Nuprl Lemma : differentiable-functional2
∀I:Interval. ∀f,g:I ⟶ℝ.
((∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (g[x] = g[y])))
⇒ d(f[x])/dx = λx.g[x] on I
⇒ iproper(I)
⇒ (∀a,b:{x:ℝ| x ∈ I} . ((a = b) ⇒ (f[a] = f[b]))))
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,
req: x = y,
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,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
label: ...$L... t,
rfun: I ⟶ℝ,
guard: {T}
Lemmas referenced :
differentiable-continuous,
proper-continuous-implies-functional,
req_wf,
set_wf,
real_wf,
i-member_wf,
iproper_wf,
derivative_wf,
all_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,
independent_functionElimination,
isectElimination,
setElimination,
rename,
sqequalRule,
lambdaEquality,
applyEquality,
dependent_set_memberEquality,
setEquality,
because_Cache,
functionEquality
Latex:
\mforall{}I:Interval. \mforall{}f,g:I {}\mrightarrow{}\mBbbR{}.
((\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} I\} . ((x = y) {}\mRightarrow{} (g[x] = g[y])))
{}\mRightarrow{} d(f[x])/dx = \mlambda{}x.g[x] on I
{}\mRightarrow{} iproper(I)
{}\mRightarrow{} (\mforall{}a,b:\{x:\mBbbR{}| x \mmember{} I\} . ((a = b) {}\mRightarrow{} (f[a] = f[b]))))
Date html generated:
2018_05_22-PM-02_45_22
Last ObjectModification:
2017_10_21-PM-07_45_57
Theory : reals
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