Nuprl Lemma : countable-Heine-Borel-weak
∀a:ℝ. ∀b:{b:ℝ| a ≤ b} .
∀[C:ℕ ⟶ {x:ℝ| x ∈ [a, b]} ⟶ ℙ]
((∀n:ℕ. ∀x:{x:ℝ| x ∈ [a, b]} . ∀y:{y:{x:ℝ| x ∈ [a, b]} | x = y} . (C[n;x] ⇒ C[n;y]))
⇒ (∀x:{x:ℝ| x ∈ [a, b]} . ∃n:ℕ. C[n;x])
⇒ (∃k:ℕ. ∀x:{x:ℝ| x ∈ [a, b]} . (¬¬(∃n:ℕk. C[n;x]))))
Proof
Definitions occuring in Statement :
rccint: [l, u],
i-member: r ∈ I,
rleq: x ≤ y,
req: x = y,
real: ℝ,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uimplies: b supposing a,
sq_stable: SqStable(P),
squash: ↓T,
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
not: ¬A,
nat: ℕ,
so_apply: x[s1;s2],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than: a < b,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
prop: ℙ,
i-member: r ∈ I,
rccint: [l, u],
pi1: fst(t),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
bfalse: ff,
less_than': less_than'(a;b),
powerset: powerset(T),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
equipollent: A ~ B,
cand: A c∧ B,
subtract: n - m,
true: True,
sq_type: SQType(T),
compose: f o g,
biject: Bij(A;B;f),
surject: Surj(A;B;f),
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
stable: Stable{P}
Latex:
\mforall{}a:\mBbbR{}. \mforall{}b:\{b:\mBbbR{}| a \mleq{} b\} .
\mforall{}[C:\mBbbN{} {}\mrightarrow{} \{x:\mBbbR{}| x \mmember{} [a, b]\} {}\mrightarrow{} \mBbbP{}]
((\mforall{}n:\mBbbN{}. \mforall{}x:\{x:\mBbbR{}| x \mmember{} [a, b]\} . \mforall{}y:\{y:\{x:\mBbbR{}| x \mmember{} [a, b]\} | x = y\} . (C[n;x] {}\mRightarrow{} C[n;y]))
{}\mRightarrow{} (\mforall{}x:\{x:\mBbbR{}| x \mmember{} [a, b]\} . \mexists{}n:\mBbbN{}. C[n;x])
{}\mRightarrow{} (\mexists{}k:\mBbbN{}. \mforall{}x:\{x:\mBbbR{}| x \mmember{} [a, b]\} . (\mneg{}\mneg{}(\mexists{}n:\mBbbN{}k. C[n;x]))))
Date html generated:
2020_05_20-PM-00_11_17
Last ObjectModification:
2019_11_25-PM-00_23_54
Theory : reals
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