Nuprl Lemma : integral-is-Riemann
∀[a,b:ℝ]. ∀[f:{f:[rmin(a;b), rmax(a;b)] ⟶ℝ| ifun(f;[rmin(a;b), rmax(a;b)])} ].
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],
ifun: ifun(f;I),
rfun: I ⟶ℝ,
rccint: [l, u],
rleq: x ≤ y,
rmin: rmin(x;y),
rmax: rmax(x;y),
req: x = y,
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
integral: a_∫-b f[x] dx,
prop: ℙ,
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
so_apply: x[s],
rfun: I ⟶ℝ,
label: ...$L... t,
subtype_rel: A ⊆r B,
guard: {T},
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rsub: x - y
Lemmas referenced :
sq_stable__rleq,
rleq_wf,
set_wf,
rfun_wf,
rccint_wf,
rmin_wf,
rmax_wf,
ifun_wf,
rccint-icompact,
rmin-rleq-rmax,
real_wf,
rmin-req2,
rmax-req,
rsub_wf,
i-member_wf,
squash_wf,
icompact_wf,
interval_wf,
eta_conv,
iff_weakening_equal,
ifun_subtype_3,
rleq_weakening_equal,
rmin-rleq,
rleq-rmax,
Riemann-integral_wf,
int-to-real_wf,
req_wf,
radd_wf,
rminus_wf,
req_weakening,
req_functionality,
rsub_functionality,
Riemann-integral_functionality_endpoints,
Riemann-integral-same-endpoints,
uiff_transitivity,
radd_functionality,
rminus-zero,
radd_comm,
radd-zero-both
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
lambdaEquality,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
productElimination,
setElimination,
rename,
dependent_set_memberEquality,
applyEquality,
setEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
natural_numberEquality
Latex:
\mforall{}[a,b:\mBbbR{}]. \mforall{}[f:\{f:[rmin(a;b), rmax(a;b)] {}\mrightarrow{}\mBbbR{}| ifun(f;[rmin(a;b), rmax(a;b)])\} ].
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_14
Last ObjectModification:
2016_09_12-PM-05_38_32
Theory : reals_2
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