Nuprl Lemma : integral_wf
∀[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 ∈ ℝ)
Proof
Definitions occuring in Statement :
integral: a_∫-b f[x] dx,
ifun: ifun(f;I),
rfun: I ⟶ℝ,
rccint: [l, u],
rmin: rmin(x;y),
rmax: rmax(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,
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,
subtype_rel: A ⊆r B,
guard: {T},
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
integral: a_∫-b f[x] dx
Lemmas referenced :
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,
set_wf,
ifun_subtype_1,
rmin-rleq,
rleq-rmax,
rsub_wf,
Riemann-integral_wf,
rleq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
setElimination,
thin,
rename,
dependent_set_memberEquality,
sqequalRule,
lambdaEquality,
applyEquality,
sqequalHypSubstitution,
hypothesisEquality,
hypothesis,
extract_by_obid,
isectElimination,
setEquality,
imageElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
dependent_functionElimination,
productElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
universeEquality,
axiomEquality,
isect_memberEquality
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 \mmember{} \mBbbR{})
Date html generated:
2016_10_26-PM-00_07_10
Last ObjectModification:
2016_09_12-PM-05_38_29
Theory : reals_2
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