Nuprl Lemma : fund-theorem-of-calculus
∀I:Interval
(iproper(I)
⇒ (∀a:{a:ℝ| a ∈ I} . ∀f:{f:I ⟶ℝ| ∀x,y:{a:ℝ| a ∈ I} . ((x = y) ⇒ ((f x) = (f y)))} . ∀g:I ⟶ℝ.
(d(g[x])/dx = λx.f[x] on I ⇒ (∃c:ℝ. ∀x:{a:ℝ| a ∈ I} . (a_∫-x f[t] dt = (g[x] + c))))))
Proof
Definitions occuring in Statement :
integral: a_∫-b f[x] dx,
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,
radd: a + b,
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃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,
so_lambda: λ2x.t[x],
so_apply: x[s],
rfun: I ⟶ℝ,
uall: ∀[x:A]. B[x],
prop: ℙ,
label: ...$L... t,
integrate: a_∫- f[t] dt,
subtype_rel: A ⊆r B
Lemmas referenced :
antiderivatives-differ-by-constant,
i-member_wf,
derivative_wf,
real_wf,
rfun_wf,
set_wf,
all_wf,
req_wf,
iproper_wf,
interval_wf,
integrate_wf,
derivative-of-integral
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
setElimination,
rename,
sqequalRule,
lambdaEquality,
applyEquality,
dependent_set_memberEquality,
isectElimination,
because_Cache,
setEquality,
functionEquality
Latex:
\mforall{}I:Interval
(iproper(I)
{}\mRightarrow{} (\mforall{}a:\{a:\mBbbR{}| a \mmember{} I\} . \mforall{}f:\{f:I {}\mrightarrow{}\mBbbR{}| \mforall{}x,y:\{a:\mBbbR{}| a \mmember{} I\} . ((x = y) {}\mRightarrow{} ((f x) = (f y)))\} . \mforall{}g:I {}\mrightarrow{}\mBbbR{}.
(d(g[x])/dx = \mlambda{}x.f[x] on I {}\mRightarrow{} (\mexists{}c:\mBbbR{}. \mforall{}x:\{a:\mBbbR{}| a \mmember{} I\} . (a\_\mint{}\msupminus{}x f[t] dt = (g[x] + c))))))
Date html generated:
2016_10_26-PM-00_10_10
Last ObjectModification:
2016_09_12-PM-05_39_09
Theory : reals_2
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