Nuprl Lemma : continuous-range-totally-bounded
∀I:Interval. ∀f:I ⟶ℝ.
(f[x] continuous for x ∈ I ⇒ (∀m:ℕ+. (i-nonvoid(i-approx(I;m)) ⇒ totally-bounded(f[x](x∈i-approx(I;m))))))
Proof
Definitions occuring in Statement :
continuous: f[x] continuous for x ∈ I,
rrange: f[x](x∈I),
i-nonvoid: i-nonvoid(I),
rfun: I ⟶ℝ,
i-approx: i-approx(I;n),
interval: Interval,
totally-bounded: totally-bounded(A),
nat_plus: ℕ+,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
continuous: f[x] continuous for x ∈ I,
member: t ∈ T,
icompact: icompact(I),
and: P ∧ Q,
cand: A c∧ B,
uall: ∀[x:A]. B[x],
prop: ℙ,
totally-bounded: totally-bounded(A),
exists: ∃x:A. B[x],
sq_exists: ∃x:{A| B[x]},
subtype_rel: A ⊆r B,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
rfun: I ⟶ℝ,
nat_plus: ℕ+,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rless: x < y,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
label: ...$L... t,
sq_stable: SqStable(P),
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
squash: ↓T,
subinterval: I ⊆ J ,
int_seg: {i..j-},
lelt: i ≤ j < k,
real: ℝ,
less_than: a < b,
full-partition: full-partition(I;p),
partition: partition(I),
less_than': less_than'(a;b),
true: True,
uiff: uiff(P;Q),
l_all: (∀x∈L.P[x]),
rrange: f[x](x∈I),
rset-member: x ∈ A,
r-ap: f(x)
Lemmas referenced :
i-approx-closed,
i-approx-finite,
icompact_wf,
i-approx_wf,
small-reciprocal-real,
rless_wf,
int-to-real_wf,
i-approx-is-subinterval,
interval_wf,
rfun_subtype,
rfun_wf,
all_wf,
i-member_wf,
rleq_wf,
rabs_wf,
rsub_wf,
rdiv_wf,
rless-int,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
equal_wf,
subinterval_wf,
real_wf,
i-nonvoid_wf,
nat_plus_wf,
continuous_wf,
less_than'_wf,
squash_wf,
sq_stable__and,
sq_stable__rless,
sq_stable__all,
sq_stable__rleq,
partition-exists,
r-ap_wf,
select_wf,
full-partition_wf,
int_seg_properties,
length_wf,
sq_stable__less_than,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
int_seg_wf,
rset-member_wf,
rrange_wf,
exists_wf,
length_of_cons_lemma,
append_wf,
cons_wf,
right-endpoint_wf,
add_nat_plus,
length_wf_nat,
nil_wf,
less_than_wf,
length-append,
length_of_nil_lemma,
add-is-int-iff,
itermAdd_wf,
intformeq_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
false_wf,
full-partition-point-member,
req_weakening,
req_wf,
mesh-property,
rless_transitivity2,
rleq_functionality,
rabs_functionality,
rsub_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
dependent_set_memberEquality,
hypothesisEquality,
cut,
hypothesis,
independent_pairFormation,
introduction,
extract_by_obid,
because_Cache,
isectElimination,
natural_numberEquality,
productElimination,
setElimination,
rename,
applyEquality,
independent_isectElimination,
sqequalRule,
productEquality,
lambdaEquality,
functionEquality,
inrFormation,
independent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
equalityTransitivity,
equalitySymmetry,
setEquality,
minusEquality,
independent_pairEquality,
axiomEquality,
imageMemberEquality,
baseClosed,
imageElimination,
addEquality,
functionExtensionality,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion
Latex:
\mforall{}I:Interval. \mforall{}f:I {}\mrightarrow{}\mBbbR{}.
(f[x] continuous for x \mmember{} I
{}\mRightarrow{} (\mforall{}m:\mBbbN{}\msupplus{}. (i-nonvoid(i-approx(I;m)) {}\mRightarrow{} totally-bounded(f[x](x\mmember{}i-approx(I;m))))))
Date html generated:
2017_10_03-AM-10_23_25
Last ObjectModification:
2017_07_28-AM-08_07_41
Theory : reals
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