Nuprl Lemma : ideal-detach-equiv
∀r:CRng. ∀a:Ideal(r){i}. ((∀x:|r|. SqStable(a x)) ⇒ (∀d:detach_fun(|r|;a). EquivRel(|r|;x,y.↑(d (x +r (-r y))))))
Proof
Definitions occuring in Statement :
ideal: Ideal(r){i},
crng: CRng,
rng_minus: -r,
rng_plus: +r,
rng_car: |r|,
detach_fun: detach_fun(T;A),
equiv_rel: EquivRel(T;x,y.E[x; y]),
assert: ↑b,
sq_stable: SqStable(P),
infix_ap: x f y,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
crng: CRng,
rng: Rng,
ideal: Ideal(r){i},
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
det_ideal_defines_eqv,
detach_fun_wf,
rng_car_wf,
all_wf,
sq_stable_wf,
ideal_wf,
crng_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
setElimination,
rename,
sqequalRule,
lambdaEquality,
applyEquality
Latex:
\mforall{}r:CRng. \mforall{}a:Ideal(r)\{i\}.
((\mforall{}x:|r|. SqStable(a x)) {}\mRightarrow{} (\mforall{}d:detach\_fun(|r|;a). EquivRel(|r|;x,y.\muparrow{}(d (x +r (-r y))))))
Date html generated:
2016_05_15-PM-00_23_29
Last ObjectModification:
2015_12_27-AM-00_00_46
Theory : rings_1
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