Nuprl Lemma : rexp-unique
∀f:ℝ ⟶ ℝ
((∀x,y:ℝ. ((x = y) ⇒ (f[x] = f[y]))) ⇒ (f[r0] = r1) ⇒ d(f[x])/dx = λx.f[x] on (-∞, ∞) ⇒ (∀x:ℝ. (f[x] = e^x)))
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
riiint: (-∞, ∞),
rexp: e^x,
req: x = y,
int-to-real: r(n),
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_lambda: λ2x y.t[x; y],
member: t ∈ T,
so_apply: x[s],
so_apply: x[s1;s2],
uall: ∀[x:A]. B[x],
prop: ℙ,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
infinite-deriv-seq: infinite-deriv-seq(I;i,x.F[i; x]),
so_lambda: λ2x.t[x],
label: ...$L... t,
rfun: I ⟶ℝ,
guard: {T},
exists: ∃x:A. B[x],
nat: ℕ,
ifun: ifun(f;I),
top: Top,
real-fun: real-fun(f;a;b),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
le: A ≤ B,
less_than': less_than'(a;b),
int_upper: {i...},
subtype_rel: A ⊆r B,
cand: A c∧ B,
squash: ↓T,
true: True
Lemmas referenced :
equal-functions-by-Taylor,
real_wf,
nat_wf,
rexp_wf,
req_wf,
req_functionality,
rexp_functionality,
req_weakening,
derivative-rexp,
derivative_wf,
riiint_wf,
i-member_wf,
int-to-real_wf,
all_wf,
rccint_wf,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
set_wf,
ifun_wf,
rccint-icompact,
rleq-int,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermMinus_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_minus_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
I-norm_wf,
icompact_wf,
false_wf,
le_wf,
I-norm-bound,
int_upper_properties,
member_rccint_lemma,
subtype_rel_dep_function,
rleq_wf,
subtype_rel_self,
rabs-rleq-iff,
squash_wf,
true_wf,
rminus-int,
iff_weakening_equal,
rabs_wf,
int_upper_wf,
exists_wf,
rexp0
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
hypothesis,
isectElimination,
independent_functionElimination,
because_Cache,
independent_isectElimination,
productElimination,
setElimination,
rename,
setEquality,
natural_numberEquality,
functionEquality,
dependent_pairFormation,
dependent_set_memberEquality,
minusEquality,
isect_memberEquality,
voidElimination,
voidEquality,
unionElimination,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
equalityTransitivity,
equalitySymmetry,
productEquality,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}f:\mBbbR{} {}\mrightarrow{} \mBbbR{}
((\mforall{}x,y:\mBbbR{}. ((x = y) {}\mRightarrow{} (f[x] = f[y])))
{}\mRightarrow{} (f[r0] = r1)
{}\mRightarrow{} d(f[x])/dx = \mlambda{}x.f[x] on (-\minfty{}, \minfty{})
{}\mRightarrow{} (\mforall{}x:\mBbbR{}. (f[x] = e\^{}x)))
Date html generated:
2016_10_26-PM-00_11_49
Last ObjectModification:
2016_09_12-PM-05_39_21
Theory : reals_2
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