Nuprl Lemma : mean-value-for-bounded-derivative
∀I:Interval
(iproper(I)
⇒ (∀f,f':I ⟶ℝ.
((∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f'[x] = f'[y])))
⇒ d(f[x])/dx = λx.f'[x] on I
⇒ (∀c:ℝ. ((∀x:{x:ℝ| x ∈ I} . (|f'[x]| ≤ c)) ⇒ (∀x,y:{x:ℝ| x ∈ I} . (|f[x] - f[y]| ≤ (c * |x - y|))))))))
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
rfun: I ⟶ℝ,
i-member: r ∈ I,
iproper: iproper(I),
interval: Interval,
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
req: x = y,
rmul: a * b,
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
so_apply: x[s],
uall: ∀[x:A]. B[x],
prop: ℙ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
label: ...$L... t,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
sq_stable: SqStable(P),
squash: ↓T,
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
rev_uimplies: rev_uimplies(P;Q),
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
not: ¬A,
false: False,
guard: {T},
subinterval: I ⊆ J ,
rsub: x - y,
rge: x ≥ y,
or: P ∨ Q,
i-nonvoid: i-nonvoid(I),
rneq: x ≠ y
Lemmas referenced :
function-is-continuous,
differentiable-continuous,
i-member_wf,
real_wf,
proper-continuous-is-continuous,
rleq-iff-all-rless,
rabs_wf,
rsub_wf,
rmul_wf,
set_wf,
rless_wf,
int-to-real_wf,
all_wf,
rleq_wf,
derivative_wf,
req_wf,
rfun_wf,
iproper_wf,
interval_wf,
rcc-subinterval,
sq_stable__i-member,
continuous_functionality_wrt_subinterval,
rccint_wf,
mean-value-theorem,
rfun_subtype,
sq_stable__rless,
derivative_functionality_wrt_subinterval,
radd_wf,
rleq_functionality,
rabs-difference-symmetry,
radd_functionality,
req_weakening,
rmul_functionality,
rmul_preserves_rleq2,
zero-rleq-rabs,
less_than'_wf,
nat_plus_wf,
equal_wf,
rabs-rmul,
rminus_wf,
uiff_transitivity,
req_functionality,
radd_comm,
radd-rminus-assoc,
rabs_functionality,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
rleq_transitivity,
r-triangle-inequality,
radd_functionality_wrt_rleq,
function-diff-small-or-interval-proper,
iproper-nonvoid,
rmul-nonneg-case1,
trivial-rleq-radd,
rneq-if-rabs
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
because_Cache,
sqequalRule,
lambdaEquality,
applyEquality,
setElimination,
rename,
dependent_set_memberEquality,
isectElimination,
setEquality,
productElimination,
independent_isectElimination,
natural_numberEquality,
functionEquality,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
isect_memberFormation,
independent_pairEquality,
voidElimination,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
unionElimination
Latex:
\mforall{}I:Interval
(iproper(I)
{}\mRightarrow{} (\mforall{}f,f':I {}\mrightarrow{}\mBbbR{}.
((\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} I\} . ((x = y) {}\mRightarrow{} (f'[x] = f'[y])))
{}\mRightarrow{} d(f[x])/dx = \mlambda{}x.f'[x] on I
{}\mRightarrow{} (\mforall{}c:\mBbbR{}
((\mforall{}x:\{x:\mBbbR{}| x \mmember{} I\} . (|f'[x]| \mleq{} c))
{}\mRightarrow{} (\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} I\} . (|f[x] - f[y]| \mleq{} (c * |x - y|))))))))
Date html generated:
2017_10_03-PM-00_21_39
Last ObjectModification:
2017_07_28-AM-08_40_26
Theory : reals
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