Nuprl Lemma : arcsin-is-arcsine
∀[x:{x:ℝ| x ∈ (r(-1), r1)} ]. (arcsin(x) = arcsine(x))
Proof
Definitions occuring in Statement :
arcsin: arcsin(a),
arcsine: arcsine(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],
top: Top,
and: P ∧ Q,
cand: A c∧ B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
implies: P ⇒ Q
Lemmas referenced :
arcsin-unique,
member_rooint_lemma,
istype-void,
member_rccint_lemma,
rleq_weakening_rless,
int-to-real_wf,
rleq_wf,
arcsine-bounds,
arcsine_wf,
rminus_wf,
halfpi_wf,
rsin-arcsine,
real_wf,
i-member_wf,
rooint_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
dependent_set_memberEquality_alt,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
hypothesis,
hypothesisEquality,
productElimination,
minusEquality,
natural_numberEquality,
independent_isectElimination,
independent_pairFormation,
sqequalRule,
productIsType,
universeIsType,
independent_functionElimination,
setIsType
Latex:
\mforall{}[x:\{x:\mBbbR{}| x \mmember{} (r(-1), r1)\} ]. (arcsin(x) = arcsine(x))
Date html generated:
2019_10_31-AM-06_15_37
Last ObjectModification:
2019_05_24-PM-04_55_25
Theory : reals_2
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