Nuprl Lemma : derivative-rdiv-const
∀a:ℝ. (a ≠ r0 ⇒ (∀I:Interval. ∀f,f':I ⟶ℝ. (λx.f'[x] = d(f[x])/dx on I ⇒ λx.(f'[x]/a) = d((f[x]/a))/dx on I)))
Proof
Definitions occuring in Statement :
derivative: λz.g[z] = d(f[x])/dx on I,
rfun: I ⟶ℝ,
interval: Interval,
rdiv: (x/y),
rneq: x ≠ y,
int-to-real: r(n),
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
rdiv: (x/y),
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
so_apply: x[s],
prop: ℙ,
label: ...$L... t
Lemmas referenced :
derivative-const-mul2,
rinv_wf2,
real_wf,
i-member_wf,
derivative_wf,
rfun_wf,
interval_wf,
rneq_wf,
int-to-real_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalRule,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
lambdaEquality,
applyEquality,
setEquality,
because_Cache,
natural_numberEquality
Latex:
\mforall{}a:\mBbbR{}
(a \mneq{} r0
{}\mRightarrow{} (\mforall{}I:Interval. \mforall{}f,f':I {}\mrightarrow{}\mBbbR{}.
(\mlambda{}x.f'[x] = d(f[x])/dx on I {}\mRightarrow{} \mlambda{}x.(f'[x]/a) = d((f[x]/a))/dx on I)))
Date html generated:
2016_05_18-AM-10_13_18
Last ObjectModification:
2015_12_27-PM-10_58_35
Theory : reals
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