Nuprl Lemma : compact-inf_wf
∀[X:Type]. ∀d:metric(X). ∀c:mcompact(X;d). ∀f:FUN(X ⟶ ℝ). (compact-inf{i:l}(d;c;f) ∈ ℝ)
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),
real: ℝ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
universe: Type
Definitions unfolded in proof :
mfun: FUN(X ⟶ Y),
pi2: snd(t),
mcompact: mcompact(X;d),
compact-inf: compact-inf{i:l}(d;c;f),
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
istype-universe,
metric_wf,
mcompact_wf,
rmetric_wf,
real_wf,
mfun_wf,
compact-mc_wf,
m-inf_wf
Rules used in proof :
universeEquality,
instantiate,
inhabitedIsType,
functionIsTypeImplies,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
lambdaEquality_alt,
universeIsType,
dependent_functionElimination,
because_Cache,
hypothesis,
rename,
setElimination,
productElimination,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalRule,
lambdaFormation_alt,
cut,
introduction,
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{}). (compact-inf\{i:l\}(d;c;f) \mmember{} \mBbbR{})
Date html generated:
2019_10_30-AM-07_09_08
Last ObjectModification:
2019_10_25-PM-02_39_29
Theory : reals
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