Nuprl Lemma : integration-by-parts
∀I:Interval. ∀u,v,h:I ⟶ℝ. ∀u',v':{h:I ⟶ℝ| ∀x,y:{t:ℝ| t ∈ I} . ((x = y) ⇒ ((h x) = (h y)))} .
(d(u[t])/dt = λt.u'[t] on I
⇒ d(v[t])/dt = λt.v'[t] on I
⇒ d(h[t])/dt = λt.u'[t] * v[t] on I
⇒ d((u[t] * v[t]) - h[t])/dt = λt.u[t] * v'[t] 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,
rsub: x - y,
req: x = y,
rmul: 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,
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],
uimplies: b supposing a,
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
rsub: x - y,
and: P ∧ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
derivative_wf,
i-member_wf,
real_wf,
rmul_wf,
set_wf,
rfun_wf,
all_wf,
req_wf,
interval_wf,
radd_wf,
req_witness,
rsub_wf,
req_weakening,
rminus_wf,
derivative-sub,
derivative-mul,
derivative_functionality,
uiff_transitivity,
req_functionality,
radd_functionality,
rmul_comm,
req_inversion,
radd-assoc,
req_transitivity,
radd-ac,
radd_comm,
radd-rminus-assoc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
setElimination,
rename,
dependent_set_memberEquality,
hypothesis,
setEquality,
because_Cache,
functionEquality,
dependent_functionElimination,
independent_functionElimination,
independent_isectElimination,
productElimination
Latex:
\mforall{}I:Interval. \mforall{}u,v,h:I {}\mrightarrow{}\mBbbR{}. \mforall{}u',v':\{h:I {}\mrightarrow{}\mBbbR{}| \mforall{}x,y:\{t:\mBbbR{}| t \mmember{} I\} . ((x = y) {}\mRightarrow{} ((h x) = (h y)))\} .
(d(u[t])/dt = \mlambda{}t.u'[t] on I
{}\mRightarrow{} d(v[t])/dt = \mlambda{}t.v'[t] on I
{}\mRightarrow{} d(h[t])/dt = \mlambda{}t.u'[t] * v[t] on I
{}\mRightarrow{} d((u[t] * v[t]) - h[t])/dt = \mlambda{}t.u[t] * v'[t] on I)
Date html generated:
2017_10_04-PM-10_54_02
Last ObjectModification:
2017_07_28-AM-08_52_07
Theory : reals_2
Home
Index