Nuprl Lemma : quot_ring_car_qinc
∀[r:CRng]. ∀[a:Ideal(r){i}]. ((∀x:|r|. SqStable(a x)) ⇒ (∀[d:detach_fun(|r|;a)]. (|r| ⊆r Carrier(r/d))))
Proof
Definitions occuring in Statement :
quot_ring_car: Carrier(r/d),
ideal: Ideal(r){i},
crng: CRng,
rng_car: |r|,
detach_fun: detach_fun(T;A),
sq_stable: SqStable(P),
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
quot_ring_car: Carrier(r/d),
crng: CRng,
rng: Rng,
so_lambda: λ2x y.t[x; y],
detach_fun: detach_fun(T;A),
infix_ap: x f y,
so_apply: x[s1;s2],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
ideal: Ideal(r){i},
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x]
Lemmas referenced :
subtype_quotient,
rng_car_wf,
assert_wf,
rng_plus_wf,
rng_minus_wf,
detach_fun_wf,
all_wf,
sq_stable_wf,
ideal_wf,
crng_wf,
ideal-detach-equiv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
lambdaEquality,
applyEquality,
because_Cache,
independent_isectElimination,
axiomEquality,
dependent_functionElimination,
isect_memberEquality,
independent_functionElimination
Latex:
\mforall{}[r:CRng]. \mforall{}[a:Ideal(r)\{i\}].
((\mforall{}x:|r|. SqStable(a x)) {}\mRightarrow{} (\mforall{}[d:detach\_fun(|r|;a)]. (|r| \msubseteq{}r Carrier(r/d))))
Date html generated:
2019_10_15-AM-10_33_31
Last ObjectModification:
2018_09_17-PM-06_20_49
Theory : rings_1
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