Nuprl Lemma : derivative-implies-strictly-increasing-closed
∀a:ℝ. ∀b:{b:ℝ| a < b} . ∀f,f':[a, b] ⟶ℝ.
(d(f[x])/dx = λx.f'[x] on [a, b]
⇒ ifun(λx.f'[x];[a, b])
⇒ (∀x:{x:ℝ| x ∈ [a, b]} . (r0 ≤ f'[x]))
⇒ (∀x:{x:ℝ| x ∈ (a, b)} . (r0 < f'[x]))
⇒ f[x] strictly-increasing for x ∈ [a, b])
Proof
Definitions occuring in Statement :
strictly-increasing-on-interval: f[x] strictly-increasing for x ∈ I,
derivative: d(f[x])/dx = λz.g[z] on I,
ifun: ifun(f;I),
rfun: I ⟶ℝ,
rooint: (l, u),
rccint: [l, u],
i-member: r ∈ I,
rleq: x ≤ y,
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]} ,
lambda: λx.A[x],
natural_number: $n
Definitions unfolded in proof :
label: ...$L... t,
guard: {T},
and: P ∧ Q,
iff: P ⇐⇒ Q,
uimplies: b supposing a,
rfun: I ⟶ℝ,
real-fun: real-fun(f;a;b),
ifun: ifun(f;I),
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
squash: ↓T,
sq_stable: SqStable(P),
top: Top,
iproper: iproper(I),
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
i-member: r ∈ I,
subtype_rel: A ⊆r B,
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),
cand: A c∧ B,
subinterval: I ⊆ J
Lemmas referenced :
rless_wf,
rfun_wf,
derivative_wf,
rleq_weakening_rless,
sq_stable__rleq,
rccint-icompact,
ifun_wf,
int-to-real_wf,
rleq_wf,
all_wf,
i-member_wf,
real_wf,
set_wf,
req_wf,
member_rccint_lemma,
function-is-continuous,
i-finite_wf,
sq_stable__rless,
right_endpoint_rccint_lemma,
left_endpoint_rccint_lemma,
rccint_wf,
derivative-implies-increasing,
rfun_subtype,
rooint_wf,
derivative-implies-strictly-increasing,
member_rooint_lemma,
derivative_functionality_wrt_subinterval,
continuous_functionality_wrt_subinterval,
strictly-increasing-on-closed-interval
Rules used in proof :
productElimination,
independent_isectElimination,
dependent_set_memberEquality,
applyEquality,
natural_numberEquality,
setEquality,
lambdaEquality,
because_Cache,
imageElimination,
baseClosed,
imageMemberEquality,
voidEquality,
voidElimination,
isect_memberEquality,
sqequalRule,
independent_functionElimination,
hypothesis,
rename,
setElimination,
hypothesisEquality,
isectElimination,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
productEquality,
independent_pairFormation
Latex:
\mforall{}a:\mBbbR{}. \mforall{}b:\{b:\mBbbR{}| a < b\} . \mforall{}f,f':[a, b] {}\mrightarrow{}\mBbbR{}.
(d(f[x])/dx = \mlambda{}x.f'[x] on [a, b]
{}\mRightarrow{} ifun(\mlambda{}x.f'[x];[a, b])
{}\mRightarrow{} (\mforall{}x:\{x:\mBbbR{}| x \mmember{} [a, b]\} . (r0 \mleq{} f'[x]))
{}\mRightarrow{} (\mforall{}x:\{x:\mBbbR{}| x \mmember{} (a, b)\} . (r0 < f'[x]))
{}\mRightarrow{} f[x] strictly-increasing for x \mmember{} [a, b])
Date html generated:
2017_10_03-PM-00_32_16
Last ObjectModification:
2017_07_31-PM-04_27_55
Theory : reals
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