Nuprl Lemma : fun-converges-rmul
∀I:Interval. ∀f:ℕ ⟶ I ⟶ℝ.
(λn.f[n;x]↓ for x ∈ I) ⇒ (∀g:I ⟶ℝ. (g[x] continuous for x ∈ I ⇒ λn.f[n;x] * g[x]↓ for x ∈ I))))
Proof
Definitions occuring in Statement :
fun-converges: λn.f[n; x]↓ for x ∈ I),
continuous: f[x] continuous for x ∈ I,
rfun: I ⟶ℝ,
interval: Interval,
rmul: a * b,
nat: ℕ,
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 :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
rfun: I ⟶ℝ,
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
so_apply: x[s],
rev_implies: P ⇐ Q,
fun-cauchy: λn.f[n; x] is cauchy for x ∈ I,
so_lambda: λ2x.t[x],
label: ...$L... t,
nat_plus: ℕ+,
exists: ∃x:A. B[x],
uimplies: b supposing a,
subinterval: I ⊆ J ,
guard: {T},
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
rneq: x ≠ y,
or: P ∨ Q,
int_upper: {i...},
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
le: A ≤ B,
uiff: uiff(P;Q),
cand: A c∧ B,
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
fun-converges-iff-cauchy,
nat_wf,
i-member_wf,
real_wf,
rmul_wf,
rfun_wf,
nat_plus_wf,
set_wf,
icompact_wf,
i-approx_wf,
continuous_wf,
fun-converges_wf,
interval_wf,
i-approx-is-subinterval,
less_than_wf,
continuous_functionality_wrt_subinterval,
r-bound_wf,
Inorm_wf,
subtype_rel_sets,
all_wf,
rleq_wf,
rabs_wf,
int-to-real_wf,
Inorm-bound,
rfun_subtype,
r-bound-property,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
mul_nat_plus,
int_upper_wf,
rsub_wf,
int_upper_subtype_nat,
nat_plus_subtype_nat,
rdiv_wf,
rless-int,
int_upper_properties,
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,
rless_wf,
uiff_transitivity,
rleq_functionality,
rabs_functionality,
req_inversion,
rmul-rsub-distrib,
req_weakening,
rabs-rmul,
mul_bounds_1b,
zero-rleq-rabs,
rleq-int,
sq_stable__icompact,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
multiply_nat_plus,
itermMultiply_wf,
intformeq_wf,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
equal_wf,
rmul_functionality_wrt_rleq2,
rmul_comm,
rleq-int-fractions,
rmul-int-rdiv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesis,
because_Cache,
setElimination,
rename,
dependent_set_memberEquality,
isectElimination,
setEquality,
productElimination,
independent_functionElimination,
functionEquality,
natural_numberEquality,
dependent_pairFormation,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
inrFormation,
unionElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
multiplyEquality,
inlFormation,
imageMemberEquality,
baseClosed,
imageElimination,
productEquality,
applyLambdaEquality
Latex:
\mforall{}I:Interval. \mforall{}f:\mBbbN{} {}\mrightarrow{} I {}\mrightarrow{}\mBbbR{}.
(\mlambda{}n.f[n;x]\mdownarrow{} for x \mmember{} I) {}\mRightarrow{} (\mforall{}g:I {}\mrightarrow{}\mBbbR{}. (g[x] continuous for x \mmember{} I {}\mRightarrow{} \mlambda{}n.f[n;x] * g[x]\mdownarrow{} for x \mmember{} I))))
Date html generated:
2017_10_03-PM-00_03_04
Last ObjectModification:
2017_07_28-AM-08_31_24
Theory : reals
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