Nuprl Lemma : Riemann-integral-rleq
∀[a:ℝ]. ∀[b:{b:ℝ| a ≤ b} ]. ∀[f,g:{f:[a, b] ⟶ℝ| ifun(f;[a, b])} ].
∫ f[x] dx on [a, b] ≤ ∫ g[x] dx on [a, b] supposing ∀x:ℝ. ((x ∈ [a, b]) ⇒ (f[x] ≤ g[x]))
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,
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
so_apply: x[s],
prop: ℙ,
nat_plus: ℕ+,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
uiff: uiff(P;Q),
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
less_than': less_than'(a;b),
true: True,
label: ...$L... t,
ifun: ifun(f;I),
real-fun: real-fun(f;a;b),
rev_uimplies: rev_uimplies(P;Q),
rleq: x ≤ y,
rnonneg: rnonneg(x),
sq_stable: SqStable(P),
squash: ↓T,
i-member: r ∈ I,
rccint: [l, u]
Lemmas referenced :
Riemann-sums-converge-to,
rleq-limit,
Riemann-sum_wf,
i-member_wf,
rccint_wf,
real_wf,
decidable__lt,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
less_than_wf,
nat_wf,
Riemann-integral_wf,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
req_functionality,
req_weakening,
req_wf,
set_wf,
ifun_wf,
rccint-icompact,
less_than'_wf,
rsub_wf,
nat_plus_wf,
all_wf,
rleq_wf,
rfun_wf,
sq_stable__rleq,
Riemann-sum-rleq,
member_rccint_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
sqequalRule,
lambdaEquality,
because_Cache,
setElimination,
rename,
dependent_set_memberEquality,
hypothesis,
applyEquality,
setEquality,
addEquality,
natural_numberEquality,
productElimination,
unionElimination,
independent_pairFormation,
lambdaFormation,
voidElimination,
independent_functionElimination,
independent_isectElimination,
isect_memberEquality,
voidEquality,
intEquality,
minusEquality,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
imageMemberEquality,
baseClosed,
imageElimination,
productEquality
Latex:
\mforall{}[a:\mBbbR{}]. \mforall{}[b:\{b:\mBbbR{}| a \mleq{} b\} ]. \mforall{}[f,g:\{f:[a, b] {}\mrightarrow{}\mBbbR{}| ifun(f;[a, b])\} ].
\mint{} f[x] dx on [a, b] \mleq{} \mint{} g[x] dx on [a, b] supposing \mforall{}x:\mBbbR{}. ((x \mmember{} [a, b]) {}\mRightarrow{} (f[x] \mleq{} g[x]))
Date html generated:
2016_10_26-PM-00_02_57
Last ObjectModification:
2016_09_12-PM-05_38_06
Theory : reals_2
Home
Index