Nuprl Lemma : module_properties
∀A:RngSig. ∀m:A-Module.
(IsGroup(m.car;m.plus;m.zero;m.minus)
∧ Comm(m.car;m.plus)
∧ IsAction(|A|;*;1;m.car;m.act)
∧ IsBilinear(|A|;m.car;m.car;+A;m.plus;m.plus;m.act))
Proof
Definitions occuring in Statement :
module: A-Module,
alg_act: a.act,
alg_minus: a.minus,
alg_zero: a.zero,
alg_plus: a.plus,
alg_car: a.car,
comm: Comm(T;op),
all: ∀x:A. B[x],
and: P ∧ Q,
rng_one: 1,
rng_times: *,
rng_plus: +r,
rng_car: |r|,
rng_sig: RngSig,
group_p: IsGroup(T;op;id;inv),
action_p: IsAction(A;x;e;S;f),
bilinear_p: IsBilinear(A;B;C;+a;+b;+c;f)
Definitions unfolded in proof :
all: ∀x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
module: A-Module,
uall: ∀[x:A]. B[x],
member: t ∈ T,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T
Lemmas referenced :
sq_stable__bilinear_p,
rng_plus_wf,
rng_sig_wf,
module_wf,
alg_act_wf,
rng_one_wf,
rng_times_wf,
sq_stable__action_p,
sq_stable__comm,
alg_minus_wf,
alg_zero_wf,
alg_plus_wf,
rng_car_wf,
alg_car_wf,
sq_stable__group_p
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
lemma_by_obid,
isectElimination,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
introduction,
productElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
because_Cache
Latex:
\mforall{}A:RngSig. \mforall{}m:A-Module.
(IsGroup(m.car;m.plus;m.zero;m.minus)
\mwedge{} Comm(m.car;m.plus)
\mwedge{} IsAction(|A|;*;1;m.car;m.act)
\mwedge{} IsBilinear(|A|;m.car;m.car;+A;m.plus;m.plus;m.act))
Date html generated:
2016_05_16-AM-07_26_29
Last ObjectModification:
2016_01_16-PM-09_59_59
Theory : algebras_1
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