Nuprl Lemma : rexp-functional-equation
∀f:ℝ ⟶ ℝ
∀x,y:ℝ. (f(x + y) = (f(x) * f(y))) ⇐⇒ (∃c:ℝ. ∀x:ℝ. (f(x) = e^c * x)) ∨ (∀x:ℝ. (f(x) = r0))
supposing ∀x,y:ℝ. ((x = y) ⇒ ((f x) = (f y)))
Proof
Definitions occuring in Statement :
rfun-ap: f(x),
rexp: e^x,
req: x = y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
or: P ∨ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
cand: A c∧ B,
req_int_terms: t1 ≡ t2,
rdiv: (x/y),
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
sq_type: SQType(T),
false: False,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
decidable: Dec(P),
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
rfun-ap: f(x),
rev_implies: P ⇐ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
and: P ∧ Q,
iff: P ⇐⇒ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x]
Lemmas referenced :
rexp-radd,
rmul-distrib1,
rmul-zero,
rexp-rlog,
rexp_functionality,
rlog-rmul,
rlog_functionality,
rlog_wf,
Cauchy-equation-iff,
equal_wf,
rneq_wf,
rmul-is-positive,
real_term_value_minus_lemma,
rinv-mul-as-rdiv,
rminus_functionality,
rmul_functionality,
itermMinus_wf,
radd-rminus,
rminus_wf,
rless_functionality,
rfun-ap_functionality,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
int-rinv-cancel,
req_transitivity,
rmul_comm,
nequal_wf,
true_wf,
equal-wf-base,
int_formula_prop_wf,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
intformeq_wf,
intformnot_wf,
full-omega-unsat,
decidable__equal_int,
int_subtype_base,
subtype_base_sq,
req-iff-rsub-is-0,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
itermMultiply_wf,
itermSubtract_wf,
rinv_wf2,
rmul_preserves_req,
rless_wf,
rless-int,
rdiv_wf,
req_weakening,
req_inversion,
req_functionality,
radd-zero,
square-req-self-iff,
int-to-real_wf,
rexp_wf,
exists_wf,
or_wf,
rmul_wf,
radd_wf,
rfun-ap_wf,
all_wf,
real_wf,
req_wf,
req_witness
Rules used in proof :
productEquality,
functionExtensionality,
int_eqEquality,
addLevel,
dependent_set_memberEquality,
equalitySymmetry,
equalityTransitivity,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_pairFormation,
approximateComputation,
intEquality,
cumulativity,
instantiate,
baseClosed,
imageMemberEquality,
inrFormation,
inlFormation,
unionElimination,
independent_isectElimination,
because_Cache,
productElimination,
functionEquality,
natural_numberEquality,
independent_pairFormation,
rename,
hypothesis,
independent_functionElimination,
applyEquality,
isectElimination,
extract_by_obid,
hypothesisEquality,
thin,
dependent_functionElimination,
lambdaEquality,
sqequalHypSubstitution,
sqequalRule,
introduction,
cut,
isect_memberFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}f:\mBbbR{} {}\mrightarrow{} \mBbbR{}
\mforall{}x,y:\mBbbR{}. (f(x + y) = (f(x) * f(y))) \mLeftarrow{}{}\mRightarrow{} (\mexists{}c:\mBbbR{}. \mforall{}x:\mBbbR{}. (f(x) = e\^{}c * x)) \mvee{} (\mforall{}x:\mBbbR{}. (f(x) = r0))
supposing \mforall{}x,y:\mBbbR{}. ((x = y) {}\mRightarrow{} ((f x) = (f y)))
Date html generated:
2018_05_22-PM-03_08_46
Last ObjectModification:
2018_05_20-PM-11_38_31
Theory : reals_2
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