### Nuprl Lemma : gcd-reduce-coprime

`∀p,q:ℤ.  ∃x,y:ℤ. (((x * p) + (y * q)) = 1 ∈ ℤ) supposing CoPrime(p,q)`

Proof

Definitions occuring in Statement :  coprime: `CoPrime(a,b)` uimplies: `b supposing a` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` multiply: `n * m` add: `n + m` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` member: `t ∈ T` uimplies: `b supposing a` exists: `∃x:A. B[x]` and: `P ∧ Q` prop: `ℙ` uall: `∀[x:A]. B[x]` sq_type: `SQType(T)` implies: `P `` Q` guard: `{T}` nat: `ℕ` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` top: `Top` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` coprime: `CoPrime(a,b)` gcd_p: `GCD(a;b;y)` cand: `A c∧ B` divides: `b | a` nat_plus: `ℕ+` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` true: `True`
Lemmas referenced :  gcd-reduce-ext coprime_wf subtype_base_sq int_subtype_base nat_properties decidable__equal_int full-omega-unsat intformand_wf intformnot_wf intformeq_wf itermAdd_wf itermMultiply_wf itermVar_wf itermConstant_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_add_lemma int_term_value_mul_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_wf equal-wf-base exists_wf equal-wf-base-T divisor_bound less_than_wf intformle_wf int_formula_prop_le_lemma
Rules used in proof :  cut introduction extract_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation hypothesis sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality isect_memberFormation productElimination isectElimination intEquality promote_hyp instantiate cumulativity independent_isectElimination equalityTransitivity equalitySymmetry independent_functionElimination because_Cache dependent_pairFormation setElimination rename unionElimination natural_numberEquality approximateComputation lambdaEquality int_eqEquality isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation baseApply closedConclusion baseClosed applyEquality multiplyEquality dependent_set_memberEquality imageMemberEquality

Latex:
\mforall{}p,q:\mBbbZ{}.    \mexists{}x,y:\mBbbZ{}.  (((x  *  p)  +  (y  *  q))  =  1)  supposing  CoPrime(p,q)

Date html generated: 2018_05_21-PM-00_59_22
Last ObjectModification: 2018_05_19-AM-06_35_23

Theory : num_thy_1

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