pathComponentOne — Mathlib

English

The path component of the identity in a topological monoid is a submonoid.

Русский

Путьовая компонента единицы в топологическом моноиде является подпроизнанным моноидом.

LaTeX

$$$\text{pathComponent}(1) \text{ is a submonoid of } M$$$

Lean4

/-- The path component of the identity in a topological monoid, as a submonoid. -/
@[to_additive (attr := simps) /-- The path component of the identity in an additive topological
    monoid, as an additive submonoid. -/
    ]
def pathComponentOne (M : Type*) [Monoid M] [TopologicalSpace M] [ContinuousMul M] : Submonoid M
    where
  carrier := pathComponent (1 : M)
  mul_mem' {m₁ m₂} hm₁ hm₂ := by simpa using hm₁.mul hm₂
  one_mem' := mem_pathComponent_self 1

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