Nuprl Lemma : arcsine-bounds
∀x:{x:ℝ| x ∈ (r(-1), r1)} . (arcsine(x) ∈ (-(π/2), π/2))
Proof
Definitions occuring in Statement :
arcsine: arcsine(x),
halfpi: π/2,
rooint: (l, u),
i-member: r ∈ I,
rminus: -(x),
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
rfun: I ⟶ℝ,
uimplies: b supposing a,
implies: P ⇒ Q,
iproper: iproper(I),
right-endpoint: right-endpoint(I),
left-endpoint: left-endpoint(I),
i-finite: i-finite(I),
rooint: (l, u),
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
endpoints: endpoints(I),
outl: outl(x),
pi1: fst(t),
pi2: snd(t),
and: P ∧ Q,
cand: A c∧ B,
true: True,
so_apply: x[s],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
r-ap: f(x),
top: Top,
sq_stable: SqStable(P),
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x]
Lemmas referenced :
real_wf,
i-member_wf,
rooint_wf,
int-to-real_wf,
IVT-strictly-increasing-open,
rminus_wf,
halfpi_wf,
rsin_wf,
rccint_wf,
req_wf,
halfpi-interval-proper,
rless_wf,
req_weakening,
req_functionality,
rsin_functionality,
rsin-strictly-increasing,
member_rooint_lemma,
istype-void,
sq_stable__rless,
squash_wf,
true_wf,
rminus-int,
subtype_rel_self,
iff_weakening_equal,
rless_functionality,
rminus_functionality,
rsin-halfpi,
rsin-rminus,
req_inversion,
rsin-strict-bound,
arcsine_wf,
rless_transitivity1,
rleq_weakening,
rless_transitivity2,
arcsine_functionality,
arcsine-rsin
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
setIsType,
universeIsType,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
minusEquality,
natural_numberEquality,
hypothesisEquality,
dependent_functionElimination,
sqequalRule,
lambdaEquality_alt,
setElimination,
rename,
independent_isectElimination,
because_Cache,
independent_functionElimination,
independent_pairFormation,
dependent_set_memberEquality_alt,
productElimination,
isect_memberEquality_alt,
voidElimination,
imageMemberEquality,
baseClosed,
imageElimination,
applyEquality,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
instantiate,
universeEquality,
dependent_pairFormation_alt,
productIsType
Latex:
\mforall{}x:\{x:\mBbbR{}| x \mmember{} (r(-1), r1)\} . (arcsine(x) \mmember{} (-(\mpi{}/2), \mpi{}/2))
Date html generated:
2019_10_31-AM-06_12_14
Last ObjectModification:
2019_04_03-AM-01_15_21
Theory : reals_2
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