Nuprl Lemma : coprime_elim
∀a,b:ℤ. (CoPrime(a,b) ⇐⇒ ∀c:ℤ. ((c | a) ⇒ (c | b) ⇒ (c ~ 1)))
Proof
Definitions occuring in Statement :
coprime: CoPrime(a,b),
assoced: a ~ b,
divides: b | a,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
assoced: a ~ b,
coprime: CoPrime(a,b),
gcd_p: GCD(a;b;y),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
cand: A c∧ B,
uall: ∀[x:A]. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
guard: {T}
Lemmas referenced :
one_divs_any,
divides_wf,
istype-int
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :lambdaFormation_alt,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
introduction,
extract_by_obid,
Error :universeIsType,
isectElimination,
Error :inhabitedIsType,
Error :productIsType,
natural_numberEquality,
Error :functionIsType
Latex:
\mforall{}a,b:\mBbbZ{}. (CoPrime(a,b) \mLeftarrow{}{}\mRightarrow{} \mforall{}c:\mBbbZ{}. ((c | a) {}\mRightarrow{} (c | b) {}\mRightarrow{} (c \msim{} 1)))
Date html generated:
2019_06_20-PM-02_23_24
Last ObjectModification:
2018_10_03-AM-00_12_41
Theory : num_thy_1
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