Nuprl Lemma : arcsin_wf
∀[a:{a:ℝ| a ∈ [r(-1), r1]} ]. (arcsin(a) ∈ {x:ℝ| (x ∈ [-(π/2), π/2]) ∧ (rsin(x) = a)} )
Proof
Definitions occuring in Statement :
arcsin: arcsin(a),
halfpi: π/2,
rsin: rsin(x),
rccint: [l, u],
i-member: r ∈ I,
req: x = y,
rminus: -(x),
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
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,
arcsin: arcsin(a),
all: ∀x:A. B[x],
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
i-member: r ∈ I,
rccint: [l, u],
and: P ∧ Q
Lemmas referenced :
member_rccint_lemma,
istype-void,
full-arcsin_wf,
i-member_wf,
rccint_wf,
int-to-real_wf,
subtype_rel_self,
real_wf,
rleq_wf,
rminus_wf,
halfpi_wf,
req_wf,
rsin_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
setElimination,
thin,
rename,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
hypothesis,
isectElimination,
dependent_set_memberEquality_alt,
hypothesisEquality,
universeIsType,
minusEquality,
natural_numberEquality,
applyEquality,
setEquality,
productEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
setIsType
Latex:
\mforall{}[a:\{a:\mBbbR{}| a \mmember{} [r(-1), r1]\} ]. (arcsin(a) \mmember{} \{x:\mBbbR{}| (x \mmember{} [-(\mpi{}/2), \mpi{}/2]) \mwedge{} (rsin(x) = a)\} )
Date html generated:
2019_10_31-AM-06_14_18
Last ObjectModification:
2019_05_21-PM-11_18_38
Theory : reals_2
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