Nuprl Lemma : rabs-integral
∀[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]||_x:[rmin(a;b), rmax(a;b)] * |a - b|))
Proof
Definitions occuring in Statement :
integral: a_∫-b f[x] dx,
I-norm: ||f[x]||_x:I,
ifun: ifun(f;I),
rfun: I ⟶ℝ,
rccint: [l, u],
rleq: x ≤ y,
rabs: |x|,
rmin: rmin(x;y),
rmax: rmax(x;y),
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,
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
rfun: I ⟶ℝ,
so_apply: x[s],
prop: ℙ,
squash: ↓T,
uimplies: b supposing a,
label: ...$L... t,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
guard: {T},
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
cand: A c∧ B,
uiff: uiff(P;Q),
exists: ∃x:A. B[x],
integral: a_∫-b f[x] dx,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
subinterval: I ⊆ J ,
top: Top,
or: P ∨ Q,
rsub: x - y,
sq_stable: SqStable(P),
real: ℝ,
rneq: x ≠ y,
true: True
Lemmas referenced :
less_than'_wf,
rsub_wf,
rmul_wf,
i-member_wf,
rccint_wf,
rmin_wf,
rmax_wf,
real_wf,
ifun_wf,
squash_wf,
icompact_wf,
rfun_wf,
interval_wf,
eta_conv,
rccint-icompact,
rmin-rleq-rmax,
iff_weakening_equal,
I-norm_wf,
rabs_wf,
integral_wf,
nat_plus_wf,
set_wf,
rleq_weakening_equal,
rmin_ub,
rmax_lb,
rleq-iff-all-rless,
ifun_subtype_3,
rless_wf,
int-to-real_wf,
radd_wf,
rleq_wf,
r-triangle-inequality-rsub,
rmin-rleq,
rleq-rmax,
Riemann-integral_wf,
rleq_functionality_wrt_implies,
radd_functionality_wrt_rleq,
rabs-Riemann-integral,
I-norm_functionality_wrt_subinterval,
member_rccint_lemma,
rmul_functionality_wrt_rleq2,
I-norm-non-neg,
radd-preserves-rleq,
rminus_wf,
uiff_transitivity,
rleq_functionality,
radd_comm,
radd_functionality,
req_weakening,
radd-rminus-assoc,
radd-zero-both,
equal_wf,
sq_stable__rleq,
rsub-rmin-rleq-rabs,
rabs-difference-symmetry,
rsub_functionality,
rmin-com,
rmul_preserves_rleq2,
rmul_comm,
req_inversion,
rmul-distrib2,
rmul_functionality,
req_transitivity,
rmul-identity1,
radd-int,
rmul-assoc,
rless-cases,
sq_stable__rless,
rmul-is-positive,
rneq-iff-rabs,
rmin-req2,
rleq_weakening_rless,
rmax-req,
rabs_functionality,
integral_functionality_endpoints,
rminus-as-rmul,
radd-assoc,
rmul-zero-both,
rmin-req,
rmax-req2,
rmin_lb,
rmax_ub,
integral-reverse,
true_wf,
rabs-rminus,
zero-rleq-rabs,
rless_transitivity1,
rless_irreflexivity,
radd-ac,
radd-rminus-both,
rless_transitivity2,
integral-is-Riemann,
rmax-minus-rmin
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairEquality,
because_Cache,
extract_by_obid,
isectElimination,
applyEquality,
setElimination,
rename,
dependent_set_memberEquality,
hypothesis,
setEquality,
imageElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
imageMemberEquality,
baseClosed,
universeEquality,
minusEquality,
natural_numberEquality,
axiomEquality,
isect_memberEquality,
voidElimination,
independent_pairFormation,
lambdaFormation,
dependent_pairFormation,
voidEquality,
productEquality,
inlFormation,
addEquality,
unionElimination,
inrFormation
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| \mleq{} (||f[x]||\_x:[rmin(a;b), rmax(a;b)] * |a - b|))
Date html generated:
2017_10_04-PM-10_16_10
Last ObjectModification:
2017_07_28-AM-08_47_51
Theory : reals_2
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