Nuprl Lemma : arcsin-bounds
∀[a:{a:ℝ| a ∈ [r(-1), r1]} ]. (arcsin(a) ∈ [-(π/2), π/2])
Proof
Definitions occuring in Statement :
arcsin: arcsin(a),
halfpi: π/2,
rccint: [l, u],
i-member: r ∈ I,
rminus: -(x),
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
squash: ↓T,
sq_stable: SqStable(P),
implies: P ⇒ Q,
and: P ∧ Q,
prop: ℙ,
all: ∀x:A. B[x],
top: Top,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
uimplies: b supposing a
Lemmas referenced :
arcsin_wf,
real_wf,
i-member_wf,
rccint_wf,
int-to-real_wf,
member_rccint_lemma,
istype-void,
sq_stable__and,
rleq_wf,
rminus_wf,
halfpi_wf,
sq_stable__rleq,
le_witness_for_triv
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyLambdaEquality,
setElimination,
rename,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
independent_functionElimination,
productElimination,
setIsType,
universeIsType,
minusEquality,
natural_numberEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
equalityTransitivity,
equalitySymmetry,
lambdaFormation_alt,
because_Cache,
lambdaEquality_alt,
independent_isectElimination,
functionIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[a:\{a:\mBbbR{}| a \mmember{} [r(-1), r1]\} ]. (arcsin(a) \mmember{} [-(\mpi{}/2), \mpi{}/2])
Date html generated:
2019_10_31-AM-06_14_43
Last ObjectModification:
2019_05_21-PM-11_23_06
Theory : reals_2
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