Theorems · Theorem · group theory
MulAction.is_two_pretransitive_iff
∀ {G : Type u_1} {α : Type u_2} [inst : Group G] [inst_1 : MulAction G α],
MulAction.IsMultiplyPretransitive G α 2 ↔ ∀ {a b c d : α}, a ≠ b → c ≠ d → ∃ g, g • a = c ∧ g • b = dAn action is 2-pretransitive iff
it can move any two distinct elements to any two distinct elements.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 56 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Groupstatement and proof · cited by 6,238
- MulActionstatement and proof · cited by 1,294
- Function.Embeddingproof · cited by 988
- Function.Embedding.injectiveproof · cited by 111
- MulAction.IsMultiplyPretransitivestatement and proof · cited by 33
- MulAction.exists_smul_eqproof · cited by 32
- Function.Embedding.extproof · cited by 27
- Function.Embedding.smul_applyproof · cited by 4
- Function.Embedding.embFinTwoproof · cited by 4
- Fin.eq_one_of_ne_zeroproof · cited by 2
- Function.Embedding.embFinTwo_apply_oneproof · cited by 2
Cited by2
Results whose statement or proof uses this declaration.
- MulAction.isPreprimitive_of_is_two_pretransitiveproof · cited by 3
- MulAction.isPretransitive_of_is_two_pretransitiveproof · cited by 1