Nuprl Lemma : alg_act_is_rng_chom
∀a:CRng. ∀m:algebra{i:l}(a). rng_chom_p(a;m↓rg;λx:|a|. m.act x m.one)
Proof
Definitions occuring in Statement :
algebra: algebra{i:l}(A),
rng_of_alg: a↓rg,
alg_act: a.act,
alg_one: a.one,
tlambda: λx:T. b[x],
all: ∀x:A. B[x],
apply: f a,
rng_chom_p: rng_chom_p(r;s;f),
crng: CRng,
rng_car: |r|
Definitions unfolded in proof :
all: ∀x:A. B[x],
rng_chom_p: rng_chom_p(r;s;f),
and: P ∧ Q,
rng_hom_p: rng_hom_p(r;s;f),
fun_thru_2op: FunThru2op(A;B;opa;opb;f),
uall: ∀[x:A]. B[x],
member: t ∈ T,
rng_of_alg: a↓rg,
rng_car: |r|,
pi1: fst(t),
tlambda: λx:T. b[x],
rng_plus: +r,
pi2: snd(t),
crng: CRng,
rng: Rng,
rng_times: *,
rng_one: 1,
squash: ↓T,
prop: ℙ,
algebra: algebra{i:l}(A),
module: A-Module,
infix_ap: x f y,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
rng_car_wf,
algebra_wf,
crng_wf,
equal_wf,
squash_wf,
true_wf,
alg_car_wf,
module_act_plus,
alg_one_wf,
infix_ap_wf,
alg_plus_wf,
alg_act_wf,
iff_weakening_equal,
rng_times_wf,
algebra_act_times_r,
module_action_p,
algebra_times_one,
crng_times_comm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
isect_memberFormation,
introduction,
cut,
sqequalRule,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
isect_memberEquality,
axiomEquality,
because_Cache,
dependent_functionElimination,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
productElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination
Latex:
\mforall{}a:CRng. \mforall{}m:algebra\{i:l\}(a). rng\_chom\_p(a;m\mdownarrow{}rg;\mlambda{}x:|a|. m.act x m.one)
Date html generated:
2017_10_01-AM-09_52_03
Last ObjectModification:
2017_03_03-PM-00_46_47
Theory : algebras_1
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