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Theorems · Definition · group theory

AddSubmonoid.op

{M : Type u_2} → [inst : AddZeroClass M] → AddSubmonoid M → AddSubmonoid Mᵃᵒᵖ

Pull an additive submonoid back to an opposite submonoid along AddOpposite.unop

Defined in
Mathlib.Algebra.Group.Submonoid.MulOpposite
Cited by
28 results in Mathlib
Foundations
Depth 19 from the axioms · uses Quot.sound
Assumes
AddZeroClass
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Around this declaration

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

AddSubmonoid.opEquiv · cited by 18AddSubmonoid.opEquivAddSubmonoid.op_injective · cited by 2AddSubmonoid.op_injectiveAddSubmonoid.op_closure · cited by 2AddSubmonoid.op_closureAddSubmonoid.equivOp · cited by 2AddSubmonoid.equivOpAddSubmonoid.op_sInf · cited by 1AddSubmonoid.op_sInfAddSubmonoid.op_top · cited by 1AddSubmonoid.op_topAddSubmonoid.op_unop · cited by 1AddSubmonoid.op_unopAddSubmonoid.op_bot · cited by 1AddSubmonoid.op_botAddSubmonoid.op_inj · cited by 1AddSubmonoid.op_injAddSubmonoid.op_le_iff · cited by 0AddSubmonoid.op_le_iffAddSubmonoid.op_le_op_iff · cited by 0AddSubmonoid.op_le_op_iffAddSubmonoid.op_sSup · cited by 0AddSubmonoid.op_sSupAddSubmonoid.op_sup · cited by 0AddSubmonoid.op_supAddSubmonoid.op_toSubsemigroup · cited by 0AddSubmonoid.op_toSubsemi…AddSubmonoid.unop_sInf · cited by 0AddSubmonoid.unop_sInfSetLike.coe · cited by 8199SetLike.coeSet.preimage · cited by 4946Set.preimageAddZeroClass · cited by 1237AddZeroClassAddSubmonoid · cited by 1178AddSubmonoidAddOpposite · cited by 452AddOppositeAddOpposite.unop · cited by 125AddOpposite.unopAddSubmonoid.zero_mem' · cited by 1AddSubmonoid.zero_mem'AddSubmonoid.opCITED BYCITES

Cites7

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Cited by30

Results whose statement or proof uses this declaration.