Nuprl Lemma : arcsine_functionality
∀[x:{x:ℝ| x ∈ (r(-1), r1)} ]. ∀[y:ℝ]. arcsine(x) = arcsine(y) supposing x = y
Proof
Definitions occuring in Statement :
arcsine: arcsine(x),
rooint: (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],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
top: Top,
arcsine: arcsine(x),
prop: ℙ,
and: P ∧ Q,
cand: A c∧ B,
guard: {T},
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
rfun: I ⟶ℝ,
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,
subtype_rel: A ⊆r B,
subinterval: I ⊆ J ,
rccint: [l, u],
ifun: ifun(f;I),
real-fun: real-fun(f;a;b)
Lemmas referenced :
member_rooint_lemma,
req_witness,
arcsine_wf,
i-member_wf,
rooint_wf,
int-to-real_wf,
rless_transitivity1,
rleq_weakening,
req_inversion,
rless_transitivity2,
rless_wf,
req_wf,
real_wf,
set_wf,
arcsine_deriv_wf,
rmin-rmax-subinterval,
rless-int,
subtype_rel_sets,
rccint_wf,
rmin_wf,
rmax_wf,
member_rccint_lemma,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
arcsine_deriv_functionality,
ifun_wf,
rccint-icompact,
rmin-rleq-rmax,
req_weakening,
integral_functionality_endpoints
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
setElimination,
thin,
rename,
sqequalHypSubstitution,
extract_by_obid,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
isectElimination,
dependent_set_memberEquality,
hypothesisEquality,
minusEquality,
natural_numberEquality,
sqequalRule,
productElimination,
independent_functionElimination,
independent_isectElimination,
because_Cache,
independent_pairFormation,
productEquality,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
imageMemberEquality,
baseClosed,
applyEquality,
setEquality,
lambdaFormation
Latex:
\mforall{}[x:\{x:\mBbbR{}| x \mmember{} (r(-1), r1)\} ]. \mforall{}[y:\mBbbR{}]. arcsine(x) = arcsine(y) supposing x = y
Date html generated:
2016_10_26-PM-00_41_31
Last ObjectModification:
2016_09_12-PM-05_45_43
Theory : reals_2
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