### Nuprl Lemma : rsqrt-irrational

`∀n:ℕ. (irrational(rsqrt(r(n))) ∨ (∃m:ℕn + 1. ((m * m) = n ∈ ℤ)))`

This theorem is one of freek's list of 100 theorems

Proof

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

Latex:
\mforall{}n:\mBbbN{}.  (irrational(rsqrt(r(n)))  \mvee{}  (\mexists{}m:\mBbbN{}n  +  1.  ((m  *  m)  =  n)))

Date html generated: 2017_10_03-AM-11_59_31
Last ObjectModification: 2017_07_28-AM-08_30_14

Theory : reals

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