Theorems · Theorem · commutative algebra
Submodule.LinearDisjoint.injective
∀ {R : Type u} {S : Type v} [inst : CommSemiring R] [inst_1 : Semiring S] [inst_2 : Algebra R S] {M N : Submodule R S},
M.LinearDisjoint N → Function.Injective ⇑(M.mulMap N)- Defined in
- Mathlib.LinearAlgebra.LinearDisjoint
- Cited by
- 11 results in Mathlib
- Foundations
- Depth 58 from the axioms · uses propext, Quot.sound
- Assumes
- CommSemiringSemiringAlgebra
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- RingHom.idstatement · cited by 18,349
- Semiringstatement and proof · cited by 13,802
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- LinearMapstatement · cited by 10,215
- Submodulestatement and proof · cited by 7,192
- TensorProductstatement · cited by 2,545
- Submodule.LinearDisjointstatement and proof · cited by 54
- Submodule.mulMapstatement · cited by 43
Cited by12
Results whose statement or proof uses this declaration.
- Submodule.LinearDisjoint.of_le_left_of_flatproof · cited by 4
- Submodule.LinearDisjoint.of_le_right_of_flatproof · cited by 4
- Submodule.LinearDisjoint.symm_of_commuteproof · cited by 3
- Submodule.LinearDisjoint.linearIndependent_left_of_flatproof · cited by 2
- Submodule.LinearDisjoint.linearIndependent_mul_of_flat_leftproof · cited by 2
- Submodule.LinearDisjoint.linearIndependent_mul_of_flat_rightproof · cited by 2
- Submodule.LinearDisjoint.linearIndependent_right_of_flatproof · cited by 2
- Submodule.LinearDisjoint.mulMapproof · cited by 2
- Subalgebra.LinearDisjoint.isDomainproof · cited by 2
- Subalgebra.LinearDisjoint.of_linearDisjoint_finite_leftproof · cited by 2
- Submodule.LinearDisjoint.of_linearDisjoint_fg_leftproof · cited by 1
- Submodule.LinearDisjoint.of_linearDisjoint_fg_rightproof · cited by 1