Nuprl Lemma : fun-converges-to-continuous
∀[I:Interval]. ∀[f:ℕ ⟶ I ⟶ℝ]. ∀[g:I ⟶ℝ].
(lim n→∞.f[n;x] = λy.g[y] for x ∈ I ⇒ (∀n:ℕ. f[n;x] continuous for x ∈ I) ⇒ g[y] continuous for y ∈ I)
Proof
Definitions occuring in Statement :
fun-converges-to: lim n→∞.f[n; x] = λy.g[y] for x ∈ I,
continuous: f[x] continuous for x ∈ I,
rfun: I ⟶ℝ,
interval: Interval,
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
continuous: f[x] continuous for x ∈ I,
all: ∀x:A. B[x],
fun-converges-to: lim n→∞.f[n; x] = λy.g[y] for x ∈ I,
member: t ∈ T,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
prop: ℙ,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
sq_exists: ∃x:{A| B[x]},
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
rfun: I ⟶ℝ,
uimplies: b supposing a,
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,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
int_upper: {i...},
real: ℝ,
sq_stable: SqStable(P),
uiff: uiff(P;Q)
Lemmas referenced :
radd-int,
rdiv_functionality,
radd-rdiv,
req_transitivity,
uiff_transitivity,
int_term_value_add_lemma,
itermAdd_wf,
rleq-int-fractions,
rabs-difference-symmetry,
req_weakening,
radd_functionality,
rleq_functionality,
le_wf,
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
sq_stable__icompact,
sq_stable__less_than,
int_term_value_mul_lemma,
itermMultiply_wf,
radd_functionality_wrt_rleq,
r-triangle-inequality2,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
radd_wf,
uimplies_transitivity,
interval_wf,
fun-converges-to_wf,
rfun_wf,
continuous_wf,
icompact_wf,
set_wf,
nat_plus_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_plus_properties,
rless-int,
rdiv_wf,
real_wf,
all_wf,
int-to-real_wf,
rless_wf,
i-approx_wf,
i-member_wf,
rsub_wf,
rabs_wf,
rleq_wf,
i-member-approx,
nat_plus_subtype_nat,
less_than_wf,
mul_nat_plus,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
cut,
hypothesis,
because_Cache,
lemma_by_obid,
isectElimination,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
introduction,
imageMemberEquality,
baseClosed,
productElimination,
applyEquality,
setElimination,
rename,
independent_functionElimination,
productEquality,
lambdaEquality,
functionEquality,
independent_isectElimination,
inrFormation,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
setEquality,
equalityTransitivity,
equalitySymmetry,
multiplyEquality,
addEquality,
imageElimination
Latex:
\mforall{}[I:Interval]. \mforall{}[f:\mBbbN{} {}\mrightarrow{} I {}\mrightarrow{}\mBbbR{}]. \mforall{}[g:I {}\mrightarrow{}\mBbbR{}].
(lim n\mrightarrow{}\minfty{}.f[n;x] = \mlambda{}y.g[y] for x \mmember{} I
{}\mRightarrow{} (\mforall{}n:\mBbbN{}. f[n;x] continuous for x \mmember{} I)
{}\mRightarrow{} g[y] continuous for y \mmember{} I)
Date html generated:
2016_05_18-AM-09_52_47
Last ObjectModification:
2016_01_17-AM-02_54_59
Theory : reals
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