English
Given subsemirings s ⊆ R and t ⊆ S, their product s.prod t is the subsemiring of R × S consisting of pairs (r,s) with r ∈ s and s ∈ t.
Русский
Пусть s ⊆ R и t ⊆ S — подпол semirings. Их произведение s.prod t — это подполе полусемиринга R × S, состоящее из пар (r,s) с r ∈ s и s ∈ t.
LaTeX
$$$\\mathrm{prod}\\ (s\\, t) = \\{(r,s) : r \\in R,\\ s \\in S \\mid r \\in s,\\ s \\in t\\}$$$
Lean4
/-- Given `Subsemiring`s `s`, `t` of semirings `R`, `S` respectively, `s.prod t` is `s × t`
as a subsemiring of `R × S`. -/
def prod (s : Subsemiring R) (t : Subsemiring S) : Subsemiring (R × S) :=
{ s.toSubmonoid.prod t.toSubmonoid, s.toAddSubmonoid.prod t.toAddSubmonoid with carrier := s ×ˢ t }
