Nuprl Lemma : princ_ideal_mem_cond
∀r:CRng. ∀u,v:|r|. (v | u in r ⇐⇒ (v)r u)
Proof
Definitions occuring in Statement :
princ_ideal: (a)r,
ring_divs: a | b in r,
crng: CRng,
rng_car: |r|,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a
Definitions unfolded in proof :
all: ∀x:A. B[x],
princ_ideal: (a)r,
member: t ∈ T,
uall: ∀[x:A]. B[x],
crng: CRng,
rng: Rng,
ring_divs: a | b in r,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
infix_ap: x f y,
so_apply: x[s],
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
guard: {T},
comm: Comm(T;op)
Lemmas referenced :
rng_car_wf,
crng_wf,
exists_wf,
equal_wf,
rng_times_wf,
crng_properties,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
independent_pairFormation,
lambdaEquality,
because_Cache,
applyEquality,
productElimination,
dependent_pairFormation,
hyp_replacement,
equalitySymmetry,
dependent_set_memberEquality,
equalityTransitivity,
setEquality
Latex:
\mforall{}r:CRng. \mforall{}u,v:|r|. (v | u in r \mLeftarrow{}{}\mRightarrow{} (v)r u)
Date html generated:
2016_10_21-AM-11_26_01
Last ObjectModification:
2016_07_12-PM-01_09_08
Theory : rings_1
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