Nuprl Lemma : arcsine-unique
∀[x:{x:ℝ| x ∈ (r(-1), r1)} ]. ∀[y:{y:ℝ| y ∈ (-(π/2), π/2)} ]. ((rsin(y) = x) ⇒ (arcsine(x) = y))
Proof
Definitions occuring in Statement :
arcsine: arcsine(x),
halfpi: π/2,
rsin: rsin(x),
rooint: (l, u),
i-member: r ∈ I,
req: x = y,
rminus: -(x),
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
not: ¬A,
rneq: x ≠ y,
or: P ∨ Q,
strictly-increasing-on-interval: f[x] strictly-increasing for x ∈ I,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
cand: A c∧ B,
guard: {T},
false: False
Lemmas referenced :
rsin-arcsine,
req_wf,
rsin_wf,
req_witness,
arcsine_wf,
i-member_wf,
rooint_wf,
int-to-real_wf,
set_wf,
real_wf,
rminus_wf,
halfpi_wf,
rsin-strictly-increasing,
not-rneq,
rneq_wf,
member_rooint_lemma,
rless-arcsine,
arcsine-rless,
rless_wf,
req_inversion,
rless_transitivity1,
rleq_weakening,
rless_irreflexivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
hypothesis,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
dependent_set_memberEquality,
minusEquality,
natural_numberEquality,
because_Cache,
independent_functionElimination,
isect_memberEquality,
independent_isectElimination,
unionElimination,
voidElimination,
voidEquality,
independent_pairFormation,
productEquality
Latex:
\mforall{}[x:\{x:\mBbbR{}| x \mmember{} (r(-1), r1)\} ]. \mforall{}[y:\{y:\mBbbR{}| y \mmember{} (-(\mpi{}/2), \mpi{}/2)\} ]. ((rsin(y) = x) {}\mRightarrow{} (arcsine(x) = y))
Date html generated:
2016_10_26-PM-00_42_58
Last ObjectModification:
2016_10_11-PM-02_14_49
Theory : reals_2
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