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Theorems · Theorem · group theory

Set.smul_set_subset_iff

∀ {α : Type u_2} {β : Type u_3} [inst : SMul α β] {s t : Set β} {a : α}, a • s ⊆ t ↔ ∀ ⦃b : β⦄, b ∈ s → a • b ∈ t
Defined in
Mathlib.Algebra.Group.Pointwise.Set.Scalar
Cited by
7 results in Mathlib
Foundations
Depth 9 from the axioms · uses propext, Quot.sound
Assumes
SMul
Mathlib docsSourceLoogleLeanSearch

Around this declaration

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

Submodule.mem_colon · cited by 5Submodule.mem_colonbalanced_iff_smul_mem · cited by 2balanced_iff_smul_memMulAction.op_smul_set_stabilizer_subset · cited by 1MulAction.op_smul_set_sta…balanced_iff_neg_mem · cited by 0balanced_iff_neg_memIsUpperSet.smul_subset · cited by 0IsUpperSet.smul_subsetInfiniteGalois.fixingSubgroup_fixedField · cited by 0InfiniteGalois.fixingSubg…IsLowerSet.smul_subset · cited by 0IsLowerSet.smul_subsetSet · cited by 53352SetSet.smulSet · cited by 608Set.smulSetSet.image_subset_iff · cited by 203Set.image_subset_iffSet.smul_set_subset_iffCITED BYCITES

Cites3

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

Cited by7

Results whose statement or proof uses this declaration.