Nuprl Lemma : cos-sin-equation-non-constant3
∀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)))))
⇒ (∃b:ℝ. (r0_∫-b -(g(x)) dx ≠ r0 ∧ f(b) ≠ r1))
⇒ (∃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),
integral: a_∫-b f[x] dx,
rneq: x ≠ y,
rsub: x - y,
req: x = y,
rmul: a * b,
rminus: -(x),
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,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
rfun-ap: f(x),
exists: ∃x:A. B[x],
and: P ∧ Q,
rneq: x ≠ y,
or: P ∨ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
prop: ℙ,
rfun: I ⟶ℝ,
ifun: ifun(f;I),
top: Top,
real-fun: real-fun(f;a;b),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
stable: Stable{P},
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
rat_term_to_real: rat_term_to_real(f;t),
rtermDivide: num "/" denom,
rat_term_ind: rat_term_ind,
rtermVar: rtermVar(var),
pi1: fst(t),
true: True,
rtermMultiply: left "*" right,
rtermConstant: "const",
pi2: snd(t),
rdiv: (x/y),
rev_implies: P ⇐ Q,
guard: {T},
rtermSubtract: left "-" right,
cand: A c∧ B,
real-sfun: real-sfun(f;a;b)
Lemmas referenced :
cos-sin-equation-non-constant1,
rless-implies-rless,
rsub_wf,
rfun-ap_wf,
int-to-real_wf,
rless_wf,
real_wf,
rneq_wf,
rminus_wf,
i-member_wf,
rccint_wf,
rmin_wf,
rmax_wf,
left_endpoint_rccint_lemma,
istype-void,
right_endpoint_rccint_lemma,
req_functionality,
rminus_functionality,
rfun-ap_functionality,
req_weakening,
req_wf,
ifun_wf,
rccint-icompact,
rmin-rleq-rmax,
integral_wf,
radd_wf,
rmul_wf,
itermSubtract_wf,
itermConstant_wf,
itermVar_wf,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
rsin_wf,
stable_req,
rdiv_wf,
false_wf,
rcos_wf,
not_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
ftc-total-integral,
derivative-function-rmul-const,
rsin_functionality,
derivative-rcos,
riiint_wf,
derivative_functionality,
rmul_functionality,
istype-true,
member_riiint_lemma,
derivative-mul,
derivative-const,
assert-rat-term-eq2,
rtermMultiply_wf,
rtermDivide_wf,
rtermConstant_wf,
rtermVar_wf,
rmul_preserves_req,
rinv_wf2,
itermMultiply_wf,
itermAdd_wf,
itermMinus_wf,
req_transitivity,
rmul-rinv3,
real_term_value_mul_lemma,
real_term_value_add_lemma,
real_term_value_minus_lemma,
iff_weakening_uiff,
req_inversion,
rcos_functionality,
derivative-rsin,
derivative_unique,
iproper-riiint,
req-implies-req,
rcos0,
rsub_functionality,
rdiv_functionality,
radd_functionality,
rmul-rinv,
rneq_functionality,
rtermSubtract_wf,
stable__and,
all_wf,
stable__all,
real-fun-implies-sfun,
rleq_wf,
member_rccint_lemma,
rmin-rleq,
rleq-rmax,
rmul-nonzero
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
sqequalRule,
productElimination,
unionElimination,
inlFormation_alt,
isectElimination,
natural_numberEquality,
independent_isectElimination,
universeIsType,
inrFormation_alt,
productIsType,
dependent_set_memberEquality_alt,
lambdaEquality_alt,
setElimination,
rename,
setIsType,
inhabitedIsType,
isect_memberEquality_alt,
voidElimination,
because_Cache,
closedConclusion,
equalityTransitivity,
equalitySymmetry,
functionIsType,
approximateComputation,
int_eqEquality,
unionEquality,
productEquality,
functionEquality,
unionIsType,
applyEquality,
independent_pairFormation,
minusEquality,
equalityIstype,
dependent_pairFormation_alt,
promote_hyp
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{} (\mexists{}b:\mBbbR{}. (r0\_\mint{}\msupminus{}b -(g(x)) dx \mneq{} r0 \mwedge{} f(b) \mneq{} r1))
{}\mRightarrow{} (\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:
2019_10_31-AM-06_25_00
Last ObjectModification:
2019_04_02-PM-10_27_45
Theory : reals_2
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