Nuprl Lemma : rationals-dense
∀x:ℝ. ∀y:{y:ℝ| x < y} . ∃n:ℕ+. ∃m:ℤ. ((x < (r(m)/r(n))) ∧ ((r(m)/r(n)) < y))
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rless: x < y,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
int: ℤ
Definitions unfolded in proof :
top: Top,
not: ¬A,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
and: P ∧ Q,
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
exists: ∃x:A. B[x],
nat_plus: ℕ+,
uimplies: b supposing a,
sq_exists: ∃x:A [B[x]],
rless: x < y,
subtype_rel: A ⊆r B,
so_apply: x[s],
prop: ℙ,
real: ℝ,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
rational-approx: (x within 1/n),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
uiff: uiff(P;Q),
rge: x ≥ y,
rdiv: (x/y),
req_int_terms: t1 ≡ t2
Latex:
\mforall{}x:\mBbbR{}. \mforall{}y:\{y:\mBbbR{}| x < y\} . \mexists{}n:\mBbbN{}\msupplus{}. \mexists{}m:\mBbbZ{}. ((x < (r(m)/r(n))) \mwedge{} ((r(m)/r(n)) < y))
Date html generated:
2020_05_20-AM-11_08_22
Last ObjectModification:
2020_03_20-PM-01_12_09
Theory : reals
Home
Index