Theorems · Theorem · combinatorics
Matroid.IsBase.compl_closure_sdiff_singleton_isCocircuit
∀ {α : Type u_1} {M : Matroid α} {e : α} {B : Set α}, M.IsBase B → e ∈ B → M.IsCocircuit (M.E \ M.closure (B \ {e}))For an element e of a base B, the complement of the closure of B \ {e} is a cocircuit.
- Defined in
- Mathlib.Combinatorics.Matroid.Circuit
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 103 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites28
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
- LE.le.transproof · cited by 3,151
- Matroidstatement and proof · cited by 1,258
- Matroid.Estatement and proof · cited by 550
- Matroid.closurestatement and proof · cited by 272
- Matroid.IsBasestatement and proof · cited by 239
- Set.sdiff_subsetproof · cited by 156
- Minimalproof · cited by 150
- Disjoint.symmproof · cited by 125
- LE.le.antisymm'proof · cited by 104
- Matroid.Spanningproof · cited by 63
- by_contraproof · cited by 60
Cited by1
Results whose statement or proof uses this declaration.
- Matroid.IsBase.compl_closure_diff_singleton_isCocircuitproof · cited by 0