Nuprl Lemma : integral-is-Riemann-on-interval
∀I:Interval
∀[f:{f:I ⟶ℝ| ∀a,b:{x:ℝ| x ∈ I} . ((a = b) ⇒ (f[a] = f[b]))} ]. ∀[a,b:{x:ℝ| x ∈ I} ].
a_∫-b f[x] dx = ∫ f[x] dx on [a, b] supposing a ≤ b
Proof
Definitions occuring in Statement :
integral: a_∫-b f[x] dx,
Riemann-integral: ∫ f[x] dx on [a, b],
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
rleq: x ≤ y,
req: x = y,
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
rfun: I ⟶ℝ,
subinterval: I ⊆ J ,
sq_stable: SqStable(P),
squash: ↓T,
i-member: r ∈ I,
rccint: [l, u],
and: P ∧ Q,
top: Top,
cand: A c∧ B,
guard: {T},
subtype_rel: A ⊆r B,
ifun: ifun(f;I),
real-fun: real-fun(f;a;b),
iff: P ⇐⇒ Q
Lemmas referenced :
rleq_wf,
set_wf,
real_wf,
i-member_wf,
rfun_wf,
all_wf,
req_wf,
interval_wf,
rccint_wf,
sq_stable__i-member,
i-member-between,
sq_stable__req,
rmin_wf,
rmin-req2,
rmax_wf,
rmax-req,
member_rccint_lemma,
req_inversion,
rleq_transitivity,
rleq_weakening,
subtype_rel_sets,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
ifun_wf,
rccint-icompact,
rmin-rleq-rmax,
integral_wf,
Riemann-integral_wf,
integral-is-Riemann
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
because_Cache,
sqequalRule,
lambdaEquality,
setEquality,
functionEquality,
applyEquality,
dependent_set_memberEquality,
dependent_functionElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}I:Interval
\mforall{}[f:\{f:I {}\mrightarrow{}\mBbbR{}| \mforall{}a,b:\{x:\mBbbR{}| x \mmember{} I\} . ((a = b) {}\mRightarrow{} (f[a] = f[b]))\} ]. \mforall{}[a,b:\{x:\mBbbR{}| x \mmember{} I\} ].
a\_\mint{}\msupminus{}b f[x] dx = \mint{} f[x] dx on [a, b] supposing a \mleq{} b
Date html generated:
2016_10_26-PM-00_07_21
Last ObjectModification:
2016_09_12-PM-05_38_34
Theory : reals_2
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