Nuprl Lemma : derivative-rexp-fun2
∀f,f':ℝ ⟶ ℝ.
((∀x,y:ℝ. ((x = y) ⇒ (f'[x] = f'[y])))
⇒ d(f[x])/dx = λx.f'[x] on (-∞, ∞)
⇒ d(e^f[x])/dx = λx.e^f[x] * f'[x] on (-∞, ∞))
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
riiint: (-∞, ∞),
rexp: e^x,
req: x = y,
rmul: a * b,
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
subtype_rel: A ⊆r B,
rfun: I ⟶ℝ,
top: Top,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
label: ...$L... t,
guard: {T}
Lemmas referenced :
derivative-rexp-fun,
riiint_wf,
member_riiint_lemma,
subtype_rel_dep_function,
real_wf,
true_wf,
subtype_rel_self,
set_wf,
iproper-riiint,
req_wf,
i-member_wf,
derivative_wf,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesis,
hypothesisEquality,
applyEquality,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
isectElimination,
lambdaEquality,
setEquality,
independent_isectElimination,
setElimination,
rename,
because_Cache,
independent_functionElimination,
functionExtensionality,
functionEquality
Latex:
\mforall{}f,f':\mBbbR{} {}\mrightarrow{} \mBbbR{}.
((\mforall{}x,y:\mBbbR{}. ((x = y) {}\mRightarrow{} (f'[x] = f'[y])))
{}\mRightarrow{} d(f[x])/dx = \mlambda{}x.f'[x] on (-\minfty{}, \minfty{})
{}\mRightarrow{} d(e\^{}f[x])/dx = \mlambda{}x.e\^{}f[x] * f'[x] on (-\minfty{}, \minfty{}))
Date html generated:
2017_10_04-PM-10_18_01
Last ObjectModification:
2017_06_24-PM-00_05_50
Theory : reals_2
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