Nuprl Lemma : antiderivatives-differ-by-constant
∀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 ⇒ (∃c:ℝ. ∀x:{x:ℝ| x ∈ I} . (g[x] = (h[x] + c))))))
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,
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]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
label: ...$L... t,
uimplies: b supposing a,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
real_term_value: real_term_value(f;t),
int_term_ind: int_term_ind,
itermSubtract: left (-) right,
itermVar: vvar,
uiff: uiff(P;Q),
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
exists: ∃x:A. B[x],
rsub: x - y
Lemmas referenced :
derivative-is-zero,
rsub_wf,
i-member_wf,
real_wf,
derivative_wf,
rfun_wf,
iproper_wf,
interval_wf,
derivative-sub,
int-to-real_wf,
req_weakening,
set_wf,
real_term_polynomial,
itermSubtract_wf,
itermVar_wf,
itermConstant_wf,
req-iff-rsub-is-0,
derivative_functionality,
radd_wf,
req_wf,
all_wf,
radd-zero-both,
radd-rminus-both,
radd_functionality,
radd-ac,
req_functionality,
uiff_transitivity,
rminus_wf,
radd-preserves-req
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
sqequalRule,
lambdaEquality,
isectElimination,
applyEquality,
setElimination,
rename,
dependent_set_memberEquality,
because_Cache,
setEquality,
productElimination,
natural_numberEquality,
independent_isectElimination,
computeAll,
int_eqEquality,
intEquality,
dependent_pairFormation
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{}c:\mBbbR{}. \mforall{}x:\{x:\mBbbR{}| x \mmember{} I\} . (g[x] = (h[x] + c))))))
Date html generated:
2017_10_03-PM-00_26_58
Last ObjectModification:
2017_07_28-AM-08_41_52
Theory : reals
Home
Index