Nuprl Lemma : compact-proper-interval-near-member
∀J:Interval
(icompact(J)
⇒ iproper(J)
⇒ (∀x:ℝ. ((x ∈ J) ⇒ (∀r:ℝ. ((r0 < r) ⇒ (∃y:ℝ. ((y ∈ J) ∧ (|y - x| ≤ r) ∧ (r0 < |y - x|))))))))
Proof
Definitions occuring in Statement :
icompact: icompact(I),
i-member: r ∈ I,
iproper: iproper(I),
interval: Interval,
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
interval: Interval,
iproper: iproper(I),
icompact: icompact(I),
i-finite: i-finite(I),
i-closed: i-closed(I),
i-nonvoid: i-nonvoid(I),
isl: isl(x),
outl: outl(x),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
btrue: tt,
assert: ↑b,
bor: p ∨bq,
bfalse: ff,
i-member: r ∈ I,
right-endpoint: right-endpoint(I),
left-endpoint: left-endpoint(I),
endpoints: endpoints(I),
pi1: fst(t),
pi2: snd(t),
implies: P ⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
true: True,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
false: False,
or: P ∨ Q,
iff: P ⇐⇒ Q,
uimplies: b supposing a,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
not: ¬A,
top: Top,
uiff: uiff(P;Q),
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T},
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
rsub: x - y
Lemmas referenced :
rless_wf,
int-to-real_wf,
real_wf,
rleq_wf,
true_wf,
exists_wf,
false_wf,
interval_wf,
rless-cases,
rmin_strict_ub,
rsub_wf,
rless-implies-rless,
real_term_polynomial,
itermSubtract_wf,
itermVar_wf,
itermConstant_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
itermMinus_wf,
rmin_wf,
real_term_value_minus_lemma,
rminus_wf,
rabs_wf,
squash_wf,
rabs-rminus,
iff_weakening_equal,
rmin-rleq,
rleq_weakening_rless,
rleq_functionality,
rabs_functionality,
req_weakening,
rless_functionality,
rabs-of-nonneg,
radd-preserves-rleq,
radd_wf,
itermAdd_wf,
real_term_value_add_lemma,
rmin_functionality,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
trivial-rsub-rleq,
rmul_wf,
uiff_transitivity,
req_transitivity,
radd_functionality,
rminus-as-rmul,
req_inversion,
rmul-identity1,
rmul-distrib2,
radd-assoc,
rmul_functionality,
radd-int,
rmul-zero-both,
radd-zero-both
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
unionElimination,
sqequalRule,
rename,
cut,
hypothesis,
independent_functionElimination,
natural_numberEquality,
independent_pairFormation,
because_Cache,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
productEquality,
functionEquality,
lambdaEquality,
voidElimination,
dependent_functionElimination,
independent_isectElimination,
computeAll,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
dependent_pairFormation,
addLevel,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
universeEquality,
addEquality,
minusEquality,
lemma_by_obid
Latex:
\mforall{}J:Interval
(icompact(J)
{}\mRightarrow{} iproper(J)
{}\mRightarrow{} (\mforall{}x:\mBbbR{}. ((x \mmember{} J) {}\mRightarrow{} (\mforall{}r:\mBbbR{}. ((r0 < r) {}\mRightarrow{} (\mexists{}y:\mBbbR{}. ((y \mmember{} J) \mwedge{} (|y - x| \mleq{} r) \mwedge{} (r0 < |y - x|))))))))
Date html generated:
2017_10_03-AM-09_35_11
Last ObjectModification:
2017_07_28-AM-07_52_47
Theory : reals
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