Nuprl Lemma : range-sup_wf
∀[I:{I:Interval| icompact(I)} ]. ∀[f:I ⟶ℝ]. ∀[mc:f[x] continuous for x ∈ I]. (sup{f[x]|x ∈ I} ∈ ℝ)
Proof
Definitions occuring in Statement :
range-sup: sup{f[x]|x ∈ I},
continuous: f[x] continuous for x ∈ I,
icompact: icompact(I),
rfun: I ⟶ℝ,
interval: Interval,
real: ℝ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
set: {x:A| B[x]}
Definitions unfolded in proof :
r-ap: f(x),
squash: ↓T,
implies: P ⇒ Q,
sq_stable: SqStable(P),
uimplies: b supposing a,
guard: {T},
prop: ℙ,
all: ∀x:A. B[x],
rfun: I ⟶ℝ,
label: ...$L... t,
so_lambda: λ2x.t[x],
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
range-sup: sup{f[x]|x ∈ I},
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_apply: x[s]
Lemmas referenced :
equal_wf,
all_wf,
sup-range,
icompact_wf,
interval_wf,
set_wf,
rfun_wf,
subtype_rel_self,
continuous_wf,
sq_stable__i-member,
i-member_wf,
r-ap_wf,
rrange_wf,
sup_wf,
exists_wf,
real_wf,
pi1_wf_top
Rules used in proof :
functionExtensionality,
functionEquality,
instantiate,
isect_memberEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
imageElimination,
baseClosed,
imageMemberEquality,
independent_functionElimination,
independent_isectElimination,
setEquality,
dependent_functionElimination,
dependent_set_memberEquality,
hypothesisEquality,
lambdaFormation,
lambdaEquality,
because_Cache,
applyEquality,
hypothesis,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
rename,
thin,
setElimination,
cut,
introduction,
isect_memberFormation,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution
Latex:
\mforall{}[I:\{I:Interval| icompact(I)\} ]. \mforall{}[f:I {}\mrightarrow{}\mBbbR{}]. \mforall{}[mc:f[x] continuous for x \mmember{} I]. (sup\{f[x]|x \mmember{} I\} \mmember{} \mBbbR{})
Date html generated:
2018_05_22-PM-02_18_01
Last ObjectModification:
2018_05_21-AM-00_33_36
Theory : reals
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