Nuprl Lemma : coprime_bezout_id
∀a,b:ℤ. (CoPrime(a,b) ⇐⇒ ∃x,y:ℤ. (((a * x) + (b * y)) = 1 ∈ ℤ))
This theorem is one of freek's list of 100 theorems
Proof
Definitions occuring in Statement :
coprime: CoPrime(a,b),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: 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],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B
Lemmas referenced :
coprime_bezout_id1,
coprime_wf,
coprime_bezout_id2,
int_subtype_base,
istype-int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
Error :universeIsType,
isectElimination,
sqequalRule,
Error :productIsType,
Error :inhabitedIsType,
Error :equalityIsType4,
addEquality,
multiplyEquality,
applyEquality,
natural_numberEquality
Latex:
\mforall{}a,b:\mBbbZ{}. (CoPrime(a,b) \mLeftarrow{}{}\mRightarrow{} \mexists{}x,y:\mBbbZ{}. (((a * x) + (b * y)) = 1))
Date html generated:
2019_06_20-PM-02_23_40
Last ObjectModification:
2018_10_03-AM-00_12_35
Theory : num_thy_1
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