Nuprl Lemma : has-minimum-maps-compact
∀I:Interval. ∀l:ℝ. ∀f:I ⟶ℝ.
((∀x,y:{t:ℝ| t ∈ I} . ((x = y) ⇒ (f[x] = f[y])))
⇒ (∀x:{t:ℝ| t ∈ I} . (l < f[x]))
⇒ (∀a:{a:ℝ| a ∈ I} . ∀b:{b:ℝ| (b ∈ I) ∧ (a ≤ b)} . ∃c:{t:ℝ| t ∈ [a, b]} . ∀x:{t:ℝ| t ∈ [a, b]} . (f[c] ≤ f[x]))
⇒ maps-compact(I;(l, ∞);x.f[x]))
Proof
Definitions occuring in Statement :
maps-compact: maps-compact(I;J;x.f[x]),
rfun: I ⟶ℝ,
roiint: (l, ∞),
rccint: [l, u],
i-member: r ∈ I,
interval: Interval,
rleq: x ≤ y,
rless: x < y,
req: x = y,
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
and: P ∧ Q,
so_apply: x[s],
rfun: I ⟶ℝ,
maps-compact: maps-compact(I;J;x.f[x]),
sq_stable: SqStable(P),
squash: ↓T,
uimplies: b supposing a,
subinterval: I ⊆ J ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
top: Top,
guard: {T},
subtype_rel: A ⊆r B,
ifun: ifun(f;I),
real-fun: real-fun(f;a;b),
cand: A c∧ B,
nat_plus: ℕ+,
icompact: icompact(I),
i-nonvoid: i-nonvoid(I),
i-member: r ∈ I,
rccint: [l, u]
Lemmas referenced :
all_wf,
real_wf,
i-member_wf,
rleq_wf,
exists_wf,
rccint_wf,
rless_wf,
req_wf,
rfun_wf,
interval_wf,
set_wf,
nat_plus_wf,
icompact_wf,
i-approx_wf,
sq_stable__icompact,
icompact-is-rccint,
i-approx-is-subinterval,
left-endpoint_wf,
i-approx-finite,
icompact-endpoints,
right-endpoint_wf,
rccint-icompact,
subinterval_wf,
equal_wf,
member_roiint_lemma,
subtype_rel_sets,
member_rccint_lemma,
rleq-range_sup,
rfun_subtype,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
ifun_wf,
rleq_weakening_equal,
rless_transitivity1,
range_sup_wf,
i-approx-containing2,
roiint_wf,
less_than_wf,
i-approx-closed,
i-member-between
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
hypothesis,
hypothesisEquality,
sqequalRule,
lambdaEquality,
setElimination,
rename,
because_Cache,
productEquality,
dependent_functionElimination,
productElimination,
applyEquality,
dependent_set_memberEquality,
functionEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_pairFormation,
natural_numberEquality
Latex:
\mforall{}I:Interval. \mforall{}l:\mBbbR{}. \mforall{}f:I {}\mrightarrow{}\mBbbR{}.
((\mforall{}x,y:\{t:\mBbbR{}| t \mmember{} I\} . ((x = y) {}\mRightarrow{} (f[x] = f[y])))
{}\mRightarrow{} (\mforall{}x:\{t:\mBbbR{}| t \mmember{} I\} . (l < f[x]))
{}\mRightarrow{} (\mforall{}a:\{a:\mBbbR{}| a \mmember{} I\} . \mforall{}b:\{b:\mBbbR{}| (b \mmember{} I) \mwedge{} (a \mleq{} b)\} .
\mexists{}c:\{t:\mBbbR{}| t \mmember{} [a, b]\} . \mforall{}x:\{t:\mBbbR{}| t \mmember{} [a, b]\} . (f[c] \mleq{} f[x]))
{}\mRightarrow{} maps-compact(I;(l, \minfty{});x.f[x]))
Date html generated:
2017_10_03-AM-10_28_03
Last ObjectModification:
2017_07_28-AM-08_11_35
Theory : reals
Home
Index