Nuprl Lemma : quot_ring_car_elim_a
∀r:CRng. ∀a:Ideal(r){i}. ∀d:detach_fun(|r|;a).
((∀w:|r|. SqStable(a w)) ⇒ (∀u,v:|r|. (u = v ∈ |r / d| ⇐⇒ a (u +r (-r v)))))
Proof
Definitions occuring in Statement :
quot_ring: r / d,
ideal: Ideal(r){i},
crng: CRng,
rng_minus: -r,
rng_plus: +r,
rng_car: |r|,
detach_fun: detach_fun(T;A),
sq_stable: SqStable(P),
infix_ap: x f y,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
crng: CRng,
rng: Rng,
prop: ℙ,
so_lambda: λ2x.t[x],
ideal: Ideal(r){i},
so_apply: x[s],
quot_ring: r / d,
rng_car: |r|,
pi1: fst(t),
iff: P ⇐⇒ Q,
and: P ∧ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
detach_fun: detach_fun(T;A),
infix_ap: x f y
Lemmas referenced :
rng_car_wf,
all_wf,
sq_stable_wf,
detach_fun_wf,
ideal_wf,
crng_wf,
quot_ring_car_elim,
equal_wf,
quot_ring_car_wf,
quot_ring_car_subtype,
iff_wf,
assert_wf,
rng_plus_wf,
rng_minus_wf,
det_ideal_ap_elim
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,
addLevel,
productElimination,
independent_pairFormation,
impliesFunctionality,
independent_functionElimination,
independent_isectElimination,
dependent_functionElimination
Latex:
\mforall{}r:CRng. \mforall{}a:Ideal(r)\{i\}. \mforall{}d:detach\_fun(|r|;a).
((\mforall{}w:|r|. SqStable(a w)) {}\mRightarrow{} (\mforall{}u,v:|r|. (u = v \mLeftarrow{}{}\mRightarrow{} a (u +r (-r v)))))
Date html generated:
2017_10_01-AM-08_18_17
Last ObjectModification:
2017_02_28-PM-02_03_12
Theory : rings_1
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