Nuprl Lemma : derivative_functionality2
∀I,J:Interval.
∀[f1,f2,g1,g2:I ⟶ℝ].
(rfun-eq(I;f1;f2) ⇒ rfun-eq(I;g1;g2) ⇒ J ⊆ I ⇒ d(f1[x])/dx = λx.g1[x] on I ⇒ d(f2[x])/dx = λx.g2[x] on J)
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
subinterval: I ⊆ J ,
rfun-eq: rfun-eq(I;f;g),
rfun: I ⟶ℝ,
interval: Interval,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
label: ...$L... t,
rfun: I ⟶ℝ,
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x)
Lemmas referenced :
derivative_functionality_wrt_subinterval,
derivative_wf,
i-member_wf,
real_wf,
subinterval_wf,
rfun-eq_wf,
rfun_wf,
interval_wf,
set_wf,
derivative_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
because_Cache,
hypothesisEquality,
isectElimination,
sqequalRule,
independent_functionElimination,
hypothesis,
lambdaEquality,
applyEquality,
setElimination,
rename,
dependent_set_memberEquality,
setEquality
Latex:
\mforall{}I,J:Interval.
\mforall{}[f1,f2,g1,g2:I {}\mrightarrow{}\mBbbR{}].
(rfun-eq(I;f1;f2)
{}\mRightarrow{} rfun-eq(I;g1;g2)
{}\mRightarrow{} J \msubseteq{} I
{}\mRightarrow{} d(f1[x])/dx = \mlambda{}x.g1[x] on I
{}\mRightarrow{} d(f2[x])/dx = \mlambda{}x.g2[x] on J)
Date html generated:
2018_05_22-PM-02_44_18
Last ObjectModification:
2017_10_20-PM-00_15_48
Theory : reals
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