Nuprl Lemma : arcsine-rminus
∀[x:{x:ℝ| x ∈ (r(-1), r1)} ]. (arcsine(-(x)) = -(arcsine(x)))
Proof
Definitions occuring in Statement :
arcsine: arcsine(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],
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,
uimplies: b supposing a,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
implies: P ⇒ Q,
not: ¬A,
uiff: uiff(P;Q),
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
member_rooint_lemma,
rminus_wf,
rless-implies-rless,
int-to-real_wf,
real_term_polynomial,
itermSubtract_wf,
itermConstant_wf,
itermVar_wf,
itermMinus_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma,
req-iff-rsub-is-0,
rsub_wf,
rless_wf,
set_wf,
real_wf,
i-member_wf,
rooint_wf,
arcsine-unique,
arcsine-bounds,
arcsine_wf,
halfpi_wf,
rsin-arcsine,
rminus_functionality,
rsin-rminus,
req_transitivity,
req_functionality,
req_weakening,
rsin_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
setElimination,
thin,
rename,
sqequalHypSubstitution,
introduction,
extract_by_obid,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
dependent_set_memberEquality,
isectElimination,
hypothesisEquality,
productElimination,
minusEquality,
natural_numberEquality,
independent_isectElimination,
sqequalRule,
computeAll,
lambdaEquality,
int_eqEquality,
intEquality,
independent_pairFormation,
because_Cache,
productEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination
Latex:
\mforall{}[x:\{x:\mBbbR{}| x \mmember{} (r(-1), r1)\} ]. (arcsine(-(x)) = -(arcsine(x)))
Date html generated:
2017_10_04-PM-10_48_30
Last ObjectModification:
2017_07_28-AM-08_51_32
Theory : reals_2
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