Nuprl Lemma : arcsine_functionality_wrt_rless
∀x,y:{x:ℝ| x ∈ (r(-1), r1)} . ((x < y) ⇒ (arcsine(x) < arcsine(y)))
Proof
Definitions occuring in Statement :
arcsine: arcsine(x),
rooint: (l, u),
i-member: r ∈ I,
rless: x < y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
so_apply: x[s],
rfun: I ⟶ℝ,
so_lambda: λ2x.t[x],
prop: ℙ,
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
pi2: snd(t),
pi1: fst(t),
outl: outl(x),
rooint: (l, u),
endpoints: endpoints(I),
left-endpoint: left-endpoint(I),
right-endpoint: right-endpoint(I),
iproper: iproper(I),
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
uimplies: b supposing a,
sq_stable: SqStable(P),
top: Top,
i-member: r ∈ I,
arcsine_deriv: arcsine_deriv(x),
rdiv: (x/y),
not: ¬A,
false: False,
req_int_terms: t1 ≡ t2,
itermConstant: "const",
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
subtype_rel: A ⊆r B,
strictly-increasing-on-interval: f[x] strictly-increasing for x ∈ I
Lemmas referenced :
set_wf,
rless_wf,
derivative-arcsine,
arcsine_deriv_wf,
real_wf,
i-member_wf,
arcsine_wf,
i-finite_wf,
rless-int,
int-to-real_wf,
rooint_wf,
derivative-implies-strictly-increasing,
req_wf,
req_weakening,
arcsine_deriv_functionality,
req_functionality,
function-is-continuous,
sq_stable__rless,
member_rooint_lemma,
arcsine-root-bounds,
rmul-rinv,
req_transitivity,
req-iff-rsub-is-0,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_sub_lemma,
real_term_value_const_lemma,
itermVar_wf,
itermConstant_wf,
itermMultiply_wf,
itermSubtract_wf,
real_term_polynomial,
rless_functionality,
rinv_wf2,
rleq_wf,
rsub_wf,
rleq_weakening_rless,
rmul_wf,
rsqrt-positive,
rsqrt_wf,
rdiv_wf,
rmul_preserves_rless
Rules used in proof :
setEquality,
dependent_set_memberEquality,
rename,
setElimination,
lambdaEquality,
baseClosed,
hypothesisEquality,
imageMemberEquality,
because_Cache,
independent_pairFormation,
productElimination,
sqequalRule,
independent_functionElimination,
hypothesis,
natural_numberEquality,
minusEquality,
isectElimination,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_isectElimination,
imageElimination,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
computeAll,
inrFormation,
applyEquality
Latex:
\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} (r(-1), r1)\} . ((x < y) {}\mRightarrow{} (arcsine(x) < arcsine(y)))
Date html generated:
2017_10_04-PM-10_47_12
Last ObjectModification:
2017_08_02-PM-00_31_34
Theory : reals_2
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