Nuprl Lemma : r-archimedean
∀x:ℝ. ∃n:ℕ. ((r(-n) ≤ x) ∧ (x ≤ r(n)))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
minus: -n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
and: P ∧ Q,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
real: ℝ,
nat: ℕ,
nat_plus: ℕ+,
int_upper: {i...},
so_apply: x[s],
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
cand: A c∧ B
Lemmas referenced :
canonical-bound-property,
canonical-bound_wf,
subtype_rel_set,
int_upper_wf,
nat_wf,
all_wf,
nat_plus_wf,
le_wf,
absval_wf,
int_upper_subtype_nat,
false_wf,
rleq_wf,
int-to-real_wf,
real_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
dependent_pairFormation,
isectElimination,
applyEquality,
natural_numberEquality,
sqequalRule,
lambdaEquality,
setElimination,
rename,
multiplyEquality,
independent_isectElimination,
dependent_set_memberEquality,
independent_pairFormation,
productEquality,
minusEquality,
because_Cache
Latex:
\mforall{}x:\mBbbR{}. \mexists{}n:\mBbbN{}. ((r(-n) \mleq{} x) \mwedge{} (x \mleq{} r(n)))
Date html generated:
2017_10_03-AM-08_55_13
Last ObjectModification:
2017_07_03-PM-05_27_11
Theory : reals
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