Nuprl Lemma : derivative-rnexp2
∀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
⇒ d(f[x]^2)/dx = λx.(r(2) * f'[x]) * 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,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s],
nat: ℕ,
le: A ≤ B,
false: False,
not: ¬A,
subtract: n - m,
uimplies: b supposing a,
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,
uiff: uiff(P;Q),
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
derivative-rnexp-function,
less_than_wf,
derivative_wf,
i-member_wf,
real_wf,
all_wf,
req_wf,
rfun_wf,
iproper_wf,
interval_wf,
rnexp_wf,
false_wf,
le_wf,
rmul_wf,
int-to-real_wf,
subtract_wf,
req_weakening,
set_wf,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
req-iff-rsub-is-0,
derivative_functionality,
req_functionality,
rmul_functionality,
rnexp1
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
baseClosed,
isectElimination,
lambdaEquality,
applyEquality,
setElimination,
rename,
setEquality,
because_Cache,
functionEquality,
independent_isectElimination,
computeAll,
int_eqEquality,
intEquality,
productElimination
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{} d(f[x]\^{}2)/dx = \mlambda{}x.(r(2) * f'[x]) * f[x] on I)))
Date html generated:
2017_10_03-PM-00_13_20
Last ObjectModification:
2017_07_28-AM-08_36_17
Theory : reals
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