Theorems · Theorem · linear algebra
LinearEquiv.rank_eq
∀ {R : Type u} {M M₁ : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
[inst_3 : AddCommMonoid M₁] [inst_4 : Module R M₁] (f : M ≃ₗ[R] M₁), Module.rank R M = Module.rank R M₁Two linearly equivalent vector spaces have the same dimension.
- Defined in
- Mathlib.LinearAlgebra.Dimension.Basic
- Cited by
- 31 results in Mathlib
- Foundations
- Depth 89 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
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.rankstatement · cited by 496
- Cardinal.lift_injproof · cited by 38
- LinearEquiv.lift_rank_eqproof · cited by 14
Cited by31
Results whose statement or proof uses this declaration.
- rank_topproof · cited by 9
- Module.nonempty_linearEquiv_iff_rank_eqproof · cited by 4
- ModularForm.rank_eq_one_add_rank_cuspFormproof · cited by 3
- Subalgebra.LinearDisjoint.rank_sup_of_freeproof · cited by 2
- Subfield.relrank_top_rightproof · cited by 2
- CuspForm.rank_eq_zero_of_weight_lt_twelveproof · cited by 2
- Field.sepDegree_eq_of_equivproof · cited by 2
- IntermediateField.adjoin_rank_le_of_isAlgebraicproof · cited by 2
- Module.rank_linearMapproof · cited by 2
- rank_range_of_injectiveproof · cited by 2
- Submodule.rank_sup_add_rank_inf_eqproof · cited by 2
- rank_dual_eq_card_dual_of_aleph0_le_rank'proof · cited by 2