Nuprl Lemma : mean-value-theorem
∀a,b:ℝ.
((a < b)
⇒ (∀f,f':[a, b] ⟶ℝ.
(f'[x] continuous for x ∈ [a, b]
⇒ d(f[x])/dx = λx.f'[x] on [a, b]
⇒ (∀e:ℝ. ((r0 < e) ⇒ (∃x:ℝ. ((x ∈ [a, b]) ∧ (|f[b] - f[a] - f'[x] * (b - a)| ≤ e))))))))
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
continuous: f[x] continuous for x ∈ I,
rfun: I ⟶ℝ,
rccint: [l, u],
i-member: r ∈ I,
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rsub: x - y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s],
top: Top,
and: P ∧ Q,
cand: A c∧ B,
guard: {T},
uimplies: b supposing a,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
exists: ∃x:A. B[x],
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
Rolles-theorem,
rless_wf,
int-to-real_wf,
real_wf,
derivative_wf,
rccint_wf,
i-member_wf,
continuous_wf,
rfun_wf,
rsub_wf,
rmul_wf,
member_rccint_lemma,
rleq_weakening_rless,
rleq_weakening_equal,
rleq_wf,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
continuous-mul,
continuous-const,
continuous-sub,
set_wf,
subtype_rel_self,
top_wf,
subtype_rel_dep_function,
derivative-sub,
derivative-const-mul,
derivative-const,
derivative-id,
req_weakening,
itermConstant_wf,
derivative_functionality,
rabs_wf,
rmul_comm,
rsub_functionality,
rabs_functionality,
rleq_functionality
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
isectElimination,
natural_numberEquality,
sqequalRule,
lambdaEquality,
applyEquality,
setElimination,
rename,
dependent_set_memberEquality,
setEquality,
because_Cache,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
independent_pairFormation,
productEquality,
computeAll,
int_eqEquality,
intEquality,
productElimination,
dependent_pairFormation
Latex:
\mforall{}a,b:\mBbbR{}.
((a < b)
{}\mRightarrow{} (\mforall{}f,f':[a, b] {}\mrightarrow{}\mBbbR{}.
(f'[x] continuous for x \mmember{} [a, b]
{}\mRightarrow{} d(f[x])/dx = \mlambda{}x.f'[x] on [a, b]
{}\mRightarrow{} (\mforall{}e:\mBbbR{}. ((r0 < e) {}\mRightarrow{} (\mexists{}x:\mBbbR{}. ((x \mmember{} [a, b]) \mwedge{} (|f[b] - f[a] - f'[x] * (b - a)| \mleq{} e))))))))
Date html generated:
2017_10_03-PM-00_21_10
Last ObjectModification:
2017_07_28-AM-08_40_10
Theory : reals
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