Nuprl Lemma : integral-same-endpoints
∀I:Interval. ∀f:{f:I ⟶ℝ| ∀x,y:{a:ℝ| a ∈ I} . ((x = y) ⇒ ((f x) = (f y)))} . ∀a:{a:ℝ| a ∈ I} . (a_∫-a f[t] dt = r0)
Proof
Definitions occuring in Statement :
integral: a_∫-b f[x] dx,
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
req: x = y,
int-to-real: r(n),
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
rfun: I ⟶ℝ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
top: Top,
guard: {T},
subinterval: I ⊆ J ,
ifun: ifun(f;I),
real-fun: real-fun(f;a;b),
i-finite: i-finite(I),
rccint: [l, u],
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
integrate: a_∫- f[t] dt
Lemmas referenced :
set_wf,
real_wf,
i-member_wf,
rfun_wf,
all_wf,
req_wf,
interval_wf,
rcc-subinterval,
rleq_wf,
subtype_rel_sets,
rccint_wf,
member_rccint_lemma,
ifun_wf,
rccint-icompact,
rleq_weakening_equal,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
left-endpoint_wf,
right-endpoint_wf,
int-to-real_wf,
req_weakening,
req_functionality,
integral-single,
integrate_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
sqequalRule,
lambdaEquality,
hypothesisEquality,
setEquality,
because_Cache,
setElimination,
rename,
functionEquality,
applyEquality,
dependent_set_memberEquality,
dependent_functionElimination,
productElimination,
independent_functionElimination,
independent_pairFormation,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
productEquality,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}I:Interval. \mforall{}f:\{f:I {}\mrightarrow{}\mBbbR{}| \mforall{}x,y:\{a:\mBbbR{}| a \mmember{} I\} . ((x = y) {}\mRightarrow{} ((f x) = (f y)))\} . \mforall{}a:\{a:\mBbbR{}| a \mmember{} I\} .
(a\_\mint{}\msupminus{}a f[t] dt = r0)
Date html generated:
2016_10_26-PM-00_08_25
Last ObjectModification:
2016_09_12-PM-05_38_58
Theory : reals_2
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