Nuprl Lemma : monotone-maps-compact
∀I,J:Interval. ∀f:I ⟶ℝ.
(((∀x,y:{t:ℝ| t ∈ I} . ((x ≤ y) ⇒ (f[x] ≤ f[y]))) ∨ (∀x,y:{t:ℝ| t ∈ I} . ((x ≤ y) ⇒ (f[y] ≤ f[x]))))
⇒ (∀x:{t:ℝ| t ∈ I} . (f[x] ∈ J))
⇒ maps-compact(I;J;x.f[x]))
Proof
Definitions occuring in Statement :
maps-compact: maps-compact(I;J;x.f[x]),
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
rleq: x ≤ y,
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
maps-compact: maps-compact(I;J;x.f[x]),
member: t ∈ T,
uall: ∀[x:A]. B[x],
sq_stable: SqStable(P),
squash: ↓T,
nat_plus: ℕ+,
prop: ℙ,
uimplies: b supposing a,
subinterval: I ⊆ J ,
and: P ∧ Q,
so_apply: x[s],
rfun: I ⟶ℝ,
exists: ∃x:A. B[x],
icompact: icompact(I),
i-nonvoid: i-nonvoid(I),
cand: A c∧ B,
so_lambda: λ2x.t[x],
iff: P ⇐⇒ Q,
guard: {T},
or: P ∨ Q,
rbetween: x≤y≤z
Lemmas referenced :
sq_stable__icompact,
i-approx_wf,
icompact-is-rccint,
less_than_wf,
i-approx-is-subinterval,
right-endpoint_wf,
i-approx-finite,
icompact-endpoints,
left-endpoint_wf,
i-member_wf,
i-approx-containing2,
i-approx-closed,
set_wf,
nat_plus_wf,
icompact_wf,
all_wf,
real_wf,
or_wf,
rleq_wf,
rfun_wf,
interval_wf,
sq_stable__i-member,
i-member-compact,
rbetween_wf,
i-member-between
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
setElimination,
thin,
rename,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
isectElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality,
natural_numberEquality,
independent_isectElimination,
because_Cache,
productElimination,
independent_pairFormation,
applyEquality,
dependent_pairFormation,
lambdaEquality,
setEquality,
functionEquality,
addLevel,
levelHypothesis,
unionElimination,
inlFormation,
promote_hyp,
inrFormation
Latex:
\mforall{}I,J:Interval. \mforall{}f:I {}\mrightarrow{}\mBbbR{}.
(((\mforall{}x,y:\{t:\mBbbR{}| t \mmember{} I\} . ((x \mleq{} y) {}\mRightarrow{} (f[x] \mleq{} f[y])))
\mvee{} (\mforall{}x,y:\{t:\mBbbR{}| t \mmember{} I\} . ((x \mleq{} y) {}\mRightarrow{} (f[y] \mleq{} f[x]))))
{}\mRightarrow{} (\mforall{}x:\{t:\mBbbR{}| t \mmember{} I\} . (f[x] \mmember{} J))
{}\mRightarrow{} maps-compact(I;J;x.f[x]))
Date html generated:
2016_10_26-AM-09_59_45
Last ObjectModification:
2016_08_27-AM-11_33_16
Theory : reals
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