Nuprl Lemma : coprime-equiv-unique

∀[p,q,a,b:ℤ].
  ({(p = a ∈ ℤ) ∧ (q = b ∈ ℤ)}) supposing 
     ((q < 0 ⇐⇒ b < 0) and 
     (p < 0 ⇐⇒ a < 0) and 
     ((p * b) = (a * q) ∈ ℤ) and 
     CoPrime(a,b) and 
     CoPrime(p,q))


Proof



Definitions occuring in Statement :  coprime: CoPrime(a,b),  less_than: a < b,  uimplies: b supposing a,  uall: ∀[x:A]. B[x],  guard: {T},  iff: P ⇐⇒ Q,  and: P ∧ Q,  multiply: n * m,  natural_number: $n,  int: ℤ,  equal: s = t ∈ T
Definitions unfolded in proof :  uall: ∀[x:A]. B[x],  member: t ∈ T,  uimplies: b supposing a,  guard: {T},  and: P ∧ Q,  prop: ℙ,  subtype_rel: A ⊆r B,  assoced: a ~ b,  cand: A c∧ B,  all: ∀x:A. B[x],  iff: P ⇐⇒ Q,  implies: P ⇒ Q,  exists: ∃x:A. B[x],  divides: b | a,  decidable: Dec(P),  or: P ∨ Q,  satisfiable_int_formula: satisfiable_int_formula(fmla),  false: False,  not: ¬A,  top: Top,  sq_type: SQType(T),  squash: ↓T,  true: True,  rev_implies: P ⇐ Q,  coprime: CoPrime(a,b),  gcd_p: GCD(a;b;y)
Lemmas referenced :  equal-wf-base,  coprime_wf,  iff_wf,  less_than_wf,  int_subtype_base,  coprime_bezout_id,  decidable__equal_int,  satisfiable-full-omega-tt,  intformnot_wf,  intformeq_wf,  itermVar_wf,  itermMultiply_wf,  itermConstant_wf,  int_formula_prop_not_lemma,  int_formula_prop_eq_lemma,  int_term_value_var_lemma,  int_term_value_mul_lemma,  int_term_value_constant_lemma,  int_formula_prop_wf,  subtype_base_sq,  equal_wf,  squash_wf,  true_wf,  mul_com,  iff_weakening_equal,  mul_add_distrib,  divides_wf,  intformand_wf,  int_formula_prop_and_lemma,  one_divs_any,  assoced_elim,  decidable__lt,  itermMinus_wf,  intformless_wf,  int_term_value_minus_lemma,  int_formula_prop_less_lemma,  intformimplies_wf,  int_formual_prop_imp_lemma
Rules used in proof :  sqequalSubstitution,  sqequalTransitivity,  computationStep,  sqequalReflexivity,  isect_memberFormation,  introduction,  cut,  sqequalRule,  sqequalHypSubstitution,  isect_memberEquality,  isectElimination,  thin,  hypothesisEquality,  productElimination,  independent_pairEquality,  axiomEquality,  hypothesis,  because_Cache,  equalityTransitivity,  equalitySymmetry,  extract_by_obid,  intEquality,  baseApply,  closedConclusion,  baseClosed,  applyEquality,  natural_numberEquality,  independent_pairFormation,  dependent_functionElimination,  independent_functionElimination,  dependent_pairFormation,  addEquality,  multiplyEquality,  unionElimination,  independent_isectElimination,  lambdaEquality,  int_eqEquality,  voidElimination,  voidEquality,  computeAll,  instantiate,  cumulativity,  hyp_replacement,  imageElimination,  universeEquality,  imageMemberEquality,  rename,  lambdaFormation,  productEquality,  promote_hyp
Latex:
\mforall{}[p,q,a,b:\mBbbZ{}].
    (\{(p  =  a)  \mwedge{}  (q  =  b)\})  supposing 
          ((q  <  0  \mLeftarrow{}{}\mRightarrow{}  b  <  0)  and 
          (p  <  0  \mLeftarrow{}{}\mRightarrow{}  a  <  0)  and 
          ((p  *  b)  =  (a  *  q))  and 
          CoPrime(a,b)  and 
          CoPrime(p,q))


Date html generated: 2018_05_21-PM-11_43_31
Last ObjectModification: 2017_07_26-PM-06_42_53

Theory : rationals


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