Lemma·L57
Addition on rational representatives is well defined
Adding representative pairs by the schoolbook fraction rule gives the same class no matter which representatives are chosen.
Let
with
,
and
. Then
In words
With denominators nonzero, if two pairs represent the same fraction, their cross-multiplied sums represent the same fraction too.
Never needed: F05 · F10 · F13 · A03 · A04 · A05 · A07 (computed from the citation graph, not asserted).
Proof
- 1The hypotheses say (i) and (ii). We must show
- 2Expand the left side via distributivity: . Using (i) (substitutivity) and commutativity/associativity to regroup: . Using (ii) similarly: .
- 3So the left side equals (factoring back out via distributivity), which is the right side.
∎