Nuprl Lemma : antiderivatives-equal
∀I:Interval
(iproper(I)
⇒ (∀f,g,h:I ⟶ℝ.
(d(g[x])/dx = λx.f[x] on I
⇒ d(h[x])/dx = λx.f[x] on I
⇒ (∃x:{x:ℝ| x ∈ I} . (g[x] = h[x]))
⇒ (∀x:{x:ℝ| x ∈ I} . (g[x] = h[x])))))
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
rfun: I ⟶ℝ,
i-member: r ∈ I,
iproper: iproper(I),
interval: Interval,
req: x = y,
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
rfun: I ⟶ℝ,
label: ...$L... t,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
antiderivatives-differ-by-constant,
set_wf,
real_wf,
i-member_wf,
exists_wf,
req_wf,
derivative_wf,
rfun_wf,
iproper_wf,
interval_wf,
radd_wf,
req_functionality,
req_weakening,
radd-preserves-req,
rminus_wf,
rmul_wf,
int-to-real_wf,
uiff_transitivity,
req_transitivity,
radd_functionality,
rminus-as-rmul,
radd-assoc,
req_inversion,
rmul-identity1,
rmul-distrib2,
rmul_functionality,
radd-int,
rmul-zero-both,
radd-zero-both,
radd-zero
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
productElimination,
isectElimination,
sqequalRule,
lambdaEquality,
setEquality,
setElimination,
rename,
applyEquality,
dependent_set_memberEquality,
because_Cache,
independent_isectElimination,
minusEquality,
natural_numberEquality,
addEquality
Latex:
\mforall{}I:Interval
(iproper(I)
{}\mRightarrow{} (\mforall{}f,g,h:I {}\mrightarrow{}\mBbbR{}.
(d(g[x])/dx = \mlambda{}x.f[x] on I
{}\mRightarrow{} d(h[x])/dx = \mlambda{}x.f[x] on I
{}\mRightarrow{} (\mexists{}x:\{x:\mBbbR{}| x \mmember{} I\} . (g[x] = h[x]))
{}\mRightarrow{} (\mforall{}x:\{x:\mBbbR{}| x \mmember{} I\} . (g[x] = h[x])))))
Date html generated:
2016_10_26-AM-11_34_03
Last ObjectModification:
2016_09_05-PM-07_38_26
Theory : reals
Home
Index