Nuprl Lemma : coprime_bezout_id1
∀a,b:ℤ. (CoPrime(a,b) ⇒ (∃x,y:ℤ. (((a * x) + (b * y)) = 1 ∈ ℤ)))
Proof
Definitions occuring in Statement :
coprime: CoPrime(a,b),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
exists: ∃x:A. B[x],
prop: ℙ,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
decidable: Dec(P),
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A
Lemmas referenced :
mul-swap,
minus-one-mul,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
itermMultiply_wf,
itermAdd_wf,
intformeq_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__equal_int,
exists_wf,
equal_wf,
assoced_elim,
coprime_wf,
coprime_bezout_id0
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
productElimination,
isectElimination,
intEquality,
addEquality,
multiplyEquality,
natural_numberEquality,
unionElimination,
dependent_pairFormation,
sqequalRule,
lambdaEquality,
minusEquality,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
independent_isectElimination,
int_eqEquality,
independent_pairFormation,
computeAll
Latex:
\mforall{}a,b:\mBbbZ{}. (CoPrime(a,b) {}\mRightarrow{} (\mexists{}x,y:\mBbbZ{}. (((a * x) + (b * y)) = 1)))
Date html generated:
2016_05_14-PM-04_20_40
Last ObjectModification:
2016_01_14-PM-11_40_12
Theory : num_thy_1
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