Theorems · Theorem · combinatorics
Matroid.isLoop_tfae
∀ {α : Type u_1} (M : Matroid α) (e : α),
[M.IsLoop e, e ∈ M.closure ∅, M.IsCircuit {e}, M.Dep {e}, ∀ ⦃B : Set α⦄, M.IsBase B → e ∈ M.E \ B].TFAE- Defined in
- Mathlib.Combinatorics.Matroid.Loop
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 88 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites18
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- Matroidstatement and proof · cited by 1,258
- Matroid.Estatement and proof · cited by 550
- Matroid.Indepproof · cited by 367
- Matroid.closurestatement and proof · cited by 272
- Matroid.IsBasestatement and proof · cited by 239
- Matroid.IsCircuitstatement and proof · cited by 108
- List.TFAEstatement · cited by 102
- List.tfae_of_cycleproof · cited by 83
- Matroid.Depstatement and proof · cited by 76
- Matroid.IsLoopstatement and proof · cited by 55
- Matroid.Indep.subsetproof · cited by 42
Cited by3
Results whose statement or proof uses this declaration.
- Matroid.singleton_depproof · cited by 8
- Matroid.singleton_isCircuitproof · cited by 3
- Matroid.isLoop_iff_forall_mem_compl_isBaseproof · cited by 0