Theorems · Theorem · commutative algebra
Submodule.ext
∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {p q : Submodule R M},
(∀ (x : M), x ∈ p ↔ x ∈ q) → p = q- Defined in
- Mathlib.Algebra.Module.Submodule.Defs
- Cited by
- 204 results in Mathlib
- Foundations
- Depth 13 from the axioms, rests on 81 definitions · uses propext, Quot.sound
- Assumes
- SemiringAddCommMonoidModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
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
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- Submodulestatement and proof · cited by 7,192
- SetLike.extproof · cited by 92
Cited by204
Results whose statement or proof uses this declaration.
- Ideal.extproof · cited by 131
- Submodule.ker_mkQproof · cited by 63
- LinearMap.range_eq_mapproof · cited by 27
- Submodule.span_zero_singletonproof · cited by 26
- Finsupp.range_linearCombinationproof · cited by 24
- Submodule.map_comap_subtypeproof · cited by 17
- AffineSubspace.direction_mk'proof · cited by 15
- Submodule.orthogonal_orthogonalproof · cited by 14
- Submodule.map_idproof · cited by 14
- Submodule.map_equiv_eq_comap_symmproof · cited by 12
- AffineSubspace.direction_topproof · cited by 11
- FractionalIdeal.coe_zeroproof · cited by 10
Showing the 200 most cited of 204.