### Nuprl Lemma : implies-real

`∀[x:ℕ+ ⟶ ℤ]. x ∈ ℝ supposing ∀n,m:ℕ+.  (|(x within 1/n) - (x within 1/m)| ≤ ((r1/r(n)) + (r1/r(m))))`

Proof

Definitions occuring in Statement :  rational-approx: `(x within 1/n)` rdiv: `(x/y)` rleq: `x ≤ y` rabs: `|x|` rsub: `x - y` radd: `a + b` int-to-real: `r(n)` real: `ℝ` nat_plus: `ℕ+` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` member: `t ∈ T` function: `x:A ⟶ B[x]` natural_number: `\$n` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` real: `ℝ` nat_plus: `ℕ+` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` true: `True` and: `P ∧ Q` prop: `ℙ` so_lambda: `λ2x.t[x]` rneq: `x ≠ y` guard: `{T}` or: `P ∨ Q` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` implies: `P `` Q` decidable: `Dec(P)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` so_apply: `x[s]`
Lemmas referenced :  rless_wf int_formula_prop_wf int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_less_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermVar_wf itermConstant_wf intformless_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__lt nat_plus_properties rless-int int-to-real_wf rdiv_wf radd_wf rational-approx_wf rsub_wf rabs_wf rleq_wf nat_plus_wf all_wf regular-int-seq_wf less_than_wf implies-regular
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut dependent_set_memberEquality hypothesisEquality lemma_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality sqequalRule independent_pairFormation imageMemberEquality baseClosed hypothesis because_Cache independent_isectElimination axiomEquality equalityTransitivity equalitySymmetry lambdaEquality setElimination rename inrFormation dependent_functionElimination productElimination independent_functionElimination unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll functionEquality

Latex:
\mforall{}[x:\mBbbN{}\msupplus{}  {}\mrightarrow{}  \mBbbZ{}]
x  \mmember{}  \mBbbR{}  supposing  \mforall{}n,m:\mBbbN{}\msupplus{}.    (|(x  within  1/n)  -  (x  within  1/m)|  \mleq{}  ((r1/r(n))  +  (r1/r(m))))

Date html generated: 2016_05_18-AM-07_35_21
Last ObjectModification: 2016_01_17-AM-02_02_09

Theory : reals

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