Nuprl Lemma : icompact-endpoints
∀I:Interval. (icompact(I) ⇒ ((left-endpoint(I) ∈ I) ∧ (right-endpoint(I) ∈ I)))
Proof
Definitions occuring in Statement :
icompact: icompact(I),
i-member: r ∈ I,
right-endpoint: right-endpoint(I),
left-endpoint: left-endpoint(I),
interval: Interval,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
icompact: icompact(I),
and: P ∧ Q,
interval: Interval,
i-finite: i-finite(I),
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
right-endpoint: right-endpoint(I),
i-member: r ∈ I,
left-endpoint: left-endpoint(I),
i-length: |I|,
endpoints: endpoints(I),
outl: outl(x),
pi1: fst(t),
pi2: snd(t),
cand: A c∧ B,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
uiff: uiff(P;Q),
prop: ℙ,
true: True,
i-closed: i-closed(I),
bnot: ¬bb,
bor: p ∨bq,
bfalse: ff
Lemmas referenced :
icompact-length-nonneg,
rleq_weakening_equal,
rleq-implies-rleq,
int-to-real_wf,
rsub_wf,
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,
rleq_wf,
i-length_wf,
real_wf,
top_wf,
icompact_wf,
interval_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
productElimination,
unionElimination,
sqequalRule,
because_Cache,
independent_pairFormation,
natural_numberEquality,
dependent_functionElimination,
computeAll,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
equalityElimination,
independent_pairEquality,
inlEquality,
inrEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination
Latex:
\mforall{}I:Interval. (icompact(I) {}\mRightarrow{} ((left-endpoint(I) \mmember{} I) \mwedge{} (right-endpoint(I) \mmember{} I)))
Date html generated:
2017_10_03-AM-09_33_53
Last ObjectModification:
2017_07_28-AM-07_51_56
Theory : reals
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