submonoid — Mathlib

English

In a finite group, IsComplement'(H,K) is equivalent to Nat.card H · Nat.card K = Nat.card G and Disjoint H K.

Русский

В конечной группе IsComplement'(H,K) эквивалентно условию |H|·|K| = |G| и Disjoint(H,K).

LaTeX

$$IsComplement'(H,K) ↔ Nat.card H * Nat.card K = Nat.card G ∧ Disjoint H K$$

Lean4

/-- The submonoid of `M × M` defined by a congruence relation on a monoid `M`. -/
@[to_additive (attr := coe) /-- The `AddSubmonoid` of `M × M` defined by an additive congruence
    relation on an `AddMonoid` `M`. -/
    ]
protected def submonoid : Submonoid (M × M)
    where
  carrier := {x | c x.1 x.2}
  one_mem' := c.iseqv.1 1
  mul_mem' := c.mul

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