Nuprl Lemma : derivative-int-rmul
∀a:ℤ. ∀I:Interval. ∀f,f':I ⟶ℝ. (d(f[x])/dx = λx.f'[x] on I ⇒ d(a * f[x])/dx = λx.a * f'[x] on I)
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
rfun: I ⟶ℝ,
interval: Interval,
int-rmul: k1 * a,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
derivative-const-mul,
int-to-real_wf,
derivative_wf,
real_wf,
i-member_wf,
rfun_wf,
interval_wf,
istype-int,
rmul_wf,
int-rmul_wf,
req_weakening,
derivative_functionality,
req_functionality,
int-rmul-req
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
universeIsType,
sqequalRule,
lambdaEquality_alt,
applyEquality,
setIsType,
because_Cache,
inhabitedIsType,
independent_isectElimination,
productElimination
Latex:
\mforall{}a:\mBbbZ{}. \mforall{}I:Interval. \mforall{}f,f':I {}\mrightarrow{}\mBbbR{}. (d(f[x])/dx = \mlambda{}x.f'[x] on I {}\mRightarrow{} d(a * f[x])/dx = \mlambda{}x.a * f'[x] on I)
Date html generated:
2019_10_30-AM-09_05_02
Last ObjectModification:
2019_04_03-PM-05_55_49
Theory : reals
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