Nuprl Lemma : compact-mc_wf
∀[X:Type]. ∀d:metric(X). ∀c:mcompact(X;d). ∀f:FUN(X ⟶ ℝ). (compact-mc{i:l}(d;c;f) ∈ UC(f:X ⟶ ℝ))
Proof
Definitions occuring in Statement :
compact-mc: compact-mc{i:l}(d;c;f),
mcompact: mcompact(X;d),
m-unif-cont: UC(f:X ⟶ Y),
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 :
prop: ℙ,
mfun: FUN(X ⟶ Y),
subtype_rel: A ⊆r B,
compact-mc: compact-mc{i:l}(d;c;f),
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
istype-universe,
compact-metric-to-real-continuity,
m-unif-cont_wf,
rmetric_wf,
real_wf,
mfun_wf,
mcompact_wf,
metric_wf
Rules used in proof :
universeEquality,
functionIsTypeImplies,
axiomEquality,
dependent_functionElimination,
because_Cache,
instantiate,
rename,
setElimination,
thin,
sqequalHypSubstitution,
extract_by_obid,
universeIsType,
functionIsType,
inhabitedIsType,
isectIsType,
hypothesis,
equalitySymmetry,
equalityTransitivity,
hypothesisEquality,
isectElimination,
lambdaEquality_alt,
applyEquality,
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-mc\{i:l\}(d;c;f) \mmember{} UC(f:X {}\mrightarrow{} \mBbbR{}))
Date html generated:
2019_10_30-AM-07_07_26
Last ObjectModification:
2019_10_25-PM-02_27_12
Theory : reals
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