Nuprl Lemma : rng_times_over_plus
∀[r:Rng]. ∀[a,b,c:|r|]. (((a * (b +r c)) = ((a * b) +r (a * c)) ∈ |r|) ∧ (((b +r c) * a) = ((b * a) +r (c * a)) ∈ |r|))
Proof
Definitions occuring in Statement :
rng: Rng,
rng_times: *,
rng_plus: +r,
rng_car: |r|,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
and: P ∧ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
bilinear: BiLinear(T;pl;tm),
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
rng: Rng
Lemmas referenced :
rng_car_wf,
rng_wf,
rng_all_properties
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairEquality,
axiomEquality,
hypothesis,
lemma_by_obid,
setElimination,
rename
Latex:
\mforall{}[r:Rng]. \mforall{}[a,b,c:|r|].
(((a * (b +r c)) = ((a * b) +r (a * c))) \mwedge{} (((b +r c) * a) = ((b * a) +r (c * a))))
Date html generated:
2016_05_15-PM-00_21_37
Last ObjectModification:
2015_12_27-AM-00_01_51
Theory : rings_1
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