Theorems · Theorem · linear algebra
Submodule.map_id
∀ {R : Type u_1} {M : Type u_5} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
(p : Submodule R M), Submodule.map LinearMap.id p = p- Defined in
- Mathlib.Algebra.Module.Submodule.Map
- Cited by
- 14 results in Mathlib
- Foundations
- Depth 27 from the axioms · uses propext, Quot.sound
- Assumes
- SemiringAddCommMonoidModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
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 · cited by 18,349
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- Submodulestatement and proof · cited by 7,192
- LinearMap.idstatement · cited by 625
- Submodule.mapstatement · cited by 614
- Submodule.extproof · cited by 204
Cited by14
Results whose statement or proof uses this declaration.
- AffineSubspace.Parallel.direction_eqproof · cited by 4
- LinearMap.range_negproof · cited by 4
- FractionalIdeal.map_idproof · cited by 3
- CliffordAlgebra.ι_range_map_reverseproof · cited by 2
- Submodule.map_unop_mulproof · cited by 2
- AffineSubspace.direction_shiftproof · cited by 1
- LinearMap.prod_eq_sup_mapproof · cited by 1
- ClosedSubmodule.map_idproof · cited by 1
- AffineSubspace.parallel_iff_direction_eq_and_eq_bot_iff_eq_botproof · cited by 1
- LieSubmodule.lcs_le_lcs_of_is_nilpotent_span_sup_eq_topproof · cited by 1
- Submodule.map_dualCoannihilator_linearEquiv_flipproof · cited by 1
- AffineSubspace.direction_smulproof · cited by 0