Theorems · Theorem · group theory
AddAction.is_two_pretransitive_iff
∀ {G : Type u_1} {α : Type u_2} [inst : AddGroup G] [inst_1 : AddAction G α],
AddAction.IsMultiplyPretransitive G α 2 ↔ ∀ {a b c d : α}, a ≠ b → c ≠ d → ∃ g, g +ᵥ a = c ∧ g +ᵥ b = dAn additive 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.
Cites14
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- AddGroupstatement and proof · cited by 4,410
- HVAdd.hVAddstatement and proof · cited by 1,820
- Function.Embeddingproof · cited by 988
- AddActionstatement and proof · cited by 820
- Function.Embedding.injectiveproof · cited by 111
- Function.Embedding.extproof · cited by 27
- AddAction.exists_vadd_eqproof · cited by 20
- AddAction.IsMultiplyPretransitivestatement and proof · cited by 17
- Function.Embedding.vadd_applyproof · cited by 4
- Function.Embedding.embFinTwoproof · cited by 4
- Fin.eq_one_of_ne_zeroproof · cited by 2
Cited by2
Results whose statement or proof uses this declaration.
- AddAction.isPretransitive_of_is_two_pretransitiveproof · cited by 1
- AddAction.isPreprimitive_of_is_two_pretransitiveproof · cited by 1