Nuprl Lemma : i-member-compact
∀I:Interval. (icompact(I) ⇒ (∀r:ℝ. (r ∈ I ⇐⇒ left-endpoint(I)≤r≤right-endpoint(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,
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],
member: t ∈ T,
implies: P ⇒ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
icompact: icompact(I),
and: P ∧ Q
Lemmas referenced :
i-member-finite-closed,
icompact_wf,
interval_wf
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
isectElimination,
productElimination
Latex:
\mforall{}I:Interval. (icompact(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_47_30
Last ObjectModification:
2015_12_27-PM-11_46_17
Theory : reals
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