### Nuprl Lemma : rleq-int-fractions2

`∀[a,b:ℤ]. ∀[d:ℕ+].  uiff(r(a) ≤ (r(b)/r(d));(a * d) ≤ b)`

Proof

Definitions occuring in Statement :  rdiv: `(x/y)` rleq: `x ≤ y` int-to-real: `r(n)` nat_plus: `ℕ+` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` le: `A ≤ B` multiply: `n * m` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` le: `A ≤ B` not: `¬A` implies: `P `` Q` false: `False` nat_plus: `ℕ+` prop: `ℙ` rneq: `x ≠ y` guard: `{T}` or: `P ∨ Q` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` decidable: `Dec(P)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` rleq: `x ≤ y` rnonneg: `rnonneg(x)` subtype_rel: `A ⊆r B` rev_uimplies: `rev_uimplies(P;Q)`
Lemmas referenced :  rmul_preserves_rleq rmul-rdiv-cancel2 req_weakening rmul-int rleq_functionality uiff_transitivity rmul_wf int_formula_prop_le_lemma intformle_wf decidable__le rleq-int rmul_preserves_rleq2 le_wf nat_plus_wf rsub_wf 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 rdiv_wf int-to-real_wf rleq_wf less_than'_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut independent_pairFormation sqequalRule sqequalHypSubstitution productElimination thin independent_pairEquality lambdaEquality dependent_functionElimination hypothesisEquality because_Cache lemma_by_obid isectElimination multiplyEquality setElimination rename hypothesis axiomEquality equalityTransitivity equalitySymmetry independent_isectElimination inrFormation independent_functionElimination natural_numberEquality unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll applyEquality minusEquality

Latex:
\mforall{}[a,b:\mBbbZ{}].  \mforall{}[d:\mBbbN{}\msupplus{}].    uiff(r(a)  \mleq{}  (r(b)/r(d));(a  *  d)  \mleq{}  b)

Date html generated: 2016_05_18-AM-07_27_26
Last ObjectModification: 2016_01_17-AM-01_58_44

Theory : reals

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