Nuprl Lemma : proper-continuous-is-continuous
∀I:Interval. (iproper(I) ⇒ (∀f:I ⟶ℝ. (f[x] (proper)continuous for x ∈ I ⇒ f[x] continuous for x ∈ I)))
Proof
Definitions occuring in Statement :
proper-continuous: f[x] (proper)continuous for x ∈ I,
continuous: f[x] continuous for x ∈ I,
rfun: I ⟶ℝ,
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,
continuous: f[x] continuous for x ∈ I,
proper-continuous: f[x] (proper)continuous for x ∈ I,
member: t ∈ T,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
label: ...$L... t,
rfun: I ⟶ℝ,
sq_stable: SqStable(P),
squash: ↓T,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
subinterval: I ⊆ J ,
guard: {T},
sq_exists: ∃x:{A| B[x]},
rneq: x ≠ y,
rless: x < y
Lemmas referenced :
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,
iproper-approx,
icompact_wf,
i-approx_wf,
iproper_wf,
set_wf,
nat_plus_wf,
proper-continuous_wf,
i-member_wf,
real_wf,
rfun_wf,
interval_wf,
i-approx-monotonic,
sq_stable__icompact,
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,
i-approx-is-subinterval,
rleq_wf,
rabs_wf,
rsub_wf,
rless_wf,
int-to-real_wf,
all_wf,
rdiv_wf,
rless-int,
intformand_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
cut,
dependent_set_memberEquality,
addEquality,
hypothesisEquality,
hypothesis,
natural_numberEquality,
introduction,
extract_by_obid,
unionElimination,
independent_pairFormation,
voidElimination,
productElimination,
independent_functionElimination,
independent_isectElimination,
isectElimination,
sqequalRule,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidEquality,
intEquality,
because_Cache,
minusEquality,
productEquality,
setEquality,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_pairFormation,
int_eqEquality,
computeAll,
functionEquality,
inrFormation
Latex:
\mforall{}I:Interval
(iproper(I) {}\mRightarrow{} (\mforall{}f:I {}\mrightarrow{}\mBbbR{}. (f[x] (proper)continuous for x \mmember{} I {}\mRightarrow{} f[x] continuous for x \mmember{} I)))
Date html generated:
2016_10_26-AM-09_51_00
Last ObjectModification:
2016_09_19-PM-00_38_31
Theory : reals
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