Nuprl Lemma : continuous-maps-compact
∀I:Interval. ∀f:I ⟶ℝ. (iproper(I) ⇒ f[x] (proper)continuous for x ∈ I ⇒ maps-compact(I;(-∞, ∞);x.f[x]))
Proof
Definitions occuring in Statement :
maps-compact: maps-compact(I;J;x.f[x]),
proper-continuous: f[x] (proper)continuous for x ∈ I,
rfun: I ⟶ℝ,
riiint: (-∞, ∞),
iproper: iproper(I),
interval: Interval,
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,
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
so_apply: x[s],
uall: ∀[x:A]. B[x],
prop: ℙ,
label: ...$L... t,
maps-compact: maps-compact(I;J;x.f[x]),
maps-compact-proper: maps-compact-proper(I;J;x.f[x]),
and: P ∧ Q,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
exists: ∃x:A. B[x],
guard: {T},
cand: A c∧ B,
sq_stable: SqStable(P),
squash: ↓T,
satisfiable_int_formula: satisfiable_int_formula(fmla),
i-member: r ∈ I,
i-approx: i-approx(I;n),
riiint: (-∞, ∞),
rccint: [l, u]
Lemmas referenced :
proper-continuous-maps-compact,
i-member_wf,
real_wf,
proper-continuous_wf,
iproper_wf,
rfun_wf,
interval_wf,
set_wf,
nat_plus_wf,
icompact_wf,
i-approx_wf,
iproper-approx,
decidable__lt,
false_wf,
not-lt-2,
less-iff-le,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
less_than_wf,
i-member-approx,
riiint_wf,
all_wf,
i-approx-monotonic,
sq_stable__and,
sq_stable__icompact,
sq_stable__iproper,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformnot_wf,
intformle_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
setElimination,
rename,
dependent_set_memberEquality,
hypothesis,
isectElimination,
setEquality,
independent_functionElimination,
productElimination,
addEquality,
natural_numberEquality,
unionElimination,
independent_pairFormation,
voidElimination,
independent_isectElimination,
isect_memberEquality,
voidEquality,
intEquality,
because_Cache,
minusEquality,
productEquality,
dependent_pairFormation,
imageMemberEquality,
baseClosed,
imageElimination,
int_eqEquality,
computeAll
Latex:
\mforall{}I:Interval. \mforall{}f:I {}\mrightarrow{}\mBbbR{}.
(iproper(I) {}\mRightarrow{} f[x] (proper)continuous for x \mmember{} I {}\mRightarrow{} maps-compact(I;(-\minfty{}, \minfty{});x.f[x]))
Date html generated:
2016_10_26-AM-09_59_16
Last ObjectModification:
2016_08_24-PM-02_14_54
Theory : reals
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