Nuprl Lemma : module_hom_action
∀[A:RngSig]. ∀[M,N:algebra_sig{i:l}(|A|)]. ∀[f:module_hom(A; M; N)]. ∀[u:|A|].
∀a:M.car. ((f (M.act u a)) = (N.act u (f a)) ∈ N.car)
Proof
Definitions occuring in Statement :
module_hom: module_hom(A; M; N),
alg_act: a.act,
alg_car: a.car,
algebra_sig: algebra_sig{i:l}(A),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
equal: s = t ∈ T,
rng_car: |r|,
rng_sig: RngSig
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
guard: {T},
module_hom_p: module_hom_p(a; m; n; f),
and: P ∧ Q,
fun_thru_1op: fun_thru_1op(A;B;opa;opb;f)
Lemmas referenced :
module_hom_properties,
alg_car_wf,
rng_car_wf,
module_hom_wf,
algebra_sig_wf,
rng_sig_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
isectElimination,
sqequalRule,
lambdaEquality,
axiomEquality,
because_Cache,
isect_memberEquality,
productElimination
Latex:
\mforall{}[A:RngSig]. \mforall{}[M,N:algebra\_sig\{i:l\}(|A|)]. \mforall{}[f:module\_hom(A; M; N)]. \mforall{}[u:|A|].
\mforall{}a:M.car. ((f (M.act u a)) = (N.act u (f a)))
Date html generated:
2016_05_16-AM-07_27_19
Last ObjectModification:
2015_12_28-PM-05_07_34
Theory : algebras_1
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