproof · cited by 6
- Matrix.GeneralLinearGroup.mem_center_iff_val_eq_scalarproof · cited by 0