Nuprl Lemma : derivative-mul
∀I:Interval. ∀f1,f2:I ⟶ℝ. ∀g1,g2:{h:I ⟶ℝ| ∀x,y:{t:ℝ| t ∈ I} . ((x = y) ⇒ ((h x) = (h y)))} .
(d(f1[x])/dx = λx.g1[x] on I
⇒ d(f2[x])/dx = λx.g2[x] on I
⇒ d(f1[x] * f2[x])/dx = λx.(f1[x] * g2[x]) + (f2[x] * g1[x]) on I)
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
req: x = y,
rmul: a * b,
radd: a + b,
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_apply: x[s],
member: t ∈ T,
true: True,
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
label: ...$L... t,
sq_stable: SqStable(P),
squash: ↓T,
r-ap: f(x),
guard: {T}
Lemmas referenced :
derivative-mul-part1,
i-member_wf,
real_wf,
derivative_wf,
set_wf,
rfun_wf,
all_wf,
req_wf,
interval_wf,
differentiable-continuous,
sq_stable__req,
function-proper-continuous,
sq_stable__all,
req_witness,
squash_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
sqequalRule,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
natural_numberEquality,
hypothesis,
lambdaEquality,
applyEquality,
setElimination,
rename,
dependent_set_memberEquality,
isectElimination,
setEquality,
functionEquality,
because_Cache,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}I:Interval. \mforall{}f1,f2:I {}\mrightarrow{}\mBbbR{}. \mforall{}g1,g2:\{h:I {}\mrightarrow{}\mBbbR{}| \mforall{}x,y:\{t:\mBbbR{}| t \mmember{} I\} . ((x = y) {}\mRightarrow{} ((h x) = (h y)))\} .
(d(f1[x])/dx = \mlambda{}x.g1[x] on I
{}\mRightarrow{} d(f2[x])/dx = \mlambda{}x.g2[x] on I
{}\mRightarrow{} d(f1[x] * f2[x])/dx = \mlambda{}x.(f1[x] * g2[x]) + (f2[x] * g1[x]) on I)
Date html generated:
2016_10_26-AM-11_22_17
Last ObjectModification:
2016_08_28-PM-10_42_53
Theory : reals
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