Nuprl Lemma : coprime_bezout_id2
∀a,b:ℤ. ((∃x,y:ℤ. (((a * x) + (b * y)) = 1 ∈ ℤ)) ⇒ CoPrime(a,b))
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,
exists: ∃x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
coprime: CoPrime(a,b),
gcd_p: GCD(a;b;y),
and: P ∧ Q,
cand: A c∧ B,
uall: ∀[x:A]. B[x],
prop: ℙ,
squash: ↓T,
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
int_subtype_base,
istype-int,
one_divs_any,
divides_wf,
divisor_of_mul,
divisor_of_sum,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
sqequalRule,
Error :productIsType,
Error :inhabitedIsType,
hypothesisEquality,
Error :equalityIsType4,
cut,
addEquality,
multiplyEquality,
applyEquality,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
natural_numberEquality,
productElimination,
thin,
dependent_functionElimination,
independent_pairFormation,
Error :universeIsType,
isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
because_Cache,
Error :lambdaEquality_alt,
imageElimination,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
independent_isectElimination
Latex:
\mforall{}a,b:\mBbbZ{}. ((\mexists{}x,y:\mBbbZ{}. (((a * x) + (b * y)) = 1)) {}\mRightarrow{} CoPrime(a,b))
Date html generated:
2019_06_20-PM-02_23_37
Last ObjectModification:
2018_10_03-AM-00_12_53
Theory : num_thy_1
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