Nuprl Lemma : det_ideal_defines_eqv
∀[r:CRng]. ∀[a:Ideal(r){i}]. ∀[d:detach_fun(|r|;a)]. ((∀w:|r|. SqStable(a w)) ⇒ EquivRel(|r|;u,v.↑(d (u +r (-r v)))))
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),
uall: ∀[x:A]. B[x],
infix_ap: x f y,
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,
prop: ℙ,
crng: CRng,
rng: Rng,
so_lambda: λ2x.t[x],
ideal: Ideal(r){i},
so_apply: x[s],
equiv_rel: EquivRel(T;x,y.E[x; y]),
and: P ∧ Q,
refl: Refl(T;x,y.E[x; y]),
all: ∀x:A. B[x],
detach_fun: detach_fun(T;A),
infix_ap: x f y,
sym: Sym(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
guard: {T},
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
sq_stable__ideal_p,
ideal_defines_eqv,
equiv_rel_functionality_wrt_iff,
detach_fun_properties,
crng_wf,
ideal_wf,
detach_fun_wf,
assert_wf,
rng_minus_wf,
rng_plus_wf,
assert_witness,
sq_stable_wf,
rng_car_wf,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
dependent_functionElimination,
productElimination,
independent_pairEquality,
independent_functionElimination,
isect_memberEquality,
because_Cache,
independent_isectElimination,
independent_pairFormation,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}[r:CRng]. \mforall{}[a:Ideal(r)\{i\}]. \mforall{}[d:detach\_fun(|r|;a)].
((\mforall{}w:|r|. SqStable(a w)) {}\mRightarrow{} EquivRel(|r|;u,v.\muparrow{}(d (u +r (-r v)))))
Date html generated:
2016_05_15-PM-00_23_23
Last ObjectModification:
2016_01_15-AM-08_51_39
Theory : rings_1
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