Nuprl Lemma : derivative-rnexp-function
∀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
⇒ (∀n:ℕ+. d(f[x]^n)/dx = λx.(r(n) * f[x]^n - 1) * f'[x] on I))))
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,
rnexp: x^k1,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
subtract: n - m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat_plus: ℕ+,
prop: ℙ,
so_lambda: λ2x.t[x],
label: ...$L... t,
rfun: I ⟶ℝ,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
so_apply: x[s],
subtype_rel: A ⊆r B,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
r-ap: f(x),
rfun-eq: rfun-eq(I;f;g),
less_than': less_than'(a;b),
le: A ≤ B,
subtract: n - m,
guard: {T},
itermConstant: "const",
req_int_terms: t1 ≡ t2,
real_term_value: real_term_value(f;t),
int_term_ind: int_term_ind,
itermSubtract: left (-) right,
itermMultiply: left (*) right,
itermVar: vvar,
true: True,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q
Lemmas referenced :
nat_plus_properties,
derivative_wf,
rnexp_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
i-member_wf,
rmul_wf,
int-to-real_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
primrec-wf-nat-plus,
nat_plus_subtype_nat,
nat_plus_wf,
real_wf,
all_wf,
req_wf,
rfun_wf,
iproper_wf,
interval_wf,
rmul-one-both,
rmul-int,
rmul_functionality,
uiff_transitivity,
rpower-one,
req_functionality,
derivative_functionality,
set_wf,
req_weakening,
false_wf,
rnexp_zero_lemma,
derivative-mul,
rnexp_functionality,
continuous-implies-functional,
proper-continuous-is-continuous,
differentiable-continuous,
real_term_polynomial,
itermMultiply_wf,
req-iff-rsub-is-0,
add-subtract-cancel,
radd_wf,
int_term_value_add_lemma,
itermAdd_wf,
rmul_comm,
less_than_wf,
le-add-cancel,
add-zero,
add-associates,
add_functionality_wrt_le,
add-commutes,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
less-iff-le,
not-lt-2,
decidable__lt,
rnexp_step,
radd_functionality,
equal_wf,
rmul-ac,
req_transitivity,
rmul-assoc,
req_inversion,
rmul-distrib,
radd-int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
rename,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
because_Cache,
sqequalRule,
lambdaEquality,
dependent_set_memberEquality,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
applyEquality,
setEquality,
functionEquality,
productElimination,
independent_functionElimination,
multiplyEquality,
addEquality,
minusEquality,
equalitySymmetry,
equalityTransitivity
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{}n:\mBbbN{}\msupplus{}. d(f[x]\^{}n)/dx = \mlambda{}x.(r(n) * f[x]\^{}n - 1) * f'[x] on I))))
Date html generated:
2017_10_03-PM-00_13_00
Last ObjectModification:
2017_07_28-AM-08_35_59
Theory : reals
Home
Index