Nuprl Lemma : i-member-finite-closed
∀I:Interval. (i-closed(I) ⇒ i-finite(I) ⇒ (∀r:ℝ. (r ∈ I ⇐⇒ left-endpoint(I)≤r≤right-endpoint(I))))
Proof
Definitions occuring in Statement :
i-member: r ∈ I,
i-closed: i-closed(I),
right-endpoint: right-endpoint(I),
left-endpoint: left-endpoint(I),
i-finite: i-finite(I),
interval: Interval,
rbetween: x≤y≤z,
real: ℝ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
uimplies: b supposing a,
rbetween: x≤y≤z,
interval: Interval,
i-finite: i-finite(I),
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
i-member: r ∈ I,
left-endpoint: left-endpoint(I),
pi1: fst(t),
endpoints: endpoints(I),
outl: outl(x),
right-endpoint: right-endpoint(I),
pi2: snd(t),
i-closed: i-closed(I),
bnot: ¬bb,
bor: p ∨bq,
bfalse: ff,
false: False
Lemmas referenced :
i-member_wf,
rbetween_wf,
left-endpoint_wf,
right-endpoint_wf,
real_wf,
i-finite_wf,
i-closed_wf,
interval_wf,
i-member-finite
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_isectElimination,
productElimination,
unionElimination,
sqequalRule,
voidElimination
Latex:
\mforall{}I:Interval
(i-closed(I) {}\mRightarrow{} i-finite(I) {}\mRightarrow{} (\mforall{}r:\mBbbR{}. (r \mmember{} I \mLeftarrow{}{}\mRightarrow{} left-endpoint(I)\mleq{}r\mleq{}right-endpoint(I))))
Date html generated:
2016_05_18-AM-08_41_08
Last ObjectModification:
2015_12_27-PM-11_51_49
Theory : reals
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