Lemma·L59
Multiplicative inversion on rational representatives is well defined
Swapping the coordinates of a representative pair gives the same class no matter which nonzero representative is chosen.
Let
with
and
. If
then
, and
In words
With denominators nonzero, if two pairs represent the same fraction: if the first numerator is nonzero so is the other, and swapping each pair's coordinates gives representatives of the same value.
Never needed: F10 · A03 · A04 · A05 · A07 (computed from the citation graph, not asserted).
Proof
- 1The hypothesis says . Suppose ; if then (zero times anything is zero), so , and by L53, or , contradicting and . So , and , are both valid elements of .
- 2means . This is exactly read via commutativity and symmetry, which is the hypothesis.
∎
Remarks
The engine behind D069. Also shows that "the numerator is nonzero" is a property of the class
, not just of the chosen representative.