Nuprl Lemma : gcd_of_triple
∀a,b,c,x,y:ℤ.
(GCD(a;b;x) ⇒ GCD(x;c;y) ⇒ (((y | a) ∧ (y | b) ∧ (y | c)) ∧ (∀z:ℤ. ((z | a) ⇒ (z | b) ⇒ (z | c) ⇒ (z | y)))))
Proof
Definitions occuring in Statement :
gcd_p: GCD(a;b;y),
divides: b | a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
int: ℤ
Definitions unfolded in proof :
gcd_p: GCD(a;b;y),
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
guard: {T},
uall: ∀[x:A]. B[x],
prop: ℙ
Lemmas referenced :
divides_transitivity,
divides_wf,
istype-int
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
cut,
hypothesis,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
independent_pairFormation,
Error :universeIsType,
isectElimination,
Error :inhabitedIsType,
Error :productIsType,
Error :functionIsType
Latex:
\mforall{}a,b,c,x,y:\mBbbZ{}.
(GCD(a;b;x)
{}\mRightarrow{} GCD(x;c;y)
{}\mRightarrow{} (((y | a) \mwedge{} (y | b) \mwedge{} (y | c)) \mwedge{} (\mforall{}z:\mBbbZ{}. ((z | a) {}\mRightarrow{} (z | b) {}\mRightarrow{} (z | c) {}\mRightarrow{} (z | y)))))
Date html generated:
2019_06_20-PM-02_21_50
Last ObjectModification:
2018_10_03-AM-00_12_12
Theory : num_thy_1
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