Nuprl Lemma : second-deriv-implies-nonzero-on
∀I:Interval
(iproper(I)
⇒ (∀f,g,h:I ⟶ℝ.
((∀x,y:{a:ℝ| a ∈ I} . ((x = y) ⇒ (h[x] = h[y])))
⇒ d(f[x])/dx = λx.g[x] on I
⇒ d(g[x])/dx = λx.h[x] on I
⇒ (((∀x:{a:ℝ| a ∈ I} . (h[x] ≤ r0)) ∧ (∀x:ℝ. ((x ∈ I) ⇒ (r0 < f[x]))))
∨ ((∀x:{a:ℝ| a ∈ I} . (r0 ≤ h[x])) ∧ (∀x:ℝ. ((x ∈ I) ⇒ (f[x] < r0)))))
⇒ f[x]≠r0 for x ∈ I)))
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
nonzero-on: f[x]≠r0 for x ∈ I,
rfun: I ⟶ℝ,
i-member: r ∈ I,
iproper: iproper(I),
interval: Interval,
rleq: x ≤ y,
rless: x < y,
req: x = y,
int-to-real: r(n),
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
rfun: I ⟶ℝ,
label: ...$L... t,
guard: {T}
Lemmas referenced :
or_wf,
all_wf,
real_wf,
i-member_wf,
rleq_wf,
int-to-real_wf,
rless_wf,
derivative_wf,
req_wf,
rfun_wf,
iproper_wf,
interval_wf,
concave-positive-nonzero-on,
differentiable-functional2,
second-deriv-nonpos-concave,
convex-negative-nonzero-on,
second-deriv-nonneg-convex
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
unionElimination,
thin,
productElimination,
cut,
introduction,
extract_by_obid,
isectElimination,
productEquality,
setEquality,
hypothesis,
hypothesisEquality,
sqequalRule,
lambdaEquality,
setElimination,
rename,
applyEquality,
dependent_set_memberEquality,
natural_numberEquality,
because_Cache,
functionEquality,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}I:Interval
(iproper(I)
{}\mRightarrow{} (\mforall{}f,g,h:I {}\mrightarrow{}\mBbbR{}.
((\mforall{}x,y:\{a:\mBbbR{}| a \mmember{} I\} . ((x = y) {}\mRightarrow{} (h[x] = h[y])))
{}\mRightarrow{} d(f[x])/dx = \mlambda{}x.g[x] on I
{}\mRightarrow{} d(g[x])/dx = \mlambda{}x.h[x] on I
{}\mRightarrow{} (((\mforall{}x:\{a:\mBbbR{}| a \mmember{} I\} . (h[x] \mleq{} r0)) \mwedge{} (\mforall{}x:\mBbbR{}. ((x \mmember{} I) {}\mRightarrow{} (r0 < f[x]))))
\mvee{} ((\mforall{}x:\{a:\mBbbR{}| a \mmember{} I\} . (r0 \mleq{} h[x])) \mwedge{} (\mforall{}x:\mBbbR{}. ((x \mmember{} I) {}\mRightarrow{} (f[x] < r0)))))
{}\mRightarrow{} f[x]\mneq{}r0 for x \mmember{} I)))
Date html generated:
2018_05_22-PM-02_50_57
Last ObjectModification:
2017_10_21-PM-10_53_22
Theory : reals
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