Nuprl Lemma : cos-sin-equation-non-constant1
∀f,g:ℝ ⟶ ℝ.
((∀x,y:ℝ. ((x = y) ⇒ (f(x) = f(y))))
⇒ (∀x,y:ℝ. ((x = y) ⇒ (g(x) = g(y))))
⇒ (∃a,b:ℝ. f(a) ≠ f(b))
⇒ (∀x,y:ℝ. (f(x - y) = ((f(x) * f(y)) + (g(x) * g(y)))))
⇒ (¬¬(∃a:ℝ. (a ≠ r0 ∧ (∀x:ℝ. (f(x) = rcos(a * x))) ∧ (∀x:ℝ. (g(x) = rsin(a * x)))))))
Proof
Definitions occuring in Statement :
rfun-ap: f(x),
rcos: rcos(x),
rsin: rsin(x),
rneq: x ≠ y,
rsub: x - y,
req: x = y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
or: P ∨ Q,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
le: A ≤ B,
nat: ℕ,
so_apply: x[s],
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
exists: ∃x:A. B[x],
cand: A c∧ B,
and: P ∧ Q,
iff: P ⇐⇒ Q,
member: t ∈ T,
false: False,
not: ¬A,
implies: P ⇒ Q,
all: ∀x:A. B[x],
uimplies: b supposing a
Lemmas referenced :
double-negation-hyp-elim,
rminus_wf,
rsqrt_wf,
le_wf,
false_wf,
rnexp_wf,
rleq_wf,
or_wf,
radd_wf,
rsub_wf,
rsin_wf,
rmul_wf,
rcos_wf,
rfun-ap_wf,
req_wf,
all_wf,
int-to-real_wf,
rneq_wf,
real_wf,
exists_wf,
not_wf,
cos-sin-equation,
rneq_functionality,
rneq_irreflexivity
Rules used in proof :
unionElimination,
setEquality,
rename,
setElimination,
dependent_set_memberEquality,
functionEquality,
applyEquality,
functionExtensionality,
because_Cache,
natural_numberEquality,
productEquality,
lambdaEquality,
sqequalRule,
isectElimination,
voidElimination,
independent_pairFormation,
hypothesis,
independent_functionElimination,
productElimination,
hypothesisEquality,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
thin,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_isectElimination
Latex:
\mforall{}f,g:\mBbbR{} {}\mrightarrow{} \mBbbR{}.
((\mforall{}x,y:\mBbbR{}. ((x = y) {}\mRightarrow{} (f(x) = f(y))))
{}\mRightarrow{} (\mforall{}x,y:\mBbbR{}. ((x = y) {}\mRightarrow{} (g(x) = g(y))))
{}\mRightarrow{} (\mexists{}a,b:\mBbbR{}. f(a) \mneq{} f(b))
{}\mRightarrow{} (\mforall{}x,y:\mBbbR{}. (f(x - y) = ((f(x) * f(y)) + (g(x) * g(y)))))
{}\mRightarrow{} (\mneg{}\mneg{}(\mexists{}a:\mBbbR{}. (a \mneq{} r0 \mwedge{} (\mforall{}x:\mBbbR{}. (f(x) = rcos(a * x))) \mwedge{} (\mforall{}x:\mBbbR{}. (g(x) = rsin(a * x)))))))
Date html generated:
2017_10_04-PM-11_08_57
Last ObjectModification:
2017_08_02-PM-05_56_04
Theory : reals_2
Home
Index