Nuprl Lemma : Riemann-integral_wf
∀[a:ℝ]. ∀[b:{b:ℝ| a ≤ b} ]. ∀[f:{f:[a, b] ⟶ℝ| ifun(f;[a, b])} ]. (∫ f[x] dx on [a, b] ∈ ℝ)
Proof
Definitions occuring in Statement :
Riemann-integral: ∫ f[x] dx on [a, b],
ifun: ifun(f;I),
rfun: I ⟶ℝ,
rccint: [l, u],
rleq: x ≤ y,
real: ℝ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
Riemann-integral: ∫ f[x] dx on [a, b],
squash: ↓T,
uimplies: b supposing a,
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
guard: {T},
rev_implies: P ⇐ Q,
prop: ℙ,
converges: x[n]↓ as n→∞,
exists: ∃x:A. B[x],
top: Top,
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
nat: ℕ,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
false: False,
uiff: uiff(P;Q),
subtract: n - m,
less_than': less_than'(a;b),
true: True,
sq_stable: SqStable(P)
Lemmas referenced :
pi1_wf_top,
ifun_wf,
eta_conv,
real_wf,
rccint-icompact,
iff_weakening_equal,
i-member_wf,
rccint_wf,
exists_wf,
converges-to_wf,
Riemann-sum_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,
set_wf,
rfun_wf,
sq_stable__rleq,
rleq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
setElimination,
thin,
rename,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
applyEquality,
lambdaEquality,
imageElimination,
independent_isectElimination,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
productElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
setEquality,
independent_pairEquality,
isect_memberEquality,
voidElimination,
voidEquality,
addEquality,
natural_numberEquality,
unionElimination,
independent_pairFormation,
lambdaFormation,
minusEquality,
axiomEquality
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] \mmember{} \mBbbR{})
Date html generated:
2016_10_26-PM-00_02_04
Last ObjectModification:
2016_09_12-PM-05_37_47
Theory : reals_2
Home
Index