Nuprl Lemma : arcsine_wf
∀[x:{x:ℝ| x ∈ (r(-1), r1)} ]. (arcsine(x) ∈ ℝ)
Proof
Definitions occuring in Statement :
arcsine: arcsine(x),
rooint: (l, u),
i-member: r ∈ I,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
arcsine: arcsine(x),
rfun: I ⟶ℝ,
all: ∀x:A. B[x],
top: Top,
implies: P ⇒ Q,
i-member: r ∈ I,
rooint: (l, u),
and: P ∧ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
subinterval: I ⊆ J ,
cand: A c∧ B,
rccint: [l, u],
ifun: ifun(f;I),
real-fun: real-fun(f;a;b)
Lemmas referenced :
arcsine_deriv_wf,
member_rooint_lemma,
rmin-rmax-subinterval,
rooint_wf,
int-to-real_wf,
rless-int,
subtype_rel_sets,
real_wf,
i-member_wf,
rccint_wf,
rmin_wf,
rmax_wf,
rless_wf,
member_rccint_lemma,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
arcsine_deriv_functionality,
req_wf,
set_wf,
ifun_wf,
rccint-icompact,
rmin-rleq-rmax,
integral_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
setElimination,
thin,
rename,
sqequalRule,
dependent_set_memberEquality,
lambdaEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
minusEquality,
natural_numberEquality,
hypothesisEquality,
independent_functionElimination,
independent_pairFormation,
productElimination,
imageMemberEquality,
baseClosed,
because_Cache,
applyEquality,
productEquality,
independent_isectElimination,
setEquality,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
axiomEquality
Latex:
\mforall{}[x:\{x:\mBbbR{}| x \mmember{} (r(-1), r1)\} ]. (arcsine(x) \mmember{} \mBbbR{})
Date html generated:
2016_10_26-PM-00_41_20
Last ObjectModification:
2016_09_12-PM-05_45_35
Theory : reals_2
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