Nuprl Lemma : ftc-total-integral
∀f,g:ℝ ⟶ ℝ.
((∀x,y:ℝ. ((x = y) ⇒ (f[x] = f[y])))
⇒ d(g[x])/dx = λx.f[x] on (-∞, ∞)
⇒ (∀a,b:ℝ. (a_∫-b f[t] dt = (g[b] - g[a]))))
Proof
Definitions occuring in Statement :
integral: a_∫-b f[x] dx,
derivative: d(f[x])/dx = λz.g[z] on I,
riiint: (-∞, ∞),
rsub: x - y,
req: x = y,
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
iproper: iproper(I),
i-finite: i-finite(I),
riiint: (-∞, ∞),
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
and: P ∧ Q,
false: False,
prop: ℙ,
top: Top,
true: True,
so_apply: x[s],
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
rfun: I ⟶ℝ,
uimplies: b supposing a,
label: ...$L... t
Lemmas referenced :
ftc-integral,
riiint_wf,
false_wf,
member_riiint_lemma,
true_wf,
req_wf,
set_wf,
real_wf,
subtype_rel_dep_function,
subtype_rel_self,
all_wf,
derivative_wf,
i-member_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesis,
independent_functionElimination,
sqequalRule,
productElimination,
voidElimination,
productEquality,
isect_memberEquality,
voidEquality,
dependent_set_memberEquality,
hypothesisEquality,
natural_numberEquality,
because_Cache,
setElimination,
rename,
isectElimination,
lambdaEquality,
applyEquality,
setEquality,
independent_isectElimination,
functionEquality,
functionExtensionality
Latex:
\mforall{}f,g:\mBbbR{} {}\mrightarrow{} \mBbbR{}.
((\mforall{}x,y:\mBbbR{}. ((x = y) {}\mRightarrow{} (f[x] = f[y])))
{}\mRightarrow{} d(g[x])/dx = \mlambda{}x.f[x] on (-\minfty{}, \minfty{})
{}\mRightarrow{} (\mforall{}a,b:\mBbbR{}. (a\_\mint{}\msupminus{}b f[t] dt = (g[b] - g[a]))))
Date html generated:
2016_10_26-PM-00_11_27
Last ObjectModification:
2016_09_12-PM-05_39_18
Theory : reals_2
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