Theorems · Theorem · linear algebra
Module.nonempty_linearEquiv_iff_rank_eq
∀ {R : Type u} {M M₁ : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] [Module.Free R M]
[inst_4 : AddCommMonoid M₁] [inst_5 : Module R M₁] [Module.Free R M₁] [StrongRankCondition R],
Nonempty (M ≃ₗ[R] M₁) ↔ Module.rank R M = Module.rank R M₁Two vector spaces are isomorphic if and only if they have the same dimension.
- Defined in
- Mathlib.LinearAlgebra.Dimension.Free
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 113 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
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
- RingHom.idstatement and proof · cited by 18,349
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- LinearEquivstatement and proof · cited by 3,317
- Cardinalstatement · cited by 2,598
- Module.Freestatement and proof · cited by 597
- Module.rankstatement and proof · cited by 496
- StrongRankConditionstatement and proof · cited by 286
- LinearEquiv.rank_eqproof · cited by 31
- nonempty_linearEquiv_of_rank_eqproof · cited by 1
Cited by4
Results whose statement or proof uses this declaration.
- LinearEquiv.nonempty_equiv_iff_rank_eqproof · cited by 0
- MulOpposite.rankproof · cited by 0
- Submodule.exists_linearEquiv_restrict_eqproof · cited by 0
- Complex.nonempty_linearEquiv_realproof · cited by 0