Nuprl Lemma : not-in-compact-separated
∀[X:Type]
∀d:metric(X)
∀[A:Type]
∀c:mcompact(A;d). ∀x:X. (¬(x ∈ A) ⇐⇒ ¬¬(∃n:ℕ+. ∀a:A. ((r1/r(n)) ≤ mdist(d;x;a))))
supposing mcomplete(X with d) ∧ metric-subspace(X;d;A)
Proof
Definitions occuring in Statement :
mcompact: mcompact(X;d),
mcomplete: mcomplete(M),
metric-subspace: metric-subspace(X;d;A),
mk-metric-space: X with d,
mdist: mdist(d;x;y),
metric: metric(X),
rdiv: (x/y),
rleq: x ≤ y,
int-to-real: r(n),
nat_plus: ℕ+,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
and: P ∧ Q,
member: t ∈ T,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
sq_stable: SqStable(P),
implies: P ⇒ Q,
and: P ∧ Q,
squash: ↓T,
metric-subspace: metric-subspace(X;d;A),
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
not: ¬A,
exists: ∃x:A. B[x],
istype: istype(T),
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
prop: ℙ,
label: ...$L... t,
respects-equality: respects-equality(S;T),
stable: Stable{P},
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
rdiv: (x/y),
req_int_terms: t1 ≡ t2
Latex:
\mforall{}[X:Type]
\mforall{}d:metric(X)
\mforall{}[A:Type]
\mforall{}c:mcompact(A;d). \mforall{}x:X. (\mneg{}(x \mmember{} A) \mLeftarrow{}{}\mRightarrow{} \mneg{}\mneg{}(\mexists{}n:\mBbbN{}\msupplus{}. \mforall{}a:A. ((r1/r(n)) \mleq{} mdist(d;x;a))))
supposing mcomplete(X with d) \mwedge{} metric-subspace(X;d;A)
Date html generated:
2020_05_20-PM-00_00_57
Last ObjectModification:
2019_11_18-AM-11_40_41
Theory : reals
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