Nuprl Lemma : fun-converges-to-rminus
∀I:Interval. ∀f:ℕ ⟶ I ⟶ℝ. ∀g:I ⟶ℝ. (lim n→∞.f[n;x] = λy.g[y] for x ∈ I ⇒ lim n→∞.-(f[n;x]) = λy.-(g[y]) for x ∈ I)
Proof
Definitions occuring in Statement :
fun-converges-to: lim n→∞.f[n; x] = λy.g[y] for x ∈ I,
rfun: I ⟶ℝ,
interval: Interval,
rminus: -(x),
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,
fun-converges-to: lim n→∞.f[n; x] = λy.g[y] for x ∈ I,
member: t ∈ T,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
prop: ℙ,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
rfun: I ⟶ℝ,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
int_upper: {i...},
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
so_lambda: λ2x y.t[x; y],
label: ...$L... t,
subinterval: I ⊆ J ,
le: A ≤ B,
rsub: x - y,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
true: True,
squash: ↓T
Lemmas referenced :
i-approx-is-subinterval,
less_than_wf,
int_upper_wf,
set_wf,
real_wf,
i-member_wf,
i-approx_wf,
all_wf,
rleq_wf,
rabs_wf,
rsub_wf,
rminus_wf,
int_upper_subtype_nat,
nat_plus_subtype_nat,
rdiv_wf,
int-to-real_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,
nat_plus_wf,
icompact_wf,
fun-converges-to_wf,
nat_wf,
rfun_wf,
interval_wf,
req_wf,
subtype_rel_sets,
radd_wf,
rmul_wf,
req_weakening,
uiff_transitivity,
req_functionality,
rminus-radd,
radd_functionality,
rminus-rminus,
radd_comm,
rminus-as-rmul,
rmul_functionality,
req_inversion,
req_transitivity,
uiff_transitivity2,
rleq_functionality,
rabs_functionality,
squash_wf,
true_wf,
rabs-rminus
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
dependent_set_memberEquality,
setElimination,
rename,
hypothesis,
isectElimination,
natural_numberEquality,
productElimination,
dependent_pairFormation,
sqequalRule,
lambdaEquality,
because_Cache,
setEquality,
applyEquality,
functionExtensionality,
independent_isectElimination,
inrFormation,
independent_functionElimination,
unionElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
functionEquality,
minusEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed
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{} lim n\mrightarrow{}\minfty{}.-(f[n;x]) = \mlambda{}y.-(g[y]) for x \mmember{} I)
Date html generated:
2016_10_26-AM-11_13_43
Last ObjectModification:
2016_08_28-PM-06_43_55
Theory : reals
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