Nuprl Lemma : rexp-is-limit
∀x:ℝ. Σn.(x^n)/(n)! = e^x
Proof
Definitions occuring in Statement :
rexp: e^x,
series-sum: Σn.x[n] = a,
rnexp: x^k1,
int-rdiv: (a)/k1,
real: ℝ,
all: ∀x:A. B[x],
fact: (n)!
Definitions unfolded in proof :
all: ∀x:A. B[x],
rexp: e^x,
member: t ∈ T,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
int_nzero: ℤ-o,
so_apply: x[s],
uimplies: b supposing a,
prop: ℙ,
implies: P ⇒ Q,
nequal: a ≠ b ∈ T ,
not: ¬A,
false: False,
guard: {T},
nat: ℕ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
pi1: fst(t)
Lemmas referenced :
exp-exists-ext,
all_wf,
exists_wf,
series-sum_wf,
int-rdiv_wf,
fact_wf,
subtype_rel_sets,
less_than_wf,
nequal_wf,
nat_plus_properties,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
equal-wf-base,
int_subtype_base,
rnexp_wf,
nat_wf,
real_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
applyEquality,
thin,
instantiate,
extract_by_obid,
hypothesis,
lambdaEquality,
sqequalHypSubstitution,
sqequalRule,
hypothesisEquality,
introduction,
isectElimination,
because_Cache,
intEquality,
natural_numberEquality,
independent_isectElimination,
setElimination,
rename,
setEquality,
equalityTransitivity,
equalitySymmetry,
Error :applyLambdaEquality,
dependent_pairFormation,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
baseClosed,
independent_functionElimination,
productElimination
Latex:
\mforall{}x:\mBbbR{}. \mSigma{}n.(x\^{}n)/(n)! = e\^{}x
Date html generated:
2016_10_26-AM-09_27_10
Last ObjectModification:
2016_08_26-PM-02_51_35
Theory : reals
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