Nuprl Lemma : period-rsin-is-2pi
∀a:ℝ. ((r0 < a) ⇒ (∀x:ℝ. (rsin(x + a) = rsin(x))) ⇒ (2 * π ≤ a))
Proof
Definitions occuring in Statement :
pi: π,
rsin: rsin(x),
rleq: x ≤ y,
rless: x < y,
int-rmul: k1 * a,
req: x = y,
radd: a + b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
rcos-1-implies-at-least-2pi,
rless_wf,
int-to-real_wf,
all_wf,
real_wf,
req_wf,
rsin_wf,
radd_wf,
rcos_wf,
half-pi_wf,
req_functionality,
req_inversion,
rsin-shift-half-pi,
req_weakening,
rsin_functionality,
radd_comm,
rsin-half-pi
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
dependent_set_memberEquality,
hypothesisEquality,
hypothesis,
isectElimination,
natural_numberEquality,
independent_functionElimination,
sqequalRule,
lambdaEquality,
because_Cache,
independent_isectElimination,
productElimination
Latex:
\mforall{}a:\mBbbR{}. ((r0 < a) {}\mRightarrow{} (\mforall{}x:\mBbbR{}. (rsin(x + a) = rsin(x))) {}\mRightarrow{} (2 * \mpi{} \mleq{} a))
Date html generated:
2016_10_26-PM-00_26_51
Last ObjectModification:
2016_09_12-PM-05_44_04
Theory : reals_2
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