Nuprl Lemma : algebra_properties
∀A:Rng. ∀m:algebra{i:l}(A).
(IsMonoid(m.car;m.times;m.one) ∧ BiLinear(m.car;m.plus;m.times) ∧ (∀a:|A|. Dist1op2opLR(m.car;m.act a;m.times)))
Proof
Definitions occuring in Statement :
algebra: algebra{i:l}(A),
alg_act: a.act,
alg_one: a.one,
alg_times: a.times,
alg_plus: a.plus,
alg_car: a.car,
all: ∀x:A. B[x],
and: P ∧ Q,
apply: f a,
rng: Rng,
rng_car: |r|,
monoid_p: IsMonoid(T;op;id),
dist_1op_2op_lr: Dist1op2opLR(A;1op;2op),
bilinear: BiLinear(T;pl;tm)
Definitions unfolded in proof :
all: ∀x:A. B[x],
algebra: algebra{i:l}(A),
uall: ∀[x:A]. B[x],
member: t ∈ T,
rng: Rng,
module: A-Module,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
sq_stable: SqStable(P),
bilinear: BiLinear(T;pl;tm),
dist_1op_2op_lr: Dist1op2opLR(A;1op;2op),
squash: ↓T
Lemmas referenced :
rng_wf,
algebra_wf,
squash_wf,
sq_stable__dist_1op_2op_lr,
sq_stable__all,
sq_stable__bilinear,
sq_stable__monoid_p,
alg_act_wf,
dist_1op_2op_lr_wf,
all_wf,
alg_plus_wf,
bilinear_wf,
alg_one_wf,
alg_times_wf,
rng_car_wf,
alg_car_wf,
monoid_p_wf,
sq_stable__and
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
setElimination,
thin,
rename,
cut,
lemma_by_obid,
isectElimination,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
productElimination,
isect_memberEquality,
productEquality,
because_Cache,
sqequalRule,
lambdaEquality,
applyEquality,
independent_functionElimination,
introduction,
independent_pairEquality,
axiomEquality,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}A:Rng. \mforall{}m:algebra\{i:l\}(A).
(IsMonoid(m.car;m.times;m.one)
\mwedge{} BiLinear(m.car;m.plus;m.times)
\mwedge{} (\mforall{}a:|A|. Dist1op2opLR(m.car;m.act a;m.times)))
Date html generated:
2016_05_16-AM-07_27_27
Last ObjectModification:
2016_01_16-PM-09_59_55
Theory : algebras_1
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