Nuprl Lemma : non-zero-deriv-non-constant
∀a,b:ℝ.
((a < b)
⇒ (∀f,f':[a, b] ⟶ℝ.
(d(f(x))/dx = λx.f'(x) on [a, b]
⇒ (∃z:{z:ℝ| z ∈ [a, b]} . f'(z) ≠ r0)
⇒ (∀c:ℝ. ∃z:{z:ℝ| z ∈ [a, b]} . f(z) ≠ c))))
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
r-ap: f(x),
rfun: I ⟶ℝ,
rccint: [l, u],
i-member: r ∈ I,
rneq: x ≠ y,
rless: x < y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
i-member: r ∈ I,
rccint: [l, u],
and: P ∧ Q,
top: Top,
sq_stable: SqStable(P),
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
label: ...$L... t,
rfun: I ⟶ℝ,
iff: P ⇐⇒ Q,
guard: {T},
derivative: d(f[x])/dx = λz.g[z] on I,
nat_plus: ℕ+,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
i-approx: i-approx(I;n),
iproper: iproper(I),
i-finite: i-finite(I),
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
cand: A c∧ B,
sq_exists: ∃x:{A| B[x]},
rev_uimplies: rev_uimplies(P;Q),
rneq: x ≠ y,
or: P ∨ Q,
rev_implies: P ⇐ Q,
rless: x < y,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
rge: x ≥ y,
subtype_rel: A ⊆r B,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
uiff: uiff(P;Q),
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
rdiv: (x/y),
rsub: x - y
Lemmas referenced :
small-reciprocal-rneq-zero,
r-ap_wf,
member_rccint_lemma,
sq_stable__rleq,
real_wf,
exists_wf,
i-member_wf,
rccint_wf,
rneq_wf,
int-to-real_wf,
derivative_wf,
rfun_wf,
rless_wf,
rccint-icompact,
rleq_weakening_rless,
mul_nat_plus,
less_than_wf,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
true_wf,
icompact_wf,
rleq_functionality_wrt_implies,
rabs_wf,
rsub_wf,
rmul_wf,
rdiv_wf,
rless-int,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
i-approx-of-compact,
iff_weakening_equal,
rleq_weakening_equal,
rleq_weakening,
real_term_polynomial,
itermSubtract_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,
rleq_wf,
all_wf,
less_than'_wf,
nat_plus_wf,
squash_wf,
sq_stable__and,
sq_stable__rless,
sq_stable__all,
radd_wf,
equal_wf,
r-triangle-inequality2,
rleq_functionality,
req_weakening,
rabs_functionality,
radd_functionality_wrt_rleq,
radd_functionality,
rabs-difference-symmetry,
rmul_functionality,
req_inversion,
rmul_preserves_rleq2,
rmul-nonneg-case1,
rleq-int,
false_wf,
zero-rleq-rabs,
rinv_wf2,
uiff_transitivity,
rinv-mul-as-rdiv,
rabs-rmul,
rmul-is-positive,
rmul_preserves_rleq,
rdiv_functionality,
rmul-int,
rinv-of-rmul,
req_transitivity,
itermAdd_wf,
real_term_value_add_lemma,
rmul-rinv3,
rleq_transitivity,
rleq-implies-rleq,
rless-cases,
ravg-between,
rmin_wf,
ravg_wf,
radd-zero-both,
radd-rminus-assoc,
radd-rminus-both,
radd-ac,
rminus_functionality,
rmin_functionality,
radd_comm,
radd-assoc,
rmin_lb,
rminus_wf,
radd-preserves-rleq,
rmul-zero-both,
radd-int,
rmul-distrib2,
rmul-identity1,
rminus-as-rmul,
rless_transitivity2,
rmin_ub,
rabs-of-nonneg,
rabs-rminus,
rless_functionality,
itermMinus_wf,
real_term_value_minus_lemma,
rmin_strict_ub,
rless-implies-rless,
rless_functionality_wrt_implies,
rless-int-fractions2,
rabs-positive-iff,
radd-preserves-rless
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
isectElimination,
because_Cache,
hypothesisEquality,
setElimination,
rename,
hypothesis,
independent_isectElimination,
sqequalRule,
independent_pairFormation,
isect_memberEquality,
voidElimination,
voidEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
setEquality,
lambdaEquality,
natural_numberEquality,
dependent_set_memberEquality,
productEquality,
functionEquality,
multiplyEquality,
inrFormation,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
applyEquality,
equalityTransitivity,
equalitySymmetry,
minusEquality,
independent_pairEquality,
axiomEquality,
isect_memberFormation,
inlFormation,
addEquality,
universeEquality,
promote_hyp
Latex:
\mforall{}a,b:\mBbbR{}.
((a < b)
{}\mRightarrow{} (\mforall{}f,f':[a, b] {}\mrightarrow{}\mBbbR{}.
(d(f(x))/dx = \mlambda{}x.f'(x) on [a, b]
{}\mRightarrow{} (\mexists{}z:\{z:\mBbbR{}| z \mmember{} [a, b]\} . f'(z) \mneq{} r0)
{}\mRightarrow{} (\mforall{}c:\mBbbR{}. \mexists{}z:\{z:\mBbbR{}| z \mmember{} [a, b]\} . f(z) \mneq{} c))))
Date html generated:
2017_10_03-PM-00_34_57
Last ObjectModification:
2017_07_28-AM-08_43_35
Theory : reals
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