Nuprl Lemma : proper-continuous-maps-compact
∀I:Interval. ∀f:I ⟶ℝ. (f[x] (proper)continuous for x ∈ I ⇒ maps-compact-proper(I;(-∞, ∞);x.f[x]))
Proof
Definitions occuring in Statement :
maps-compact-proper: maps-compact-proper(I;J;x.f[x]),
proper-continuous: f[x] (proper)continuous for x ∈ I,
rfun: I ⟶ℝ,
riiint: (-∞, ∞),
interval: Interval,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
so_apply: x[s],
rfun: I ⟶ℝ,
label: ...$L... t,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
squash: ↓T,
sq_stable: SqStable(P),
prop: ℙ,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
and: P ∧ Q,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
member: t ∈ T,
maps-compact-proper: maps-compact-proper(I;J;x.f[x]),
implies: P ⇒ Q,
all: ∀x:A. B[x],
true: True,
cand: A c∧ B,
guard: {T},
subinterval: I ⊆ J ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
riiint: (-∞, ∞),
i-approx: i-approx(I;n),
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
isl: isl(x),
rccint: [l, u],
i-finite: i-finite(I),
iproper: iproper(I),
lower-bound: lower-bound(A;b),
inf: inf(A) = b,
rset-member: x ∈ A,
rrange: f[x](x∈I),
sup: sup(A) = b,
upper-bound: A ≤ b
Latex:
\mforall{}I:Interval. \mforall{}f:I {}\mrightarrow{}\mBbbR{}. (f[x] (proper)continuous for x \mmember{} I {}\mRightarrow{} maps-compact-proper(I;(-\minfty{}, \minfty{});x.f[x]))
Date html generated:
2020_05_20-PM-00_26_53
Last ObjectModification:
2019_12_28-AM-11_09_14
Theory : reals
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