Nuprl Lemma : compact-dist_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[A:Type]. ∀[c:mcompact(A;d)]. ∀[x:X]. (dist(x;A) ∈ ℝ) supposing A ⊆r X
Proof
Definitions occuring in Statement :
compact-dist: dist(x;A),
mcompact: mcompact(X;d),
metric: metric(X),
real: ℝ,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
member: t ∈ T,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
compact-dist: dist(x;A),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
istype-universe,
metric_wf,
subtype_rel_wf,
mcompact_wf,
metric-on-subtype,
compact-inf_wf,
rmetric_wf,
real_wf,
mfun-subtype2,
dist-fun_wf
Rules used in proof :
universeEquality,
instantiate,
inhabitedIsType,
isectIsTypeImplies,
isect_memberEquality_alt,
universeIsType,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
dependent_functionElimination,
sqequalRule,
independent_isectElimination,
applyEquality,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[A:Type].
\mforall{}[c:mcompact(A;d)]. \mforall{}[x:X]. (dist(x;A) \mmember{} \mBbbR{}) supposing A \msubseteq{}r X
Date html generated:
2019_10_30-AM-07_12_22
Last ObjectModification:
2019_10_25-PM-04_45_31
Theory : reals
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