Lemma·L58
Multiplication on rational representatives is well defined
Multiplying representative pairs coordinatewise 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 coordinatewise products 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 .
- 2Rearrange the left side by commutativity/associativity and substitute (i) then (ii) (substitutivity): which is the right side.
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