Nuprl Lemma : sq_stable__prime_ideal
∀r:CRng. ∀p:Ideal(r){i}. ((∀u:|r|. Dec(p u)) ⇒ SqStable(IsPrimeIdeal(r;p)))
Proof
Definitions occuring in Statement :
prime_ideal_p: IsPrimeIdeal(R;P),
ideal: Ideal(r){i},
crng: CRng,
rng_car: |r|,
sq_stable: SqStable(P),
decidable: Dec(P),
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
crng: CRng,
rng: Rng,
so_lambda: λ2x.t[x],
ideal: Ideal(r){i},
so_apply: x[s],
prime_ideal_p: IsPrimeIdeal(R;P),
subtype_rel: A ⊆r B,
or: P ∨ Q,
infix_ap: x f y
Lemmas referenced :
all_wf,
rng_car_wf,
decidable_wf,
ideal_wf,
crng_wf,
sq_stable__and,
not_wf,
rng_one_wf,
infix_ap_wf,
rng_times_wf,
or_wf,
sq_stable__not,
sq_stable__all,
sq_stable_from_decidable,
decidable__or
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
isect_memberEquality,
because_Cache,
functionEquality,
universeEquality,
independent_functionElimination,
dependent_functionElimination
Latex:
\mforall{}r:CRng. \mforall{}p:Ideal(r)\{i\}. ((\mforall{}u:|r|. Dec(p u)) {}\mRightarrow{} SqStable(IsPrimeIdeal(r;p)))
Date html generated:
2016_05_15-PM-00_24_46
Last ObjectModification:
2015_12_27-AM-00_00_15
Theory : rings_1
Home
Index