prod — Mathlib

English

The product of subrings s ⊆ R and t ⊆ S is the subring of R × S consisting of pairs with first component in s and second in t.

Русский

Произведение подкольц s ⊆ R и t ⊆ S есть подкольцо R × S состоящее из пар (r,s) with r ∈ s и s ∈ t.

LaTeX

$$$ \mathrm{prod}(s,t) = \{ (r,s) \in R \times S \mid r \in s \ & s \in t \}$$$

Lean4

/-- Given `Subring`s `s`, `t` of rings `R`, `S` respectively, `s.prod t` is `s ×̂ t`
as a subring of `R × S`. -/
def prod (s : Subring R) (t : Subring S) : Subring (R × S) :=
  { s.toSubmonoid.prod t.toSubmonoid, s.toAddSubgroup.prod t.toAddSubgroup with carrier := s ×ˢ t }

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