Nuprl Lemma : quot_by_prime_ideal
∀r:CRng. ∀p:Ideal(r){i}. ∀d:detach_fun(|r|;p). ((∀u:|r|. SqStable(p u)) ⇒ (IsPrimeIdeal(r;p) ⇐⇒ IsIntegDom(r / d)))
Proof
Definitions occuring in Statement :
prime_ideal_p: IsPrimeIdeal(R;P),
quot_ring: r / d,
ideal: Ideal(r){i},
integ_dom_p: IsIntegDom(r),
crng: CRng,
rng_car: |r|,
detach_fun: detach_fun(T;A),
sq_stable: SqStable(P),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
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],
integ_dom_p: IsIntegDom(r),
nequal: a ≠ b ∈ T ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
infix_ap: x f y,
type_inj: [x]{T},
quot_ring: r / d,
rng_one: 1,
pi2: snd(t),
pi1: fst(t),
rng_zero: 0,
rng_times: *,
true: True,
squash: ↓T,
or: P ∨ Q,
false: False,
not: ¬A,
cand: A c∧ B,
decidable: Dec(P),
prime_ideal_p: IsPrimeIdeal(R;P)
Lemmas referenced :
all_wf,
rng_car_wf,
sq_stable_wf,
detach_fun_wf,
ideal_wf,
crng_wf,
not_functionality_wrt_iff,
equal_wf,
quot_ring_wf,
rng_subtype_rng_sig,
crng_subtype_rng,
subtype_rel_transitivity,
rng_wf,
rng_sig_wf,
type_inj_wf_for_qrng,
rng_one_wf,
rng_zero_wf,
rng_plus_wf,
rng_minus_wf,
quot_ring_car_elim_b,
rng_times_wf,
not_wf,
prime_ideal_p_wf,
subtype_rel_self,
all_rng_quot_elim_a,
rev_implies_wf,
sq_stable__all,
sq_stable__equal,
squash_wf,
true_wf,
rng_plus_zero,
iff_weakening_equal,
rng_plus_comm,
rng_minus_zero,
or_wf,
dec_alt_char,
iff_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
because_Cache,
productElimination,
independent_pairFormation,
independent_functionElimination,
instantiate,
independent_isectElimination,
dependent_functionElimination,
cumulativity,
universeEquality,
productEquality,
functionEquality,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageElimination,
imageMemberEquality,
baseClosed,
voidElimination,
inlFormation,
inrFormation,
unionElimination,
addLevel
Latex:
\mforall{}r:CRng. \mforall{}p:Ideal(r)\{i\}. \mforall{}d:detach\_fun(|r|;p).
((\mforall{}u:|r|. SqStable(p u)) {}\mRightarrow{} (IsPrimeIdeal(r;p) \mLeftarrow{}{}\mRightarrow{} IsIntegDom(r / d)))
Date html generated:
2019_10_15-AM-10_33_56
Last ObjectModification:
2018_08_29-AM-10_10_15
Theory : rings_1
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