Nuprl Lemma : ufm_char
∀g:IAbMonoid
(Cancel(|g|;|g|;*)
⇒ WellFnd{i}(|g|;x,y.x p| y)
⇒ (∀a:Atom{g}. IsPrime(a))
⇒ (∀a:|g|. Dec(Reducible(a)))
⇒ (∀a,b:|g|. Dec(a | b))
⇒ IsUFM(g))
Proof
Definitions occuring in Statement :
is_ufm: IsUFM(g),
matom_ty: Atom{g},
mreducible: Reducible(a),
mprime: IsPrime(a),
mpdivides: a p| b,
mdivides: b | a,
wellfounded: WellFnd{i}(A;x,y.R[x; y]),
decidable: Dec(P),
all: ∀x:A. B[x],
implies: P ⇒ Q,
iabmonoid: IAbMonoid,
grp_op: *,
grp_car: |g|,
cancel: Cancel(T;S;op)
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
iabmonoid: IAbMonoid,
imon: IMonoid,
so_lambda: λ2x.t[x],
so_apply: x[s],
matom_ty: Atom{g},
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
is_ufm: IsUFM(g),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
permr_massoc_rel: ≡~,
ab_binrel: x,y:T. E[x; y],
binrel_ap: a [r] b,
mprime_ty: Prime(g),
guard: {T}
Lemmas referenced :
all_wf,
grp_car_wf,
decidable_wf,
mdivides_wf,
mreducible_wf,
matom_ty_wf,
mprime_wf,
wellfounded_wf,
mpdivides_wf,
cancel_wf,
grp_op_wf,
iabmonoid_wf,
not_wf,
munit_wf,
exists_uni_upto_char,
list_wf,
permr_massoc_rel_wf,
subtype_rel_dep_function,
subtype_rel_list,
subtype_rel_self,
equal_wf,
mon_reduce_wf,
mfact_exists,
unique_mfact,
mprime_ty_wf,
subtype_rel_sets,
matomic_wf,
massoc_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
because_Cache,
dependent_functionElimination,
applyEquality,
instantiate,
cumulativity,
functionEquality,
universeEquality,
independent_isectElimination,
independent_functionElimination,
setEquality,
dependent_set_memberEquality,
equalitySymmetry,
equalityTransitivity
Latex:
\mforall{}g:IAbMonoid
(Cancel(|g|;|g|;*)
{}\mRightarrow{} WellFnd\{i\}(|g|;x,y.x p| y)
{}\mRightarrow{} (\mforall{}a:Atom\{g\}. IsPrime(a))
{}\mRightarrow{} (\mforall{}a:|g|. Dec(Reducible(a)))
{}\mRightarrow{} (\mforall{}a,b:|g|. Dec(a | b))
{}\mRightarrow{} IsUFM(g))
Date html generated:
2016_05_16-AM-07_45_26
Last ObjectModification:
2015_12_28-PM-05_54_11
Theory : factor_1
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