English
There is a natural equivalence between invertible elements of tsze_R M and invertible elements of R given by mapping x to fst(x) and vice versa.
Русский
Существует естественное эквидистриум между обратимыми элементами tsze_R M и обратимыми элементами R, задаваемое переходами x ↦ fst(x) и обратно.
LaTeX
$$$ Inv(tsze_R M) \\simeq Inv(R) $$$
Lean4
/-- `x : tzre R M` is invertible when `x.fst : R` is. -/
abbrev invertibleOfInvertibleFst (x : tsze R M) [Invertible x.fst] : Invertible x
where
invOf := (⅟x.fst, -(⅟x.fst •> x.snd <• ⅟x.fst))
invOf_mul_self := by
convert mul_left_eq_one _ _ (invOf_mul_self x.fst)
ext <;> simp
mul_invOf_self := by
convert mul_right_eq_one _ _ (mul_invOf_self x.fst)
ext <;> simp [smul_comm]