Nuprl Lemma : ideal_of_prime
∀r:CRng. ∀u:|r|. (r-Prime(u) ⇐⇒ IsPrimeIdeal(r;(u)r))
Proof
Definitions occuring in Statement :
prime_ideal_p: IsPrimeIdeal(R;P),
princ_ideal: (a)r,
rprime: r-Prime(u),
crng: CRng,
rng_car: |r|,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
prime_ideal_p: IsPrimeIdeal(R;P),
member: t ∈ T,
uall: ∀[x:A]. B[x],
crng: CRng,
rng: Rng,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
ideal: Ideal(r){i},
prop: ℙ,
infix_ap: x f y,
so_lambda: λ2x.t[x],
or: P ∨ Q,
so_apply: x[s],
rprime: r-Prime(u),
false: False,
guard: {T}
Lemmas referenced :
rng_car_wf,
crng_wf,
princ_ideal_mem_cond,
rng_one_wf,
princ_ideal_wf,
ideal_wf,
rng_times_wf,
ring_divs_wf,
iff_wf,
rprime_wf,
not_wf,
all_wf,
infix_ap_wf,
or_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
addLevel,
productElimination,
independent_pairFormation,
impliesFunctionality,
dependent_functionElimination,
independent_functionElimination,
applyEquality,
lambdaEquality,
sqequalRule,
universeEquality,
allFunctionality,
orFunctionality,
because_Cache,
andLevelFunctionality,
impliesLevelFunctionality,
allLevelFunctionality,
orLevelFunctionality,
productEquality,
functionEquality,
voidElimination,
introduction
Latex:
\mforall{}r:CRng. \mforall{}u:|r|. (r-Prime(u) \mLeftarrow{}{}\mRightarrow{} IsPrimeIdeal(r;(u)r))
Date html generated:
2016_05_15-PM-00_25_34
Last ObjectModification:
2015_12_27-AM-00_00_26
Theory : rings_1
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