Nuprl Lemma : rsqrt-irrational

n:ℕ(irrational(rsqrt(r(n))) ∨ (∃m:ℕ1. ((m m) n ∈ ℤ)))

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


Definitions occuring in Statement :  irrational: irrational(x) rsqrt: rsqrt(x) int-to-real: r(n) int_seg: {i..j-} nat: all: x:A. B[x] exists: x:A. B[x] or: P ∨ Q multiply: m add: m natural_number: $n int: equal: t ∈ T
Definitions unfolded in proof :  all: x:A. B[x] member: t ∈ T nat: uall: [x:A]. B[x] so_lambda: λ2x.t[x] int_seg: {i..j-} so_apply: x[s] implies:  Q decidable: Dec(P) or: P ∨ Q guard: {T} prop: iff: ⇐⇒ Q and: P ∧ Q rev_implies:  Q ge: i ≥  exists: x:A. B[x] uimplies: supposing a satisfiable_int_formula: satisfiable_int_formula(fmla) false: False not: ¬A top: Top subtype_rel: A ⊆B irrational: irrational(x) nat_plus: + rneq: x ≠ y rless: x < y sq_exists: x:{A| B[x]} real: 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: y rge: x ≥ y less_than: a < b
Lemmas referenced :  decidable__exists_int_seg equal_wf int_seg_wf decidable__equal_int irrational_wf rsqrt_wf rleq-int nat_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf int-to-real_wf rleq_wf nat_plus_wf exists_wf nat_wf decidable__lt rless_wf nat_plus_properties rdiv_wf rless-int intformless_wf int_formula_prop_less_lemma rmul_wf rless_functionality req_weakening rmul-int rmul_functionality sq_stable__less_than real_wf rmul_preserves_rless rinv_wf2 req_transitivity real_term_polynomial itermSubtract_wf itermMultiply_wf real_term_value_const_lemma real_term_value_sub_lemma real_term_value_mul_lemma real_term_value_var_lemma req-iff-rsub-is-0 rmul-rinv3 rinv-as-rdiv rmul_preserves_rleq rleq_functionality rsqrt_squared rsqrt_nonneg req_wf req_inversion rless_transitivity1 rleq_weakening rmul-is-positive radd_wf rsub_wf rmul_over_rminus rmul-distrib rmul_comm radd-assoc radd-ac radd_comm radd-rminus-assoc radd-rminus-both radd_functionality radd-zero-both rminus_wf radd-preserves-rless rless-implies-rless rleq_weakening_equal radd_functionality_wrt_rleq rless_functionality_wrt_implies radd-int int_term_value_add_lemma itermAdd_wf rless_transitivity2 int_term_value_mul_lemma int_formula_prop_eq_lemma intformeq_wf irrational-sqrt-number-lemma rmul-rdiv-cancel2
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

\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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