Nuprl Lemma : exp-exists
∀x:ℝ. ∃a:ℝ. Σn.(x^n)/(n)! = a
Proof
Definitions occuring in Statement :
series-sum: Σn.x[n] = a,
rnexp: x^k1,
int-rdiv: (a)/k1,
real: ℝ,
fact: (n)!,
all: ∀x:A. B[x],
exists: ∃x:A. B[x]
Definitions unfolded in proof :
iff: P ⇐⇒ Q,
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
nat: ℕ,
guard: {T},
false: False,
not: ¬A,
nequal: a ≠ b ∈ T ,
implies: P ⇒ Q,
int_nzero: ℤ-o,
subtype_rel: A ⊆r B,
series-converges: Σn.x[n]↓,
and: P ∧ Q,
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
nat_plus: ℕ+,
exists: ∃x:A. B[x],
real: ℝ,
uimplies: b supposing a,
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
iff_weakening_equal,
subtype_rel_self,
int_nzero_wf,
true_wf,
squash_wf,
nat_wf,
rnexp_wf,
int_subtype_base,
equal-wf-base,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
nat_plus_properties,
nequal_wf,
subtype_rel_sets,
fact_wf,
int-rdiv_wf,
series-sum_wf,
real_wf,
value-type_wf,
int-value-type,
less_than_wf,
function-value-type,
regular-int-seq_wf,
nat_plus_wf,
set-value-type,
equal_wf,
exp-series-converges
Rules used in proof :
universeEquality,
instantiate,
imageElimination,
voidEquality,
voidElimination,
isect_memberEquality,
int_eqEquality,
independent_functionElimination,
approximateComputation,
applyLambdaEquality,
setEquality,
applyEquality,
productElimination,
dependent_functionElimination,
rename,
setElimination,
baseClosed,
imageMemberEquality,
independent_pairFormation,
dependent_pairFormation,
natural_numberEquality,
intEquality,
functionEquality,
independent_isectElimination,
lambdaEquality,
sqequalRule,
equalitySymmetry,
hypothesis,
equalityTransitivity,
thin,
isectElimination,
sqequalHypSubstitution,
dependent_set_memberEquality,
cutEval,
hypothesisEquality,
because_Cache,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}x:\mBbbR{}. \mexists{}a:\mBbbR{}. \mSigma{}n.(x\^{}n)/(n)! = a
Date html generated:
2018_05_22-PM-02_03_58
Last ObjectModification:
2018_05_21-AM-00_16_30
Theory : reals
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