Nuprl Lemma : derivative-sub
∀I:Interval. ∀f1,f2,g1,g2:I ⟶ℝ.
(λx.g1[x] = d(f1[x])/dx on I ⇒ λx.g2[x] = d(f2[x])/dx on I ⇒ λx.g1[x] - g2[x] = d(f1[x] - f2[x])/dx on I)
Proof
Definitions occuring in Statement :
derivative: λz.g[z] = d(f[x])/dx on I,
rfun: I ⟶ℝ,
interval: Interval,
rsub: x - y,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
rsub: x - y,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s]
Lemmas referenced :
derivative_wf,
real_wf,
i-member_wf,
rfun_wf,
interval_wf,
rminus_wf,
derivative-add,
derivative-minus
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalRule,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality,
applyEquality,
setEquality,
hypothesis,
because_Cache,
independent_functionElimination,
dependent_functionElimination
Latex:
\mforall{}I:Interval. \mforall{}f1,f2,g1,g2:I {}\mrightarrow{}\mBbbR{}.
(\mlambda{}x.g1[x] = d(f1[x])/dx on I
{}\mRightarrow{} \mlambda{}x.g2[x] = d(f2[x])/dx on I
{}\mRightarrow{} \mlambda{}x.g1[x] - g2[x] = d(f1[x] - f2[x])/dx on I)
Date html generated:
2016_05_18-AM-10_07_22
Last ObjectModification:
2015_12_27-PM-11_03_18
Theory : reals
Home
Index