Nuprl Lemma : arcsin_functionality
∀[x:{x:ℝ| x ∈ [r(-1), r1]} ]. ∀[x':ℝ]. arcsin(x) = arcsin(x') supposing x = x'
Proof
Definitions occuring in Statement :
arcsin: arcsin(a),
rccint: [l, u],
i-member: r ∈ I,
req: x = y,
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
cand: A c∧ B,
guard: {T},
prop: ℙ,
subtype_rel: A ⊆r B,
i-member: r ∈ I,
rccint: [l, u],
implies: P ⇒ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
member_rccint_lemma,
istype-void,
rleq_transitivity,
int-to-real_wf,
rleq_weakening,
req_inversion,
rleq_wf,
arcsin-unique,
arcsin_wf,
subtype_rel_self,
real_wf,
i-member_wf,
rccint_wf,
req_wf,
rsin_wf,
req_functionality,
rsin-arcsin,
req_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
setElimination,
thin,
rename,
sqequalHypSubstitution,
introduction,
extract_by_obid,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
hypothesis,
dependent_set_memberEquality_alt,
hypothesisEquality,
productElimination,
isectElimination,
minusEquality,
natural_numberEquality,
independent_isectElimination,
because_Cache,
independent_pairFormation,
sqequalRule,
productIsType,
universeIsType,
equalityTransitivity,
equalitySymmetry,
applyEquality,
setEquality,
independent_functionElimination,
setIsType
Latex:
\mforall{}[x:\{x:\mBbbR{}| x \mmember{} [r(-1), r1]\} ]. \mforall{}[x':\mBbbR{}]. arcsin(x) = arcsin(x') supposing x = x'
Date html generated:
2019_10_31-AM-06_14_59
Last ObjectModification:
2019_05_24-PM-02_11_26
Theory : reals_2
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