Nuprl Lemma : compact-inf-property
∀[X:Type]
∀d:metric(X). ∀c:mcompact(X;d). ∀f:FUN(X ⟶ ℝ).
((∀x:X. (compact-inf{i:l}(d;c;f) ≤ (f x))) ∧ (∀e:ℝ. ((r0 < e) ⇒ (∃x:X. ((f x) < (compact-inf{i:l}(d;c;f) + e))))))
Proof
Definitions occuring in Statement :
compact-inf: compact-inf{i:l}(d;c;f),
mcompact: mcompact(X;d),
mfun: FUN(X ⟶ Y),
rmetric: rmetric(),
metric: metric(X),
rleq: x ≤ y,
rless: x < y,
radd: a + b,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
compact-inf: compact-inf{i:l}(d;c;f),
mfun: FUN(X ⟶ Y),
pi2: snd(t),
mcompact: mcompact(X;d),
member: t ∈ T,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x]
Lemmas referenced :
istype-universe,
metric_wf,
mcompact_wf,
rmetric_wf,
real_wf,
mfun_wf,
compact-mc_wf,
m-inf-property
Rules used in proof :
universeEquality,
instantiate,
universeIsType,
hypothesis,
rename,
setElimination,
sqequalRule,
productElimination,
dependent_functionElimination,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation_alt,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[X:Type]
\mforall{}d:metric(X). \mforall{}c:mcompact(X;d). \mforall{}f:FUN(X {}\mrightarrow{} \mBbbR{}).
((\mforall{}x:X. (compact-inf\{i:l\}(d;c;f) \mleq{} (f x)))
\mwedge{} (\mforall{}e:\mBbbR{}. ((r0 < e) {}\mRightarrow{} (\mexists{}x:X. ((f x) < (compact-inf\{i:l\}(d;c;f) + e))))))
Date html generated:
2019_10_30-AM-07_09_29
Last ObjectModification:
2019_10_25-PM-02_41_45
Theory : reals
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