Nuprl Lemma : cos-sin-equation-nc
∀f,g:ℝ ⟶ ℝ.
((∀x,y:ℝ. ((x = y) ⇒ (f(x) = f(y))))
⇒ (∀x,y:ℝ. ((x = y) ⇒ (g(x) = g(y))))
⇒ (∀x,y:ℝ. (f(x - y) = ((f(x) * f(y)) + (g(x) * g(y)))))
⇒ (∃a:ℝ. f(a) ≠ f(r0))
⇒ (∃b:ℝ. g(-(b)) ≠ g(b))
⇒ ((∀x,y:ℝ. (f(x - y) = f(y - x)))
∧ (∀y:ℝ. (f(-(y)) = f(y)))
∧ (∀y:ℝ. (g(-(y)) = -(g(y))))
∧ (g(r0) = r0)
∧ (∃d:ℝ. f(d) ≠ r0)
∧ (f(r0) = r1)
∧ (∀x:ℝ. ((f(x)^2 + g(x)^2) = r1))
∧ (∀x:ℝ. (|f(x)| ≤ r1))
∧ (∀x,y:ℝ. ((f(x - -(y)) + f(x - y)) = (r(2) * f(x) * f(y))))
∧ (∀x,y:ℝ. ((f(x + y) + f(x - y)) = (r(2) * f(x) * f(y))))))
Proof
Definitions occuring in Statement :
rfun-ap: f(x),
rneq: x ≠ y,
rleq: x ≤ y,
rabs: |x|,
rnexp: x^k1,
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 :
true: True,
squash: ↓T,
less_than: a < b,
rev_implies: P ⇐ Q,
rneq: x ≠ y,
iff: P ⇐⇒ Q,
or: P ∨ Q,
exists: ∃x:A. B[x],
less_than': less_than'(a;b),
le: A ≤ B,
nat: ℕ,
guard: {T},
rfun-ap: f(x),
top: Top,
not: ¬A,
false: False,
req_int_terms: t1 ≡ t2,
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
uiff: uiff(P;Q),
so_apply: x[s],
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
cand: A c∧ B,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
radd-preserves-rleq,
radd_comm,
rabs-of-nonneg,
rabs-rnexp,
rnexp-one,
rleq_functionality,
rnexp2-nonneg,
rleq-int,
rabs_wf,
square-rleq-implies,
rmul-identity1,
rmul-zero-both,
rminus-zero,
rless_wf,
rless-int,
rneq_irreflexivity,
rneq_functionality,
req-implies-req,
rmul_preserves_req,
squares-req,
rneq-cases,
rmul_comm,
rnexp2,
le_wf,
false_wf,
rnexp_wf,
radd-rminus-assoc,
radd-preserves-req,
req_inversion,
rmul_functionality,
radd_functionality,
req_transitivity,
req_weakening,
real_term_value_minus_lemma,
rfun-ap_functionality,
itermMinus_wf,
itermConstant_wf,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_add_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
req_functionality,
req-iff-rsub-is-0,
itermVar_wf,
itermMultiply_wf,
itermAdd_wf,
itermSubtract_wf,
rmul_wf,
radd_wf,
rsub_wf,
req_wf,
all_wf,
int-to-real_wf,
rminus_wf,
rfun-ap_wf,
rneq_wf,
exists_wf,
real_wf
Rules used in proof :
equalitySymmetry,
equalityTransitivity,
inrFormation,
baseClosed,
imageMemberEquality,
inlFormation,
minusEquality,
dependent_pairFormation,
unionElimination,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
promote_hyp,
allFunctionality,
levelHypothesis,
addLevel,
independent_functionElimination,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
approximateComputation,
dependent_functionElimination,
independent_isectElimination,
productElimination,
functionEquality,
natural_numberEquality,
hypothesisEquality,
lambdaEquality,
sqequalRule,
thin,
isectElimination,
sqequalHypSubstitution,
because_Cache,
independent_pairFormation,
hypothesis,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
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{} (\mforall{}x,y:\mBbbR{}. (f(x - y) = ((f(x) * f(y)) + (g(x) * g(y)))))
{}\mRightarrow{} (\mexists{}a:\mBbbR{}. f(a) \mneq{} f(r0))
{}\mRightarrow{} (\mexists{}b:\mBbbR{}. g(-(b)) \mneq{} g(b))
{}\mRightarrow{} ((\mforall{}x,y:\mBbbR{}. (f(x - y) = f(y - x)))
\mwedge{} (\mforall{}y:\mBbbR{}. (f(-(y)) = f(y)))
\mwedge{} (\mforall{}y:\mBbbR{}. (g(-(y)) = -(g(y))))
\mwedge{} (g(r0) = r0)
\mwedge{} (\mexists{}d:\mBbbR{}. f(d) \mneq{} r0)
\mwedge{} (f(r0) = r1)
\mwedge{} (\mforall{}x:\mBbbR{}. ((f(x)\^{}2 + g(x)\^{}2) = r1))
\mwedge{} (\mforall{}x:\mBbbR{}. (|f(x)| \mleq{} r1))
\mwedge{} (\mforall{}x,y:\mBbbR{}. ((f(x - -(y)) + f(x - y)) = (r(2) * f(x) * f(y))))
\mwedge{} (\mforall{}x,y:\mBbbR{}. ((f(x + y) + f(x - y)) = (r(2) * f(x) * f(y))))))
Date html generated:
2018_05_22-PM-03_09_53
Last ObjectModification:
2018_05_20-PM-11_53_20
Theory : reals_2
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