Nuprl Lemma : req-rcos-and-rsin-implies
∀x,y:ℝ. ((rsin(x) = rsin(y)) ⇒ (rcos(x) = rcos(y)) ⇒ (∃n:ℤ. ((x - y) = 2 * n * π)))
Proof
Definitions occuring in Statement :
pi: π,
rcos: rcos(x),
rsin: rsin(x),
int-rmul: k1 * a,
rsub: x - y,
req: x = y,
real: ℝ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
multiply: n * m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
prop: ℙ,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
false: False,
uimplies: b supposing a,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
rcos-is-1-iff,
rsub_wf,
req_wf,
rcos_wf,
rsin_wf,
real_wf,
int-to-real_wf,
radd_wf,
rmul_wf,
rnexp_wf,
istype-void,
istype-le,
rsin-rcos-pythag,
uiff_transitivity,
req_functionality,
req_transitivity,
rcos-rsub,
radd_functionality,
rmul_functionality,
req_weakening,
req_inversion,
rnexp2,
radd_comm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
productElimination,
independent_functionElimination,
universeIsType,
inhabitedIsType,
natural_numberEquality,
dependent_set_memberEquality_alt,
independent_pairFormation,
sqequalRule,
voidElimination,
because_Cache,
independent_isectElimination
Latex:
\mforall{}x,y:\mBbbR{}. ((rsin(x) = rsin(y)) {}\mRightarrow{} (rcos(x) = rcos(y)) {}\mRightarrow{} (\mexists{}n:\mBbbZ{}. ((x - y) = 2 * n * \mpi{})))
Date html generated:
2019_10_31-AM-06_06_45
Last ObjectModification:
2019_05_17-PM-03_55_26
Theory : reals_2
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