Nuprl Lemma : Cauchy-equation-iff
∀f:ℝ ⟶ ℝ
∀x,y:ℝ. (f(x + y) = (f(x) + f(y))) ⇐⇒ ∃c:ℝ. ∀x:ℝ. (f(x) = (c * x)) supposing ∀x,y:ℝ. ((x = y) ⇒ ((f x) = (f y)))
Proof
Definitions occuring in Statement :
rfun-ap: f(x),
req: x = y,
rmul: a * b,
radd: a + b,
real: ℝ,
uimplies: b supposing a,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
exists: ∃x:A. B[x],
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
req_witness,
real_wf,
req_wf,
all_wf,
rfun-ap_wf,
radd_wf,
exists_wf,
rmul_wf,
Cauchy-equation-1-iff,
int-to-real_wf,
rmul_comm,
req_functionality,
req_weakening,
radd_functionality,
rmul-distrib1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
extract_by_obid,
isectElimination,
applyEquality,
functionExtensionality,
hypothesis,
independent_functionElimination,
rename,
independent_pairFormation,
functionEquality,
independent_isectElimination,
productElimination,
dependent_pairFormation,
natural_numberEquality,
because_Cache,
addLevel,
existsFunctionality,
allFunctionality,
allLevelFunctionality
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) = (c * x))
supposing \mforall{}x,y:\mBbbR{}. ((x = y) {}\mRightarrow{} ((f x) = (f y)))
Date html generated:
2017_10_04-PM-11_02_41
Last ObjectModification:
2017_06_30-PM-11_42_18
Theory : reals_2
Home
Index