Nuprl Lemma : totally-bounded-inf
∀[A:Set(ℝ)]. (totally-bounded(A) ⇒ (∃b:ℝ. inf(A) = b))
Proof
Definitions occuring in Statement :
totally-bounded: totally-bounded(A),
inf: inf(A) = b,
rset: Set(ℝ),
real: ℝ,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
rev_implies: P ⇐ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
prop: ℙ,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T}
Lemmas referenced :
inf-as-sup,
inf_wf,
totally-bounded_wf,
rset_wf,
totally-bounded-sup,
rset-neg_wf,
totally-bounded-neg,
rminus_wf,
sup_wf,
squash_wf,
true_wf,
real_wf,
rminus-rminus-eq,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation_alt,
hypothesisEquality,
introduction,
extract_by_obid,
isectElimination,
dependent_functionElimination,
hypothesis,
independent_functionElimination,
universeIsType,
applyEquality,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
independent_isectElimination
Latex:
\mforall{}[A:Set(\mBbbR{})]. (totally-bounded(A) {}\mRightarrow{} (\mexists{}b:\mBbbR{}. inf(A) = b))
Date html generated:
2019_10_29-AM-10_45_02
Last ObjectModification:
2019_04_19-PM-06_33_53
Theory : reals
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