Nuprl Lemma : function-maps-compact
∀I:Interval. ∀f:I ⟶ℝ.
(iproper(I) ⇒ (∀x,y:{t:ℝ| t ∈ I} . ((x = y) ⇒ (f[x] = f[y]))) ⇒ maps-compact(I;(-∞, ∞);x.f[x]))
Proof
Definitions occuring in Statement :
maps-compact: maps-compact(I;J;x.f[x]),
rfun: I ⟶ℝ,
riiint: (-∞, ∞),
i-member: r ∈ I,
iproper: iproper(I),
interval: Interval,
req: x = y,
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
so_apply: x[s],
uall: ∀[x:A]. B[x],
prop: ℙ
Lemmas referenced :
continuous-maps-compact,
function-proper-continuous,
i-member_wf,
real_wf,
all_wf,
req_wf,
iproper_wf,
rfun_wf,
interval_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
sqequalRule,
lambdaEquality,
applyEquality,
setElimination,
rename,
dependent_set_memberEquality,
isectElimination,
setEquality,
because_Cache,
functionEquality
Latex:
\mforall{}I:Interval. \mforall{}f:I {}\mrightarrow{}\mBbbR{}.
(iproper(I) {}\mRightarrow{} (\mforall{}x,y:\{t:\mBbbR{}| t \mmember{} I\} . ((x = y) {}\mRightarrow{} (f[x] = f[y]))) {}\mRightarrow{} maps-compact(I;(-\minfty{}, \minfty{});x.f[x])\000C)
Date html generated:
2016_10_26-AM-09_59_29
Last ObjectModification:
2016_08_27-AM-10_14_59
Theory : reals
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