Nuprl Lemma : derivative-rexp-fun
∀I:Interval. ∀f,f':I ⟶ℝ.
(iproper(I)
⇒ (∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f'[x] = f'[y])))
⇒ d(f[x])/dx = λx.f'[x] on I
⇒ d(e^f[x])/dx = λx.e^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,
rexp: e^x,
req: x = y,
rmul: a * b,
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
rfun: I ⟶ℝ,
uall: ∀[x:A]. B[x],
prop: ℙ,
implies: P ⇒ Q,
so_apply: x[s],
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
so_lambda: λ2x.t[x],
label: ...$L... t,
subtype_rel: A ⊆r B
Lemmas referenced :
simple-chain-rule,
rexp_wf,
real_wf,
i-member_wf,
riiint_wf,
req_functionality,
rexp_functionality,
req_weakening,
req_wf,
derivative-rexp,
derivative_wf,
all_wf,
iproper_wf,
rfun_wf,
interval_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
isectElimination,
setElimination,
rename,
setEquality,
because_Cache,
independent_functionElimination,
independent_isectElimination,
productElimination,
applyEquality,
dependent_set_memberEquality,
functionEquality
Latex:
\mforall{}I:Interval. \mforall{}f,f':I {}\mrightarrow{}\mBbbR{}.
(iproper(I)
{}\mRightarrow{} (\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(e\^{}f[x])/dx = \mlambda{}x.e\^{}f[x] * f'[x] on I)
Date html generated:
2017_10_04-PM-10_17_58
Last ObjectModification:
2017_06_24-PM-00_01_17
Theory : reals_2
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