Nuprl Lemma : converges-to_functionality
∀x1,x2:ℕ ⟶ ℝ. ∀y1,y2:ℝ. ({lim n→∞.x1[n] = y1 ⇒ lim n→∞.x2[n] = y2}) supposing ((y1 = y2) and (∀n:ℕ. (x1[n] = x2[n])))
Proof
Definitions occuring in Statement :
converges-to: lim n→∞.x[n] = y,
req: x = y,
real: ℝ,
nat: ℕ,
uimplies: b supposing a,
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
guard: {T},
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_apply: x[s],
implies: P ⇒ Q,
converges-to: lim n→∞.x[n] = y,
nat_plus: ℕ+,
rneq: x ≠ y,
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
sq_exists: ∃x:{A| B[x]},
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
prop: ℙ,
so_lambda: λ2x.t[x],
uiff: uiff(P;Q)
Lemmas referenced :
req_weakening,
rsub_functionality,
rabs_functionality,
rleq_functionality,
rsub_wf,
rabs_wf,
rleq_wf,
le_wf,
real_wf,
all_wf,
req_wf,
converges-to_wf,
nat_plus_wf,
rless_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,
nat_properties,
rless-int,
int-to-real_wf,
rdiv_wf,
nat_wf,
req_witness
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
lemma_by_obid,
isectElimination,
applyEquality,
independent_functionElimination,
hypothesis,
rename,
natural_numberEquality,
setElimination,
independent_isectElimination,
inrFormation,
because_Cache,
productElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
promote_hyp,
functionEquality,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}x1,x2:\mBbbN{} {}\mrightarrow{} \mBbbR{}. \mforall{}y1,y2:\mBbbR{}.
(\{lim n\mrightarrow{}\minfty{}.x1[n] = y1 {}\mRightarrow{} lim n\mrightarrow{}\minfty{}.x2[n] = y2\}) supposing ((y1 = y2) and (\mforall{}n:\mBbbN{}. (x1[n] = x2[n])))
Date html generated:
2016_05_18-AM-07_35_42
Last ObjectModification:
2016_01_17-AM-02_02_40
Theory : reals
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