Theorems · Theorem · group theory
AddSubmonoid.zero_mem
∀ {M : Type u_1} [inst : AddZeroClass M] (S : AddSubmonoid M), 0 ∈ SAn AddSubmonoid contains the monoid's 0.
- Defined in
- Mathlib.Algebra.Group.Submonoid.Defs
- Cited by
- 17 results in Mathlib
- Foundations
- Depth 12 from the axioms · uses no axioms
- Assumes
- AddZeroClass
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites3
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- AddZeroClassstatement and proof · cited by 1,237
- AddSubmonoidstatement and proof · cited by 1,178
- ZeroMemClass.zero_memproof · cited by 162
Cited by17
Results whose statement or proof uses this declaration.
- AddSubgroup.closure_toAddSubmonoidproof · cited by 6
- AddSubmonoid.prod_le_iffproof · cited by 5
- AddSubmonoid.eq_bot_iff_forallproof · cited by 4
- Finsupp.add_closure_setOfPred_eq_singleproof · cited by 2
- DividedPowers.dpow_sum'proof · cited by 2
- AddSubmonoid.nontrivial_iff_exists_ne_zeroproof · cited by 2
- AddSubmonoid.addSubmonoid_smul_supproof · cited by 2
- Submodule.iSup_toAddSubmonoidproof · cited by 2
- AddMonoidAlgebra.addSubmonoidClosure_singleproof · cited by 1
- PresentedAddMonoid.closure_range_ofproof · cited by 1
- HahnSeries.SummableFamily.support_pow_subset_closureproof · cited by 1
- DFinsupp.add_closure_iUnion_range_singleproof · cited by 1