invertibleFstOfInvertible — Mathlib

English

If x ∈ tsze_R M is invertible, then its fst component is invertible in R; moreover its inverse is given by the fst-component of the inverse of x.

Русский

Если x ∈ tsze_R M непереступим, то первая координата x обращуема в R, и обратное к fst(x) равно fst(inv x).

LaTeX

$$$ x \\text{ invertible } \\Rightarrow \\ fst(x) \\text{ invertible in } R$$$

Lean4

/-- `x.fst : R` is invertible when `x : tzre R M` is. -/
abbrev invertibleFstOfInvertible (x : tsze R M) [Invertible x] : Invertible x.fst
    where
  invOf := (⅟x).fst
  invOf_mul_self := by rw [← fst_mul, invOf_mul_self, fst_one]
  mul_invOf_self := by rw [← fst_mul, mul_invOf_self, fst_one]

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