Theorems · Theorem · linear algebra
Submodule.eq_top_of_finrank_eq
∀ {K : Type u} {V : Type v} [inst : DivisionRing K] [inst_1 : AddCommGroup V] [inst_2 : Module K V]
[FiniteDimensional K V] {S : Submodule K V}, Module.finrank K ↥S = Module.finrank K V → S = ⊤If a submodule has maximal dimension in a finite-dimensional space, then it is equal to the whole space.
- Cited by
- 15 results in Mathlib
- Foundations
- Depth 117 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites46
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Setproof · cited by 53,352
- Modulestatement and proof · cited by 20,661
- AddCommGroupstatement and proof · cited by 12,871
- Top.topstatement and proof · cited by 9,680
- Fintypeproof · cited by 7,736
- Submodulestatement and proof · cited by 7,192
- Set.Elemproof · cited by 7,166
- Set.imageproof · cited by 5,609
- Set.rangeproof · cited by 4,705
- Disjointproof · cited by 2,201
- le_reflproof · cited by 2,061
Cited by15
Results whose statement or proof uses this declaration.
- Submodule.eq_of_le_of_finrank_leproof · cited by 10
- LinearMap.surjective_of_injectiveproof · cited by 4
- LinearMap.BilinForm.isCompl_orthogonal_of_restrict_nondegenerateproof · cited by 3
- Submodule.eq_top_of_disjointproof · cited by 3
- LinearMap.continuous_of_isClosed_kerproof · cited by 2
- LinearMap.BilinForm.isCompl_orthogonal_iff_disjointproof · cited by 2
- is_simple_module_of_finrank_eq_oneproof · cited by 2
- LieAlgebra.engel_isBot_of_isMinproof · cited by 1
- RootPairing.rootSpan_eq_top_iffproof · cited by 1
- Subalgebra.isSimpleOrder_of_finrankproof · cited by 1
- LinearMap.injective_iff_surjective_of_finrank_eq_finrankproof · cited by 1
- exists_smul_eq_of_finrank_eq_oneproof · cited by 1