Nuprl Lemma : Riemann-integral-bounds
∀[a:ℝ]. ∀[b:{b:ℝ| a ≤ b} ]. ∀[f:{f:[a, b] ⟶ℝ| ifun(f;[a, b])} ].
∀c1,c2:ℝ.
((c1 * (b - a)) ≤ ∫ f[x] dx on [a, b]) ∧ (∫ f[x] dx on [a, b] ≤ (c2 * (b - a)))
supposing ∀x:ℝ. ((x ∈ [a, b]) ⇒ ((c1 ≤ f[x]) ∧ (f[x] ≤ c2)))
Proof
Definitions occuring in Statement :
Riemann-integral: ∫ f[x] dx on [a, b],
ifun: ifun(f;I),
rfun: I ⟶ℝ,
rccint: [l, u],
i-member: r ∈ I,
rleq: x ≤ y,
rsub: x - y,
rmul: a * b,
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
so_lambda: λ2x.t[x],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s],
prop: ℙ,
ifun: ifun(f;I),
top: Top,
real-fun: real-fun(f;a;b),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
sq_stable: SqStable(P),
squash: ↓T,
guard: {T}
Lemmas referenced :
less_than'_wf,
rsub_wf,
Riemann-integral_wf,
i-member_wf,
rccint_wf,
real_wf,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
member_rccint_lemma,
req_functionality,
req_weakening,
req_wf,
set_wf,
ifun_wf,
rccint-icompact,
rmul_wf,
nat_plus_wf,
all_wf,
rleq_wf,
rfun_wf,
sq_stable__rleq,
top_wf,
subtype_rel_dep_function,
subtype_rel_self,
Riemann-integral-rleq,
rleq_functionality,
req_inversion,
Riemann-integral-const
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
independent_pairFormation,
hypothesis,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
because_Cache,
extract_by_obid,
isectElimination,
applyEquality,
setElimination,
rename,
dependent_set_memberEquality,
setEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_functionElimination,
independent_isectElimination,
minusEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
productEquality,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}[a:\mBbbR{}]. \mforall{}[b:\{b:\mBbbR{}| a \mleq{} b\} ]. \mforall{}[f:\{f:[a, b] {}\mrightarrow{}\mBbbR{}| ifun(f;[a, b])\} ].
\mforall{}c1,c2:\mBbbR{}.
((c1 * (b - a)) \mleq{} \mint{} f[x] dx on [a, b]) \mwedge{} (\mint{} f[x] dx on [a, b] \mleq{} (c2 * (b - a)))
supposing \mforall{}x:\mBbbR{}. ((x \mmember{} [a, b]) {}\mRightarrow{} ((c1 \mleq{} f[x]) \mwedge{} (f[x] \mleq{} c2)))
Date html generated:
2016_10_26-PM-00_03_06
Last ObjectModification:
2016_09_12-PM-05_38_12
Theory : reals_2
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