Nuprl Lemma : integer-between-reals
∀a,b:ℝ. ((r(2) ≤ (b - a)) ⇒ (∃k:ℤ. ((a < r(k)) ∧ (r(k) < b))))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rless: x < y,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
or: P ∨ Q,
prop: ℙ,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
int_upper: {i...},
cand: A c∧ B,
rge: x ≥ y,
guard: {T},
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
req_int_terms: t1 ≡ t2,
so_lambda: λ2x.t[x],
nat: ℕ,
so_apply: x[s],
subtract: n - m,
top: Top
Latex:
\mforall{}a,b:\mBbbR{}. ((r(2) \mleq{} (b - a)) {}\mRightarrow{} (\mexists{}k:\mBbbZ{}. ((a < r(k)) \mwedge{} (r(k) < b))))
Date html generated:
2020_05_20-AM-11_04_43
Last ObjectModification:
2020_03_14-AM-09_31_56
Theory : reals
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