Nuprl Lemma : module_plus_inv
∀[A:Rng]. ∀[m:A-Module]. ∀[x:m.car].
(((x m.plus (m.minus x)) = m.zero ∈ m.car) ∧ (((m.minus x) m.plus x) = m.zero ∈ m.car))
Proof
Definitions occuring in Statement :
module: A-Module,
alg_minus: a.minus,
alg_zero: a.zero,
alg_plus: a.plus,
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
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,
isect_memberEquality,
isectElimination,
independent_pairEquality,
axiomEquality,
because_Cache
Latex:
\mforall{}[A:Rng]. \mforall{}[m:A-Module]. \mforall{}[x:m.car].
(((x m.plus (m.minus x)) = m.zero) \mwedge{} (((m.minus x) m.plus x) = m.zero))
Date html generated:
2016_05_16-AM-07_26_34
Last ObjectModification:
2015_12_28-PM-05_07_57
Theory : algebras_1
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