Nuprl Lemma : continuous-compact-range-totally-bounded
∀I:Interval. ∀f:I ⟶ℝ. (icompact(I) ⇒ f[x] continuous for x ∈ I ⇒ totally-bounded(f[x](x∈I)))
Proof
Definitions occuring in Statement :
continuous: f[x] continuous for x ∈ I,
rrange: f[x](x∈I),
icompact: icompact(I),
rfun: I ⟶ℝ,
interval: Interval,
totally-bounded: totally-bounded(A),
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
label: ...$L... t,
rfun: I ⟶ℝ,
icompact: icompact(I)
Lemmas referenced :
continuous-range-totally-bounded,
less_than_wf,
i-nonvoid_wf,
squash_wf,
true_wf,
i-approx-of-compact,
iff_weakening_equal,
continuous_wf,
i-member_wf,
real_wf,
icompact_wf,
rfun_wf,
interval_wf,
set_wf,
subtype_rel-equal,
i-approx_wf,
equal_wf,
totally-bounded_wf,
rset_wf,
rrange_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
baseClosed,
isectElimination,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
universeEquality,
independent_isectElimination,
productElimination,
setElimination,
rename,
setEquality,
functionEquality,
instantiate
Latex:
\mforall{}I:Interval. \mforall{}f:I {}\mrightarrow{}\mBbbR{}. (icompact(I) {}\mRightarrow{} f[x] continuous for x \mmember{} I {}\mRightarrow{} totally-bounded(f[x](x\mmember{}I)))
Date html generated:
2017_10_03-AM-10_23_42
Last ObjectModification:
2017_07_28-AM-08_07_54
Theory : reals
Home
Index