Nuprl Lemma : full-partition-point-member
∀I:Interval. (icompact(I) ⇒ (∀p:partition(I). (∀x∈full-partition(I;p).x ∈ I)))
Proof
Definitions occuring in Statement :
full-partition: full-partition(I;p),
partition: partition(I),
icompact: icompact(I),
i-member: r ∈ I,
interval: Interval,
l_all: (∀x∈L.P[x]),
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
full-partition: full-partition(I;p),
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
partition: partition(I),
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
prop: ℙ,
icompact: icompact(I)
Lemmas referenced :
l_all_cons,
real_wf,
i-member_wf,
left-endpoint_wf,
append_wf,
cons_wf,
right-endpoint_wf,
nil_wf,
icompact-endpoints,
l_all_append,
partition-point-member,
l_all_single,
equal_wf,
partition_wf,
icompact_wf,
interval_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
sqequalRule,
lambdaEquality,
hypothesisEquality,
dependent_functionElimination,
independent_isectElimination,
setElimination,
rename,
because_Cache,
productElimination,
independent_functionElimination,
independent_pairFormation,
setEquality
Latex:
\mforall{}I:Interval. (icompact(I) {}\mRightarrow{} (\mforall{}p:partition(I). (\mforall{}x\mmember{}full-partition(I;p).x \mmember{} I)))
Date html generated:
2017_10_03-AM-09_40_42
Last ObjectModification:
2017_07_28-AM-07_56_07
Theory : reals
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