### Nuprl Lemma : rless-iff-rleq

`∀x,y:ℝ.  (x < y `⇐⇒` ∃m:ℕ+. (x ≤ (y - (r1/r(m)))))`

Proof

Definitions occuring in Statement :  rdiv: `(x/y)` rleq: `x ≤ y` rless: `x < y` rsub: `x - y` int-to-real: `r(n)` real: `ℝ` nat_plus: `ℕ+` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` iff: `P `⇐⇒` Q` natural_number: `\$n`
Definitions unfolded in proof :  all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` uall: `∀[x:A]. B[x]` rev_implies: `P `` Q` exists: `∃x:A. B[x]` so_lambda: `λ2x.t[x]` nat_plus: `ℕ+` uimplies: `b supposing a` rneq: `x ≠ y` guard: `{T}` or: `P ∨ Q` decidable: `Dec(P)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` so_apply: `x[s]` itermConstant: `"const"` req_int_terms: `t1 ≡ t2` uiff: `uiff(P;Q)` rev_uimplies: `rev_uimplies(P;Q)` rless: `x < y` sq_exists: `∃x:{A| B[x]}` rge: `x ≥ y` rsub: `x - y`
Lemmas referenced :  rless_wf exists_wf nat_plus_wf rleq_wf rsub_wf rdiv_wf int-to-real_wf rless-int nat_plus_properties decidable__lt satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf real_wf radd-preserves-rless rminus_wf rless_functionality radd_wf real_term_polynomial itermSubtract_wf itermAdd_wf itermMinus_wf real_term_value_const_lemma real_term_value_sub_lemma real_term_value_add_lemma real_term_value_minus_lemma real_term_value_var_lemma req-iff-rsub-is-0 req_weakening small-reciprocal-real rleq_functionality_wrt_implies rleq_weakening_equal rsub_functionality_wrt_rleq rleq_weakening_rless rleq_weakening radd-preserves-rleq rless_transitivity1 uiff_transitivity rleq_functionality radd_functionality radd-rminus-assoc radd_comm radd-zero-both rmul_preserves_rless itermMultiply_wf int_term_value_mul_lemma rmul_wf rmul-rdiv-cancel2 rmul-int
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation independent_pairFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis productElimination sqequalRule lambdaEquality natural_numberEquality setElimination rename because_Cache independent_isectElimination inrFormation dependent_functionElimination independent_functionElimination unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll addLevel levelHypothesis promote_hyp dependent_set_memberEquality equalityTransitivity equalitySymmetry multiplyEquality lemma_by_obid

Latex:
\mforall{}x,y:\mBbbR{}.    (x  <  y  \mLeftarrow{}{}\mRightarrow{}  \mexists{}m:\mBbbN{}\msupplus{}.  (x  \mleq{}  (y  -  (r1/r(m)))))

Date html generated: 2017_10_03-AM-09_05_59
Last ObjectModification: 2017_07_28-AM-07_41_51

Theory : reals

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