Nuprl Lemma : sublevelset-closed
∀I:Interval. ∀[f:I ⟶ℝ]. ∀[c:ℝ]. (i-closed(I) ⇒ f(x) continuous for x ∈ I ⇒ closed-rset(sublevelset(I;f;c)))
Proof
Definitions occuring in Statement :
continuous: f[x] continuous for x ∈ I,
sublevelset: sublevelset(I;f;c),
r-ap: f(x),
rfun: I ⟶ℝ,
i-closed: i-closed(I),
interval: Interval,
closed-rset: closed-rset(A),
real: ℝ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
closed-rset: closed-rset(A),
member-closure: y ∈ closure(A),
exists: ∃x:A. B[x],
sublevelset: sublevelset(I;f;c),
and: P ∧ Q,
member: t ∈ T,
cand: A c∧ B,
prop: ℙ,
so_lambda: λ2x.t[x],
label: ...$L... t,
rfun: I ⟶ℝ,
uimplies: b supposing a,
sq_stable: SqStable(P),
squash: ↓T,
so_apply: x[s],
guard: {T}
Lemmas referenced :
continuous-limit,
rleq-limit-constant,
all_wf,
converges-to_wf,
and_wf,
nat_wf,
interval_wf,
rfun_wf,
i-closed_wf,
i-member_wf,
real_wf,
sq_stable__i-member,
r-ap_wf,
continuous_wf,
sublevelset_wf,
member-closure_wf,
i-closed-closed
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
cut,
lemma_by_obid,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
independent_pairFormation,
isectElimination,
because_Cache,
lambdaEquality,
setElimination,
rename,
independent_isectElimination,
introduction,
imageMemberEquality,
baseClosed,
imageElimination,
setEquality,
dependent_pairFormation,
applyEquality
Latex:
\mforall{}I:Interval
\mforall{}[f:I {}\mrightarrow{}\mBbbR{}]. \mforall{}[c:\mBbbR{}]. (i-closed(I) {}\mRightarrow{} f(x) continuous for x \mmember{} I {}\mRightarrow{} closed-rset(sublevelset(I;f;c)))
Date html generated:
2016_05_18-AM-09_21_22
Last ObjectModification:
2016_01_17-AM-02_41_34
Theory : reals
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