Nuprl Lemma : chain-rule
∀I,J:Interval. ∀f,f':I ⟶ℝ. ∀g,g':J ⟶ℝ.
(iproper(J)
⇒ maps-compact(I;J;x.f[x])
⇒ (∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f'[x] = f'[y])))
⇒ (∀x,y:{x:ℝ| x ∈ J} . ((x = y) ⇒ (g'[x] = g'[y])))
⇒ d(f[x])/dx = λx.f'[x] on I
⇒ d(g[x])/dx = λx.g'[x] on J
⇒ d(g[f[x]])/dx = λx.g'[f[x]] * f'[x] on I)
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
maps-compact: maps-compact(I;J;x.f[x]),
rfun: I ⟶ℝ,
i-member: r ∈ I,
iproper: iproper(I),
interval: Interval,
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,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
so_apply: x[s],
uall: ∀[x:A]. B[x],
prop: ℙ,
label: ...$L... t
Lemmas referenced :
chain-rule_0,
function-proper-continuous,
i-member_wf,
real_wf,
derivative_wf,
all_wf,
req_wf,
maps-compact_wf,
iproper_wf,
rfun_wf,
interval_wf,
differentiable-continuous
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
because_Cache,
sqequalRule,
lambdaEquality,
applyEquality,
setElimination,
rename,
dependent_set_memberEquality,
isectElimination,
setEquality,
functionEquality
Latex:
\mforall{}I,J:Interval. \mforall{}f,f':I {}\mrightarrow{}\mBbbR{}. \mforall{}g,g':J {}\mrightarrow{}\mBbbR{}.
(iproper(J)
{}\mRightarrow{} maps-compact(I;J;x.f[x])
{}\mRightarrow{} (\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} I\} . ((x = y) {}\mRightarrow{} (f'[x] = f'[y])))
{}\mRightarrow{} (\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} J\} . ((x = y) {}\mRightarrow{} (g'[x] = g'[y])))
{}\mRightarrow{} d(f[x])/dx = \mlambda{}x.f'[x] on I
{}\mRightarrow{} d(g[x])/dx = \mlambda{}x.g'[x] on J
{}\mRightarrow{} d(g[f[x]])/dx = \mlambda{}x.g'[f[x]] * f'[x] on I)
Date html generated:
2016_10_26-AM-11_30_38
Last ObjectModification:
2016_09_05-AM-10_09_21
Theory : reals
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