Nuprl Lemma : rcos-1-implies-at-least-2pi
∀x:{x:ℝ| r0 < x} . ((rcos(x) = r1) ⇒ (2 * π ≤ x))
Proof
Definitions occuring in Statement :
pi: π,
rcos: rcos(x),
rleq: x ≤ y,
rless: x < y,
int-rmul: k1 * a,
req: x = y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
uimplies: b supposing a,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
rfun: I ⟶ℝ,
true: True,
and: P ∧ Q,
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
isl: isl(x),
rccint: [l, u],
i-finite: i-finite(I),
top: Top,
iproper: iproper(I),
subinterval: I ⊆ J ,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
real-fun: real-fun(f;a;b),
ifun: ifun(f;I),
rsub: x - y,
squash: ↓T,
cand: A c∧ B,
sq_stable: SqStable(P),
or: P ∨ Q,
guard: {T},
strictly-increasing-on-interval: f[x] strictly-increasing for x ∈ I,
not: ¬A,
false: False,
strictly-decreasing-on-interval: f[x] strictly-decreasing for x ∈ I
Lemmas referenced :
rless_wf,
real_wf,
set_wf,
int-to-real_wf,
rcos_wf,
req_wf,
rcos-strictly-decreasing,
rmul-distrib2,
rmul-identity1,
req_inversion,
rminus-as-rmul,
radd_functionality,
req_transitivity,
radd-int,
rmul_functionality,
rmul-zero-both,
rmul-one-both,
int-rmul-req,
req_weakening,
rless_functionality,
radd_wf,
pi-positive,
radd-preserves-rless,
rmul_wf,
rooint_wf,
rsin_wf,
rminus_wf,
i-member_wf,
derivative-implies-strictly-increasing-closed,
int-rmul_wf,
pi_wf,
rccint_wf,
i-finite_wf,
right_endpoint_rccint_lemma,
left_endpoint_rccint_lemma,
deriviative-rcos,
rleq_wf,
member_riiint_lemma,
member_rccint_lemma,
riiint_wf,
derivative_functionality_wrt_subinterval,
rsin_functionality,
rminus_functionality,
req_functionality,
right-endpoint_wf,
left-endpoint_wf,
radd-zero-both,
radd_comm,
rleq_functionality,
uiff_transitivity,
radd-preserves-rleq,
radd-rminus-assoc,
radd-ac,
radd-assoc,
rsub_wf,
rsin-shift-pi,
sq_stable__rleq,
rsin-nonneg,
member_rooint_lemma,
rsin-positive,
radd-rminus-both,
sq_stable__rless,
rless-cases,
rleq-iff-all-rless,
rleq_weakening_rless,
rleq_weakening_equal,
rless_transitivity2,
rcos_functionality,
rcos0,
rcos-shift-2pi,
rless_irreflexivity,
rleq_weakening,
rless_transitivity1,
not-rless
Rules used in proof :
lambdaEquality,
sqequalRule,
natural_numberEquality,
hypothesis,
hypothesisEquality,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
addLevel,
independent_isectElimination,
addEquality,
minusEquality,
productElimination,
because_Cache,
setEquality,
dependent_set_memberEquality,
independent_pairFormation,
independent_functionElimination,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
productEquality,
imageElimination,
baseClosed,
imageMemberEquality,
levelHypothesis,
unionElimination,
promote_hyp
Latex:
\mforall{}x:\{x:\mBbbR{}| r0 < x\} . ((rcos(x) = r1) {}\mRightarrow{} (2 * \mpi{} \mleq{} x))
Date html generated:
2016_10_26-PM-00_26_37
Last ObjectModification:
2016_10_12-PM-03_52_26
Theory : reals_2
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