Nuprl Lemma : derivative-is-zero
∀I:Interval. (iproper(I) ⇒ (∀f:I ⟶ℝ. (d(f[x])/dx = λx.r0 on I ⇐⇒ ∃c:ℝ. ∀x:{x:ℝ| x ∈ I} . (f[x] = c))))
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,
req: x = y,
int-to-real: r(n),
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s],
rev_implies: P ⇐ Q,
cand: A c∧ B,
sq_stable: SqStable(P),
squash: ↓T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nat_plus: ℕ+,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
rsub: x - y,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
i-nonvoid: i-nonvoid(I),
subinterval: I ⊆ J ,
ifun: ifun(f;I),
real-fun: real-fun(f;a;b),
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x)
Lemmas referenced :
derivative_wf,
i-member_wf,
real_wf,
int-to-real_wf,
exists_wf,
all_wf,
req_wf,
rfun_wf,
iproper_wf,
interval_wf,
req-iff-rabs-rleq,
nat_plus_wf,
rless_wf,
set_wf,
rcc-subinterval,
sq_stable__i-member,
rleq_wf,
mean-value-theorem,
rfun_subtype,
rccint_wf,
continuous-const,
rdiv_wf,
rless-int,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_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_var_lemma,
int_formula_prop_wf,
rless-int-fractions2,
itermMultiply_wf,
int_term_value_mul_lemma,
derivative_functionality_wrt_subinterval,
rabs_wf,
rsub_wf,
rmul_wf,
radd_wf,
rminus_wf,
uiff_transitivity,
rleq_functionality,
rabs_functionality,
radd_functionality,
rminus_functionality,
req_transitivity,
rmul-distrib,
rmul_over_rminus,
req_weakening,
rmul-zero-both,
rminus-radd,
rmul_functionality,
rminus-zero,
req_inversion,
radd-assoc,
radd-ac,
radd_comm,
rmul-int,
rminus-as-rmul,
rmul-identity1,
rmul-distrib2,
radd-int,
radd-zero-both,
rabs-difference-symmetry,
iproper-nonvoid,
req-iff-not-rneq,
rneq_wf,
rless_transitivity1,
rleq_weakening,
rless_irreflexivity,
differentiable-continuous,
i-member-proper-iff,
i-approx-compact,
i-approx-is-subinterval,
proper-continuous-implies,
ifun-iff-continuous,
i-approx_wf,
subtype_rel_sets,
icompact-is-rccint,
member_rccint_lemma,
rleq_transitivity,
left-endpoint_wf,
i-approx-finite,
right-endpoint_wf,
top_wf,
subtype_rel_dep_function,
subtype_rel_self,
derivative-const,
derivative_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
setElimination,
rename,
dependent_set_memberEquality,
hypothesis,
setEquality,
natural_numberEquality,
because_Cache,
dependent_functionElimination,
productElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
independent_isectElimination,
inrFormation,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
multiplyEquality,
minusEquality,
addEquality,
productEquality
Latex:
\mforall{}I:Interval
(iproper(I) {}\mRightarrow{} (\mforall{}f:I {}\mrightarrow{}\mBbbR{}. (d(f[x])/dx = \mlambda{}x.r0 on I \mLeftarrow{}{}\mRightarrow{} \mexists{}c:\mBbbR{}. \mforall{}x:\{x:\mBbbR{}| x \mmember{} I\} . (f[x] = c))))
Date html generated:
2016_10_26-AM-11_33_27
Last ObjectModification:
2016_08_25-AM-00_31_15
Theory : reals
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