Nuprl Lemma : cos-sin-equation-non-constant2
∀f,g:ℝ ⟶ ℝ.
((∀x,y:ℝ. ((x = y) ⇒ (f(x) = f(y))))
⇒ (∀x,y:ℝ. ((x = y) ⇒ (g(x) = g(y))))
⇒ (∃u,v:ℝ. f(u) ≠ f(v))
⇒ (∀x,y:ℝ. (f(x - y) = ((f(x) * f(y)) + (g(x) * g(y)))))
⇒ (∃I:Interval. (iproper(I) ∧ (r0 ∈ I) ∧ (∃g':I ⟶ℝ. d(g(x))/dx = λx.g' x on I)))
⇒ (∃a:ℝ. ((∀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),
derivative: d(f[x])/dx = λz.g[z] on I,
rfun: I ⟶ℝ,
i-member: r ∈ I,
iproper: iproper(I),
interval: Interval,
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],
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
rev_uimplies: rev_uimplies(P;Q),
top: Top,
accelerate: accelerate(k;f),
approx-arg: approx-arg(f;B;x),
rsin: rsin(x),
guard: {T},
r-ap: f(x),
rfun-eq: rfun-eq(I;f;g),
uiff: uiff(P;Q),
or: P ∨ Q,
uimplies: b supposing a,
stable: Stable{P},
false: False,
not: ¬A,
cand: A c∧ B,
subtype_rel: A ⊆r B,
label: ...$L... t,
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
uall: ∀[x:A]. B[x],
rfun: I ⟶ℝ,
and: P ∧ Q,
exists: ∃x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
rcos0,
req_transitivity,
rmul-identity1,
rmul-zero,
subinterval-riiint,
derivative_functionality_wrt_subinterval,
rmul_comm,
derivative-id,
derivative-const-mul,
rmul-one,
derivative-rsin,
iproper-riiint,
true_wf,
subtype_rel_dep_function,
member_riiint_lemma,
top_wf,
riiint_wf,
simple-chain-rule,
derivative_functionality,
derivative_unique,
set_wf,
rsin_functionality,
rmul_functionality,
rcos_functionality,
req_weakening,
req_functionality,
minimal-not-not-excluded-middle,
minimal-double-negation-hyp-elim,
not_wf,
or_wf,
false_wf,
stable_req,
stable__all,
stable__and,
rneq_wf,
radd_wf,
rsub_wf,
subtype_rel_self,
derivative_wf,
rfun_wf,
iproper_wf,
interval_wf,
exists_wf,
rsin_wf,
rmul_wf,
rcos_wf,
rfun-ap_wf,
req_wf,
real_wf,
all_wf,
i-member_wf,
int-to-real_wf,
cos-sin-equation-non-constant1
Rules used in proof :
voidEquality,
isect_memberEquality,
allFunctionality,
addLevel,
unionElimination,
independent_isectElimination,
voidElimination,
independent_pairFormation,
functionEquality,
setEquality,
rename,
setElimination,
because_Cache,
lambdaEquality,
sqequalRule,
productEquality,
natural_numberEquality,
isectElimination,
dependent_set_memberEquality,
applyEquality,
dependent_pairFormation,
productElimination,
independent_functionElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
hypothesis,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut
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{}u,v:\mBbbR{}. f(u) \mneq{} f(v))
{}\mRightarrow{} (\mforall{}x,y:\mBbbR{}. (f(x - y) = ((f(x) * f(y)) + (g(x) * g(y)))))
{}\mRightarrow{} (\mexists{}I:Interval. (iproper(I) \mwedge{} (r0 \mmember{} I) \mwedge{} (\mexists{}g':I {}\mrightarrow{}\mBbbR{}. d(g(x))/dx = \mlambda{}x.g' x on I)))
{}\mRightarrow{} (\mexists{}a:\mBbbR{}. ((\mforall{}x:\mBbbR{}. (f(x) = rcos(a * x))) \mwedge{} (\mforall{}x:\mBbbR{}. (g(x) = rsin(a * x))))))
Date html generated:
2018_05_22-PM-03_11_20
Last ObjectModification:
2018_05_20-PM-11_54_53
Theory : reals_2
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