Nuprl Lemma : weak-continuity-principle-interval
∀u,v:ℝ. ∀x:{x:ℝ| x ∈ [u, v]} .
∃x':{x':ℝ| x' = x}
∀F:{x:ℝ| x ∈ [u, v]} ⟶ 𝔹. ∀G:n:ℕ+ ⟶ {y:ℝ| (x = y ∈ (ℕ+n ⟶ ℤ)) ∧ (y ∈ [u, v])} .
∃z:{x:ℝ| x ∈ [u, v]} . (∃n:ℕ+ [(F x' = F z ∧ (z = (G n)))])
Proof
Definitions occuring in Statement :
rccint: [l, u],
i-member: r ∈ I,
req: x = y,
real: ℝ,
int_seg: {i..j-},
nat_plus: ℕ+,
bool: 𝔹,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
exists: ∃x:A. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
i-member: r ∈ I,
rccint: [l, u],
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
prop: ℙ,
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
so_apply: x[s],
real: ℝ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
sq_exists: ∃x:A [B[x]],
interval-retraction: interval-retraction(u;v;r),
rmax: rmax(x;y),
rmin: rmin(x;y),
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
true: True,
cand: A c∧ B
Latex:
\mforall{}u,v:\mBbbR{}. \mforall{}x:\{x:\mBbbR{}| x \mmember{} [u, v]\} .
\mexists{}x':\{x':\mBbbR{}| x' = x\}
\mforall{}F:\{x:\mBbbR{}| x \mmember{} [u, v]\} {}\mrightarrow{} \mBbbB{}. \mforall{}G:n:\mBbbN{}\msupplus{} {}\mrightarrow{} \{y:\mBbbR{}| (x = y) \mwedge{} (y \mmember{} [u, v])\} .
\mexists{}z:\{x:\mBbbR{}| x \mmember{} [u, v]\} . (\mexists{}n:\mBbbN{}\msupplus{} [(F x' = F z \mwedge{} (z = (G n)))])
Date html generated:
2020_05_20-PM-00_06_14
Last ObjectModification:
2019_12_14-PM-03_11_17
Theory : reals
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