### Nuprl Lemma : rless-iff-large-diff

`∀x,y:ℝ.  (x < y `⇐⇒` ∀b:ℕ+. ∃n:ℕ+. ∀m:ℕ+. ((n ≤ m) `` (b ≤ ((y m) - x m))))`

Proof

Definitions occuring in Statement :  rless: `x < y` real: `ℝ` nat_plus: `ℕ+` le: `A ≤ B` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` iff: `P `⇐⇒` Q` implies: `P `` Q` apply: `f a` subtract: `n - m`
Definitions unfolded in proof :  rless: `x < y` all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` member: `t ∈ T` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` prop: `ℙ` rev_implies: `P `` Q` so_lambda: `λ2x.t[x]` nat_plus: `ℕ+` real: `ℝ` so_apply: `x[s]` int_upper: `{i...}` sq_exists: `∃x:{A| B[x]}` sq_stable: `SqStable(P)` squash: `↓T` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` le: `A ≤ B` less_than: `a < b` uiff: `uiff(P;Q)` subtract: `n - m` subtype_rel: `A ⊆r B` less_than': `less_than'(a;b)` true: `True` guard: `{T}`
Lemmas referenced :  subtract-is-int-iff int_upper_properties less_than_transitivity1 int_upper_wf le-add-cancel add-associates add_functionality_wrt_le add-commutes minus-one-mul-top zero-add minus-one-mul minus-add condition-implies-le not-lt-2 decidable__lt false_wf int_term_value_subtract_lemma itermSubtract_wf add-is-int-iff less_than_wf int_formula_prop_wf int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_add_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma intformless_wf itermVar_wf itermAdd_wf itermConstant_wf intformle_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__le sq_stable__less_than nat_plus_properties real_wf subtract_wf le_wf exists_wf nat_plus_wf all_wf rless_wf regular-less-iff
Rules used in proof :  cut lemma_by_obid sqequalHypSubstitution sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep lambdaFormation hypothesis isectElimination thin hypothesisEquality independent_pairFormation productElimination independent_functionElimination lambdaEquality functionEquality setElimination rename applyEquality dependent_functionElimination dependent_set_memberEquality addEquality natural_numberEquality introduction imageMemberEquality baseClosed imageElimination unionElimination independent_isectElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll because_Cache pointwiseFunctionality equalityTransitivity equalitySymmetry promote_hyp baseApply closedConclusion minusEquality

Latex:
\mforall{}x,y:\mBbbR{}.    (x  <  y  \mLeftarrow{}{}\mRightarrow{}  \mforall{}b:\mBbbN{}\msupplus{}.  \mexists{}n:\mBbbN{}\msupplus{}.  \mforall{}m:\mBbbN{}\msupplus{}.  ((n  \mleq{}  m)  {}\mRightarrow{}  (b  \mleq{}  ((y  m)  -  x  m))))

Date html generated: 2016_05_18-AM-07_03_38
Last ObjectModification: 2016_01_17-AM-01_50_09

Theory : reals

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