Theorems · Theorem · group theory
AddMonoid.le_minOrder_iff_forall_addSubgroup
∀ {G : Type u_1} [inst : AddGroup G] {n : ℕ∞},
n ≤ AddMonoid.minOrder G ↔ ∀ ⦃s : AddSubgroup G⦄, s ≠ ⊥ → (↑s).Finite → n ≤ ↑(Nat.card ↥s)- Defined in
- Mathlib.GroupTheory.Order.Min
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 98 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- AddGroup
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites22
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- SetLike.coestatement and proof · cited by 8,199
- ENatstatement and proof · cited by 4,985
- Bot.botstatement and proof · cited by 4,720
- AddGroupstatement and proof · cited by 4,410
- AddSubgroupstatement and proof · cited by 3,232
- LE.le.transproof · cited by 3,151
- Set.Finitestatement and proof · cited by 1,814
- Nat.cardstatement and proof · cited by 844
- AddSubgroup.zmultiplesproof · cited by 493
- Set.Finite.subsetproof · cited by 285
- addOrderOfproof · cited by 208
- IsOfFinAddOrderproof · cited by 105
Cited by2
Results whose statement or proof uses this declaration.
- AddMonoid.minOrder_le_natCardproof · cited by 2
- ZMod.minOrderproof · cited by 1