Theorems · Theorem · linear algebra
rank_subsingleton
∀ (R : Type u) (M : Type v) [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] [Subsingleton R], Module.rank R M = 1
- Defined in
- Mathlib.LinearAlgebra.Dimension.Basic
- Cited by
- 19 results in Mathlib
- Foundations
- Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setproof · cited by 53,352
- Modulestatement and proof · cited by 20,661
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- Cardinalstatement and proof · cited by 2,598
- Module.rankstatement · cited by 496
- LinearIndepOnproof · cited by 211
- LT.lt.trans_eqproof · cited by 65
- Module.rank_defproof · cited by 21
- Module.subsingletonproof · cited by 20
- Cardinal.mk_singletonproof · cited by 9
- ciSup_eq_of_forall_le_of_forall_lt_exists_gtproof · cited by 8
Cited by19
Results whose statement or proof uses this declaration.
- Module.finrank_subsingletonproof · cited by 11
- IsLocalizedModule.lift_rank_eqproof · cited by 7
- IsLocalization.rank_eqproof · cited by 7
- Submodule.LinearDisjoint.rank_inf_le_one_of_commute_of_flatproof · cited by 4
- Module.one_le_rank_iffproof · cited by 4
- rank_eq_zero_iffproof · cited by 3
- Subalgebra.LinearDisjoint.rank_inf_eq_one_of_commute_of_flat_of_injproof · cited by 3
- IsBaseChange.lift_rank_eqproof · cited by 3
- Module.length_of_freeproof · cited by 3
- Subalgebra.LinearDisjoint.rank_sup_of_freeproof · cited by 2
- CommSemiring.rank_selfproof · cited by 2
- Subalgebra.rank_sup_eq_rank_left_mul_rank_of_freeproof · cited by 2