Nuprl Lemma : derivative-implies-strictly-decreasing
∀I:Interval
(iproper(I)
⇒ (∀f,f':I ⟶ℝ.
(d(f[x])/dx = λx.f'[x] on I
⇒ f'[x] continuous for x ∈ I
⇒ (∀x:{x:ℝ| x ∈ I} . (f'[x] < r0))
⇒ f[x] strictly-decreasing for x ∈ I)))
Proof
Definitions occuring in Statement :
strictly-decreasing-on-interval: f[x] strictly-decreasing for x ∈ I,
derivative: d(f[x])/dx = λz.g[z] on I,
continuous: f[x] continuous for x ∈ I,
rfun: I ⟶ℝ,
i-member: r ∈ I,
iproper: iproper(I),
interval: Interval,
rless: x < y,
int-to-real: r(n),
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
rfun: I ⟶ℝ,
label: ...$L... t,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
guard: {T},
uimplies: b supposing a,
strictly-increasing-on-interval: f[x] strictly-increasing for x ∈ I,
strictly-decreasing-on-interval: f[x] strictly-decreasing for x ∈ I
Lemmas referenced :
all_wf,
real_wf,
i-member_wf,
rless_wf,
int-to-real_wf,
continuous_wf,
derivative_wf,
rfun_wf,
iproper_wf,
interval_wf,
derivative-implies-strictly-increasing,
rminus_wf,
derivative-minus,
continuous-minus,
radd-preserves-rless,
set_wf,
radd_wf,
rmul_wf,
rless_functionality,
radd-zero-both,
req_weakening,
radd_comm,
rmul-zero-both,
rmul_functionality,
radd-int,
req_transitivity,
radd_functionality,
rminus-as-rmul,
req_inversion,
rmul-identity1,
rmul-distrib2,
radd-assoc,
radd-ac,
radd-rminus-assoc,
radd-rminus-both
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
hypothesis,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
setElimination,
rename,
dependent_set_memberEquality,
natural_numberEquality,
because_Cache,
dependent_functionElimination,
independent_functionElimination,
productElimination,
minusEquality,
addEquality,
addLevel,
independent_isectElimination,
levelHypothesis,
promote_hyp
Latex:
\mforall{}I:Interval
(iproper(I)
{}\mRightarrow{} (\mforall{}f,f':I {}\mrightarrow{}\mBbbR{}.
(d(f[x])/dx = \mlambda{}x.f'[x] on I
{}\mRightarrow{} f'[x] continuous for x \mmember{} I
{}\mRightarrow{} (\mforall{}x:\{x:\mBbbR{}| x \mmember{} I\} . (f'[x] < r0))
{}\mRightarrow{} f[x] strictly-decreasing for x \mmember{} I)))
Date html generated:
2016_10_26-AM-11_39_15
Last ObjectModification:
2016_09_07-PM-10_00_29
Theory : reals
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