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Theorems · Theorem · group theory

Submonoid.ext

∀ {M : Type u_1} [inst : MulOneClass M] {S T : Submonoid M}, (∀ (x : M), x ∈ S ↔ x ∈ T) → S = T

Two submonoids are equal if they have the same elements.

Defined in
Mathlib.Algebra.Group.Submonoid.Defs
Cited by
48 results in Mathlib
Foundations
Depth 11 from the axioms · uses propext, Quot.sound
Assumes
MulOneClass
Mathlib docsSourceLoogleLeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Submonoid.powers_eq_closure · cited by 6Submonoid.powers_eq_closu…Submonoid.map_id · cited by 5Submonoid.map_idIsMulTorsion.torsion_eq_top · cited by 3IsMulTorsion.torsion_eq_t…Algebra.algebraMapSubmonoid_map_eq · cited by 3Algebra.algebraMapSubmono…nonZeroDivisorsLeft_eq_right · cited by 3nonZeroDivisorsLeft_eq_ri…Submonoid.fg_of_divisive · cited by 2Submonoid.fg_of_divisiveSubmonoid.top_prod · cited by 2Submonoid.top_prodMulEquivClass.map_nonZeroDivisors · cited by 2MulEquivClass.map_nonZero…Con.lift_range · cited by 2Con.lift_rangeUnits.unitary_eq · cited by 2Units.unitary_eqIsDiscreteValuationRing.mker_valuation_eq_isUnitSubmonoid · cited by 2IsDiscreteValuationRing.m…MonoidHom.mker_inl · cited by 1MonoidHom.mker_inlMonoidHom.mker_inr · cited by 1MonoidHom.mker_inrSubmonoid.map_inl · cited by 1Submonoid.map_inlSubmonoid.map_inr · cited by 1Submonoid.map_inrSubmonoid · cited by 3086SubmonoidMulOneClass · cited by 1018MulOneClassSetLike.ext · cited by 92SetLike.extSubmonoid.extCITED BYCITES

Cites3

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Cited by48

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