### Nuprl Lemma : rless-int-fractions

`∀a,b:ℤ. ∀c,d:ℕ+.  ((r(a)/r(c)) < (r(b)/r(d)) `⇐⇒` a * d < b * c)`

Proof

Definitions occuring in Statement :  rdiv: `(x/y)` rless: `x < y` int-to-real: `r(n)` nat_plus: `ℕ+` less_than: `a < b` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` multiply: `n * m` int: `ℤ`
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]` nat_plus: `ℕ+` uimplies: `b supposing a` rneq: `x ≠ y` guard: `{T}` or: `P ∨ Q` rev_implies: `P `` Q` decidable: `Dec(P)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` rless: `x < y` sq_exists: `∃x:{A| B[x]}` uiff: `uiff(P;Q)` rev_uimplies: `rev_uimplies(P;Q)`
Lemmas referenced :  rmul-rdiv-cancel rmul-ac rmul_functionality rmul-assoc req_inversion req_functionality uiff_transitivity rmul-int rmul_comm rmul-rdiv-cancel2 rless_functionality req_weakening req_wf rmul_wf rmul_preserves_rless nat_plus_wf less_than_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 rless_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation independent_pairFormation cut hypothesis lemma_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality setElimination rename independent_isectElimination sqequalRule inrFormation dependent_functionElimination because_Cache productElimination independent_functionElimination natural_numberEquality unionElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll multiplyEquality promote_hyp addLevel

Latex:
\mforall{}a,b:\mBbbZ{}.  \mforall{}c,d:\mBbbN{}\msupplus{}.    ((r(a)/r(c))  <  (r(b)/r(d))  \mLeftarrow{}{}\mRightarrow{}  a  *  d  <  b  *  c)

Date html generated: 2016_05_18-AM-07_27_34
Last ObjectModification: 2016_01_17-AM-02_00_26

Theory : reals

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