Nuprl Lemma : mcompact-finite-subcover
∀[X:Type]
∀d:metric(X)
(mcompact(X;d)
⇒ (∀[I:Type]. ∀[A:I ⟶ X ⟶ ℙ]. (m-open-cover(X;d;I;i,x.A[i;x]) ⇒ (∃n:ℕ+. ∃L:ℕn ⟶ I. ∀x:X. ∃j:ℕn. A[L j;x]))))
Proof
Definitions occuring in Statement :
mcompact: mcompact(X;d),
m-open-cover: m-open-cover(X;d;I;i,x.A[i; x]),
metric: metric(X),
int_seg: {i..j-},
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
iff: P ⇐⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
mcompact: mcompact(X;d),
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
prop: ℙ,
nat: ℕ,
uimplies: b supposing a,
rev_implies: P ⇐ Q,
not: ¬A,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
so_lambda: λ2x.t[x],
so_apply: x[s],
pi1: fst(t),
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
decidable: Dec(P),
squash: ↓T,
true: True,
uiff: uiff(P;Q),
le: A ≤ B,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b
Latex:
\mforall{}[X:Type]
\mforall{}d:metric(X)
(mcompact(X;d)
{}\mRightarrow{} (\mforall{}[I:Type]. \mforall{}[A:I {}\mrightarrow{} X {}\mrightarrow{} \mBbbP{}].
(m-open-cover(X;d;I;i,x.A[i;x]) {}\mRightarrow{} (\mexists{}n:\mBbbN{}\msupplus{}. \mexists{}L:\mBbbN{}n {}\mrightarrow{} I. \mforall{}x:X. \mexists{}j:\mBbbN{}n. A[L j;x]))))
Date html generated:
2020_05_20-PM-00_02_35
Last ObjectModification:
2020_01_12-PM-01_47_17
Theory : reals
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