Nuprl Lemma : algebra_bilinear
∀[A:Rng]. ∀[m:algebra{i:l}(A)]. ∀[a,x,y:m.car].
(((a m.times (x m.plus y)) = ((a m.times x) m.plus (a m.times y)) ∈ m.car)
∧ (((x m.plus y) m.times a) = ((x m.times a) m.plus (y m.times a)) ∈ m.car))
Proof
Definitions occuring in Statement :
algebra: algebra{i:l}(A),
alg_times: a.times,
alg_plus: a.plus,
alg_car: a.car,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
and: P ∧ Q,
equal: s = t ∈ T,
rng: Rng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
and: P ∧ Q,
monoid_p: IsMonoid(T;op;id),
dist_1op_2op_lr: Dist1op2opLR(A;1op;2op),
bilinear: BiLinear(T;pl;tm),
ident: Ident(T;op;id),
assoc: Assoc(T;op),
rng: Rng,
algebra: algebra{i:l}(A),
module: A-Module
Lemmas referenced :
algebra_properties,
alg_car_wf,
rng_car_wf,
algebra_wf,
rng_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
sqequalRule,
isect_memberEquality,
isectElimination,
independent_pairEquality,
axiomEquality,
setElimination,
rename,
because_Cache
Latex:
\mforall{}[A:Rng]. \mforall{}[m:algebra\{i:l\}(A)]. \mforall{}[a,x,y:m.car].
(((a m.times (x m.plus y)) = ((a m.times x) m.plus (a m.times y)))
\mwedge{} (((x m.plus y) m.times a) = ((x m.times a) m.plus (y m.times a))))
Date html generated:
2016_05_16-AM-07_27_34
Last ObjectModification:
2015_12_28-PM-05_07_42
Theory : algebras_1
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