Nuprl Lemma : derivative-const-mul2
∀a:ℝ. ∀I:Interval. ∀f,g:I ⟶ℝ. (λx.g[x] = d(f[x])/dx on I ⇒ λx.g[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,
rmul: a * b,
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q
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],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s],
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x)
Lemmas referenced :
derivative-const-mul,
derivative_wf,
real_wf,
i-member_wf,
rfun_wf,
interval_wf,
rmul_wf,
rmul_comm,
set_wf,
derivative_functionality
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
isectElimination,
sqequalRule,
lambdaEquality,
applyEquality,
setEquality,
because_Cache
Latex:
\mforall{}a:\mBbbR{}. \mforall{}I:Interval. \mforall{}f,g:I {}\mrightarrow{}\mBbbR{}. (\mlambda{}x.g[x] = d(f[x])/dx on I {}\mRightarrow{} \mlambda{}x.g[x] * a = d(f[x] * a)/dx on I)
Date html generated:
2016_05_18-AM-10_06_54
Last ObjectModification:
2015_12_27-PM-11_03_30
Theory : reals
Home
Index