Nuprl Lemma : rabs-Riemann-integral
∀[a:ℝ]. ∀[b:{b:ℝ| a ≤ b} ]. ∀[f:{f:[a, b] ⟶ℝ| ifun(f;[a, b])} ]. (|∫ f[x] dx on [a, b]| ≤ (||f[x]||_x:[a, b] * (b - a)\000C))
Proof
Definitions occuring in Statement :
Riemann-integral: ∫ f[x] dx on [a, b],
I-norm: ||f[x]||_x:I,
ifun: ifun(f;I),
rfun: I ⟶ℝ,
rccint: [l, u],
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
rmul: a * b,
real: ℝ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rfun: I ⟶ℝ,
so_apply: x[s],
prop: ℙ,
squash: ↓T,
uimplies: b supposing a,
label: ...$L... t,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
sq_stable: SqStable(P),
subtype_rel: A ⊆r B,
guard: {T},
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
cand: A c∧ B,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
top: Top,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
or: P ∨ Q,
itermConstant: "const",
req_int_terms: t1 ≡ t2
Lemmas referenced :
Riemann-integral-bounds,
sq_stable__rleq,
rabs_wf,
i-member_wf,
rccint_wf,
real_wf,
ifun_wf,
squash_wf,
icompact_wf,
rfun_wf,
interval_wf,
eta_conv,
rccint-icompact,
iff_weakening_equal,
Riemann-integral_wf,
rmul_wf,
I-norm_wf,
rsub_wf,
rminus_wf,
set_wf,
rleq_wf,
I-norm-bound,
rmul_reverses_rleq,
int-to-real_wf,
rleq-int,
false_wf,
rmax_wf,
rmin_wf,
rabs-as-rmax,
rleq_functionality,
rminus-rmax,
req_weakening,
rmin_lb,
rleq_weakening,
real_term_polynomial,
itermSubtract_wf,
itermMinus_wf,
itermVar_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_minus_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rmul-one-both,
rminus_functionality,
rmul_over_rminus,
rmul-minus,
uiff_transitivity,
rleq_transitivity,
rabs-bounds,
rmax_lb,
rleq-implies-rleq,
itermMultiply_wf,
real_term_value_mul_lemma
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
dependent_set_memberEquality,
sqequalRule,
lambdaEquality,
applyEquality,
setEquality,
imageElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
dependent_functionElimination,
productElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
universeEquality,
lambdaFormation,
independent_pairFormation,
minusEquality,
natural_numberEquality,
isect_memberEquality,
voidElimination,
voidEquality,
inrFormation,
computeAll,
int_eqEquality,
intEquality
Latex:
\mforall{}[a:\mBbbR{}]. \mforall{}[b:\{b:\mBbbR{}| a \mleq{} b\} ]. \mforall{}[f:\{f:[a, b] {}\mrightarrow{}\mBbbR{}| ifun(f;[a, b])\} ].
(|\mint{} f[x] dx on [a, b]| \mleq{} (||f[x]||\_x:[a, b] * (b - a)))
Date html generated:
2017_10_03-PM-00_55_36
Last ObjectModification:
2017_07_28-AM-08_47_42
Theory : reals_2
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