Nuprl Lemma : algebra_act_times_r
∀[A:Rng]. ∀[m:algebra{i:l}(A)]. ∀[a:|A|]. ∀[u,v:m.car].
(((m.act a (u m.times v)) = ((m.act a u) m.times v) ∈ m.car)
∧ ((m.act a (u m.times v)) = (u m.times (m.act a v)) ∈ m.car))
Proof
Definitions occuring in Statement :
algebra: algebra{i:l}(A),
alg_act: a.act,
alg_times: a.times,
alg_car: a.car,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
and: P ∧ Q,
apply: f a,
equal: s = t ∈ T,
rng: Rng,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
rng: Rng,
algebra: algebra{i:l}(A),
module: A-Module,
guard: {T},
dist_1op_2op_lr: Dist1op2opLR(A;1op;2op)
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,
independent_pairFormation,
sqequalRule,
independent_pairEquality,
axiomEquality,
isectElimination,
setElimination,
rename,
isect_memberEquality,
because_Cache
Latex:
\mforall{}[A:Rng]. \mforall{}[m:algebra\{i:l\}(A)]. \mforall{}[a:|A|]. \mforall{}[u,v:m.car].
(((m.act a (u m.times v)) = ((m.act a u) m.times v))
\mwedge{} ((m.act a (u m.times v)) = (u m.times (m.act a v))))
Date html generated:
2016_05_16-AM-07_27_36
Last ObjectModification:
2015_12_28-PM-05_07_50
Theory : algebras_1
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