Nuprl Lemma : BB-problem-21
∀a:ℝ. ∀b:{b:ℝ| a ≤ b} . ∀f,g:{f:[a, b] ⟶ℝ| ifun(f;[a, b])} .
((∀x:ℝ. ((x ∈ [a, b]) ⇒ (r0 ≤ g[x])))
⇒ (∀e:{e:ℝ| r0 < e} . ∃c:{x:ℝ| x ∈ [a, b]} . (|a_∫-b f[x] * g[x] dx - f[c] * a_∫-b g[x] dx| < e)))
Proof
Definitions occuring in Statement :
integral: a_∫-b f[x] dx,
ifun: ifun(f;I),
rfun: I ⟶ℝ,
rccint: [l, u],
i-member: r ∈ I,
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rsub: x - y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
exists: ∃x:A. B[x],
rev_implies: P ⇐ Q,
label: ...$L... t,
guard: {T},
subtype_rel: A ⊆r B,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
real-fun: real-fun(f;a;b),
top: Top,
ifun: ifun(f;I),
and: P ∧ Q,
iff: P ⇐⇒ Q,
rfun: I ⟶ℝ,
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
squash: ↓T,
sq_stable: SqStable(P),
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
true: True,
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
isl: isl(x),
rccint: [l, u],
i-finite: i-finite(I),
or: P ∨ Q,
cand: A c∧ B,
not: ¬A,
false: False,
req_int_terms: t1 ≡ t2,
le: A ≤ B,
rnonneg: rnonneg(x),
rleq: x ≤ y,
rsub: x - y,
rge: x ≥ y,
sq_exists: ∃x:{A| B[x]},
rless: x < y,
rdiv: (x/y),
rneq: x ≠ y,
less_than': less_than'(a;b),
less_than: a < b,
rset-member: x ∈ A,
rrange: f[x](x∈I),
sup: sup(A) = b,
pi2: snd(t),
right-endpoint: right-endpoint(I),
outl: outl(x),
endpoints: endpoints(I),
pi1: fst(t),
left-endpoint: left-endpoint(I),
r-ap: f(x),
i-member: r ∈ I,
subinterval: I ⊆ J
Lemmas referenced :
integral-is-Riemann,
rsub_functionality,
rabs_functionality,
rless_functionality,
exists_wf,
Riemann-integral_wf,
iff_weakening_equal,
eta_conv,
interval_wf,
icompact_wf,
squash_wf,
integral_wf,
rmin-rleq-rmax,
rleq_weakening,
req_inversion,
rmax_wf,
rmin_wf,
ifun_subtype_3,
req_wf,
req_weakening,
rmul_functionality,
req_functionality,
member_rccint_lemma,
right_endpoint_rccint_lemma,
left_endpoint_rccint_lemma,
rmul_wf,
rsub_wf,
rabs_wf,
rccint-icompact,
ifun_wf,
rfun_wf,
rleq_wf,
rccint_wf,
i-member_wf,
all_wf,
int-to-real_wf,
rless_wf,
real_wf,
set_wf,
rmax-req,
sq_stable__rleq,
rmin-req2,
right-endpoint_wf,
left-endpoint_wf,
equal_wf,
I-norm_wf,
subtype_rel_self,
sq_stable__rless,
rless-cases,
I-norm_functionality,
rleq_transitivity,
sq_stable__req,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
Riemann-integral-rsub,
Riemann-integral-rmul-const,
req-iff-rsub-is-0,
itermVar_wf,
itermMultiply_wf,
itermSubtract_wf,
Riemann-integral_functionality,
rabs-of-nonneg,
rleq_functionality,
I-norm-non-neg,
nat_plus_wf,
less_than'_wf,
I-norm-bound,
I-norm-rleq,
radd-zero-both,
radd-rminus-assoc,
radd_functionality,
radd_comm,
uiff_transitivity,
rmul_functionality_wrt_rleq2,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rabs-Riemann-integral,
rminus_wf,
radd_wf,
radd-preserves-rleq,
rmul-is-positive,
rless_transitivity1,
I-norm-positive-implies,
rabs-rmul,
rabs-positive-iff,
rmul-rinv,
req_transitivity,
rmul-one,
rinv_wf2,
rmul-zero-both,
rdiv_wf,
rmul_preserves_rless,
rless-int,
range_sup_wf,
range_sup-property,
Riemann-integral-rleq,
icompact-endpoints,
rcc-subinterval,
rleq-range_sup,
rabs-bounds,
rsub_functionality_wrt_rleq,
rless_functionality_wrt_implies,
rless-implies-rless,
true_wf,
rleq_weakening_rless,
rmul_preserves_rleq,
rmul_preserves_rleq2,
rmul_comm,
rdiv_functionality,
rmul-rinv3,
rinv-of-rmul,
Riemann-integral-const,
radd-zero,
radd-rminus,
radd-preserves-rless,
rabs-rless-iff,
rmul-rdiv-cancel2,
rmul-distrib2,
rmul-identity1,
rminus-as-rmul,
radd-int,
rmul-one-both,
rminus_functionality,
real_term_value_minus_lemma,
real_term_value_add_lemma,
itermMinus_wf,
itermAdd_wf,
radd-preserves-req,
itermConstant_wf,
rmul_over_rminus,
false_wf,
rleq-int,
radd-rminus-both,
radd-ac,
radd-assoc,
intermediate-value-theorem,
function-is-continuous,
rless_transitivity2,
rmin-rmax-subinterval,
subtype_rel_sets,
rminus-zero,
rless_irreflexivity,
zero-rleq-rabs
Rules used in proof :
existsFunctionality,
addLevel,
universeEquality,
equalitySymmetry,
equalityTransitivity,
voidEquality,
voidElimination,
isect_memberEquality,
setEquality,
productElimination,
dependent_functionElimination,
dependent_set_memberEquality,
applyEquality,
functionEquality,
natural_numberEquality,
lambdaEquality,
imageElimination,
baseClosed,
imageMemberEquality,
sqequalRule,
independent_functionElimination,
hypothesis,
hypothesisEquality,
rename,
setElimination,
independent_isectElimination,
because_Cache,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_pairFormation,
productEquality,
unionElimination,
intEquality,
int_eqEquality,
approximateComputation,
applyLambdaEquality,
hyp_replacement,
axiomEquality,
minusEquality,
independent_pairEquality,
isect_memberFormation,
inlFormation,
inrFormation,
dependent_pairFormation,
addEquality,
promote_hyp
Latex:
\mforall{}a:\mBbbR{}. \mforall{}b:\{b:\mBbbR{}| a \mleq{} b\} . \mforall{}f,g:\{f:[a, b] {}\mrightarrow{}\mBbbR{}| ifun(f;[a, b])\} .
((\mforall{}x:\mBbbR{}. ((x \mmember{} [a, b]) {}\mRightarrow{} (r0 \mleq{} g[x])))
{}\mRightarrow{} (\mforall{}e:\{e:\mBbbR{}| r0 < e\} . \mexists{}c:\{x:\mBbbR{}| x \mmember{} [a, b]\} . (|a\_\mint{}\msupminus{}b f[x] * g[x] dx - f[c] * a\_\mint{}\msupminus{}b g[x] dx| < e))\000C)
Date html generated:
2017_10_04-PM-10_58_56
Last ObjectModification:
2017_08_02-PM-00_31_40
Theory : reals_2
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