### Nuprl Lemma : small-reciprocal-real

`∀x:{x:ℝ| r0 < x} . ∃k:ℕ+. ((r1/r(k)) < x)`

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]` set: `{x:A| B[x]} ` natural_number: `\$n`
Definitions unfolded in proof :  all: `∀x:A. B[x]` member: `t ∈ T` uall: `∀[x:A]. B[x]` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` real: `ℝ` prop: `ℙ` so_apply: `x[s]` rless: `x < y` sq_exists: `∃x:{A| B[x]}` uimplies: `b supposing a` nat_plus: `ℕ+` int-to-real: `r(n)` decidable: `Dec(P)` or: `P ∨ Q` less_than: `a < b` squash: `↓T` and: `P ∧ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` implies: `P `` Q` not: `¬A` top: `Top` rational-approx: `(x within 1/n)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` subtract: `n - m` le: `A ≤ B` less_than': `less_than'(a;b)` true: `True` rneq: `x ≠ y` guard: `{T}` sq_stable: `SqStable(P)` int_nzero: `ℤ-o` nequal: `a ≠ b ∈ T ` rge: `x ≥ y` itermConstant: `"const"` req_int_terms: `t1 ≡ t2` rdiv: `(x/y)` ge: `i ≥ j ` rev_uimplies: `rev_uimplies(P;Q)` nat: `ℕ`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesis sqequalRule lambdaEquality addEquality applyEquality natural_numberEquality setElimination rename hypothesisEquality dependent_functionElimination dependent_set_memberEquality cutEval equalityTransitivity equalitySymmetry independent_isectElimination intEquality unionElimination imageElimination productElimination dependent_pairFormation int_eqEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll independent_functionElimination because_Cache minusEquality inrFormation imageMemberEquality baseClosed multiplyEquality baseApply closedConclusion addLevel levelHypothesis

Latex:
\mforall{}x:\{x:\mBbbR{}|  r0  <  x\}  .  \mexists{}k:\mBbbN{}\msupplus{}.  ((r1/r(k))  <  x)

Date html generated: 2017_10_03-AM-08_51_03
Last ObjectModification: 2017_07_28-AM-07_34_22

Theory : reals

Home Index