Nuprl Lemma : rsin-arcsine
∀[x:{x:ℝ| x ∈ (r(-1), r1)} ]. (rsin(arcsine(x)) = x)
Proof
Definitions occuring in Statement :
arcsine: arcsine(x),
rsin: rsin(x),
rooint: (l, u),
i-member: r ∈ I,
req: x = y,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
prop: ℙ,
implies: P ⇒ Q,
top: Top,
and: P ∧ Q,
cand: A c∧ B,
i-member: r ∈ I,
rooint: (l, u),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s],
subinterval: I ⊆ J ,
sq_stable: SqStable(P),
uimplies: b supposing a,
rge: x ≥ y,
guard: {T},
rfun: I ⟶ℝ,
subtype_rel: A ⊆r B,
rccint: [l, u],
ifun: ifun(f;I),
real-fun: real-fun(f;a;b),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
or: P ∨ Q,
arcsine: arcsine(x),
not: ¬A,
rsub: x - y,
arcsine_deriv: arcsine_deriv(x),
rneq: x ≠ y,
nat: ℕ,
le: A ≤ B,
false: False,
nat_plus: ℕ+,
label: ...$L... t,
i-finite: i-finite(I),
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt
Lemmas referenced :
rmin-rmax-subinterval,
rooint_wf,
int-to-real_wf,
rsin_wf,
arcsine_wf,
i-member_wf,
rsin-strict-bound,
member_rooint_lemma,
rless-arcsine,
arcsine-rless,
rless_wf,
rminus_wf,
halfpi_wf,
rless-int,
req_witness,
set_wf,
real_wf,
member_rccint_lemma,
rleq_wf,
rmin_wf,
rmax_wf,
rmin_strict_ub,
sq_stable__rless,
rleq_weakening_equal,
rmin-rleq-rmax,
rmax_strict_lb,
rless_functionality_wrt_implies,
integral-additive,
arcsine_deriv_wf,
subtype_rel_sets,
rccint_wf,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
req_functionality,
arcsine_deriv_functionality,
req_weakening,
req_wf,
ifun_wf,
rccint-icompact,
rmin_lb,
rleq-rmax,
integral-reverse,
rmin-rleq,
integral_wf,
equal_wf,
radd_wf,
radd_functionality,
arcsine-rsin,
rmul_wf,
uiff_transitivity,
req_transitivity,
rminus-as-rmul,
req_inversion,
rmul-identity1,
rmul-distrib2,
rmul_functionality,
radd-int,
rmul-zero-both,
rless_transitivity1,
rless_transitivity2,
radd-preserves-rless,
rsub_wf,
Riemann-integral-lower-bound,
rleq_weakening_rless,
Riemann-integral_wf,
rless_functionality,
radd-zero-both,
radd-rminus-assoc,
radd_comm,
rminus_functionality,
rmul-one-both,
rmul-distrib,
rmul_over_rminus,
arcsine-root-bounds,
rmul_preserves_rleq,
rdiv_wf,
rsqrt_wf,
rsqrt-positive,
rleq_functionality,
rmul-rdiv-cancel2,
rnexp_wf,
false_wf,
le_wf,
radd-preserves-rleq,
square-nonneg,
rsqrt-rnexp-2,
rnexp2,
rmul-int,
radd-assoc,
radd-ac,
radd-rminus-both,
rnexp-rleq-iff,
rsqrt_nonneg,
rleq-int,
less_than_wf,
rleq_weakening,
rless_irreflexivity,
req-iff-not-rneq,
rneq_wf,
rmin-req2,
rmax-req,
rleq_transitivity,
integral-is-Riemann,
left-endpoint_wf,
right-endpoint_wf,
rminus-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
minusEquality,
natural_numberEquality,
hypothesis,
setElimination,
rename,
dependent_set_memberEquality,
hypothesisEquality,
independent_functionElimination,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
because_Cache,
productEquality,
productElimination,
imageMemberEquality,
baseClosed,
lambdaEquality,
lambdaFormation,
imageElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
applyEquality,
setEquality,
inlFormation,
functionEquality,
addLevel,
impliesFunctionality,
addEquality,
levelHypothesis,
inrFormation,
multiplyEquality,
unionElimination
Latex:
\mforall{}[x:\{x:\mBbbR{}| x \mmember{} (r(-1), r1)\} ]. (rsin(arcsine(x)) = x)
Date html generated:
2017_10_04-PM-10_48_17
Last ObjectModification:
2017_07_28-AM-08_51_28
Theory : reals_2
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