Nuprl Lemma : type_inj_wf_for_qrng
∀[r:CRng]. ∀[a:Ideal(r){i}]. ((∀x:|r|. SqStable(a x)) ⇒ (∀[d:detach_fun(|r|;a)]. ∀[v:|r|]. ([v]{|r / d|} ∈ |r / d|)))
Proof
Definitions occuring in Statement :
quot_ring: r / d,
ideal: Ideal(r){i},
crng: CRng,
rng_car: |r|,
detach_fun: detach_fun(T;A),
type_inj: [x]{T},
sq_stable: SqStable(P),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
apply: f a
Definitions unfolded in proof :
type_inj: [x]{T},
quot_ring: r / d,
rng_car: |r|,
pi1: fst(t),
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
crng: CRng,
rng: Rng,
ideal: Ideal(r){i},
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x]
Lemmas referenced :
quot_ring_car_subtype,
rng_car_wf,
detach_fun_wf,
all_wf,
sq_stable_wf,
ideal_wf,
crng_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
hypothesisEquality,
applyEquality,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
independent_functionElimination,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
isect_memberEquality,
because_Cache,
lambdaEquality,
dependent_functionElimination
Latex:
\mforall{}[r:CRng]. \mforall{}[a:Ideal(r)\{i\}].
((\mforall{}x:|r|. SqStable(a x)) {}\mRightarrow{} (\mforall{}[d:detach\_fun(|r|;a)]. \mforall{}[v:|r|]. ([v]\{|r / d|\} \mmember{} |r / d|)))
Date html generated:
2016_05_15-PM-00_24_23
Last ObjectModification:
2015_12_27-AM-00_00_30
Theory : rings_1
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