Nuprl Lemma : module_act_plus
∀[A:Rng]. ∀[m:A-Module].
((∀[a1,a2:|A|]. ∀[b:m.car]. (((a1 +A a2) m.act b) = ((a1 m.act b) m.plus (a2 m.act b)) ∈ m.car))
∧ (∀[a:|A|]. ∀[b1,b2:m.car]. ((a m.act (b1 m.plus b2)) = ((a m.act b1) m.plus (a m.act b2)) ∈ m.car)))
Proof
Definitions occuring in Statement :
module: A-Module,
alg_act: a.act,
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,
rng_plus: +r,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
rng: Rng,
and: P ∧ Q,
group_p: IsGroup(T;op;id;inv),
monoid_p: IsMonoid(T;op;id),
bilinear_p: IsBilinear(A;B;C;+a;+b;+c;f),
action_p: IsAction(A;x;e;S;f),
comm: Comm(T;op),
inverse: Inverse(T;op;id;inv),
ident: Ident(T;op;id),
assoc: Assoc(T;op),
module: A-Module
Lemmas referenced :
module_properties,
alg_car_wf,
rng_car_wf,
module_wf,
rng_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
productElimination,
sqequalRule,
independent_pairEquality,
isect_memberEquality,
isectElimination,
axiomEquality,
because_Cache
Latex:
\mforall{}[A:Rng]. \mforall{}[m:A-Module].
((\mforall{}[a1,a2:|A|]. \mforall{}[b:m.car]. (((a1 +A a2) m.act b) = ((a1 m.act b) m.plus (a2 m.act b))))
\mwedge{} (\mforall{}[a:|A|]. \mforall{}[b1,b2:m.car]. ((a m.act (b1 m.plus b2)) = ((a m.act b1) m.plus (a m.act b2)))))
Date html generated:
2016_05_16-AM-07_26_39
Last ObjectModification:
2015_12_28-PM-05_07_53
Theory : algebras_1
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