Nuprl Lemma : divides-iff-gcd-assoced
∀x,y:ℤ. (x | y ⇐⇒ gcd(x;y) ~ x)
Proof
Definitions occuring in Statement :
assoced: a ~ b,
divides: b | a,
gcd: gcd(a;b),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
uimplies: b supposing a,
guard: {T},
assoced: a ~ b
Lemmas referenced :
divides_wf,
assoced_wf,
gcd_wf,
divides-iff-gcd,
assoced_functionality_wrt_assoced,
gcd_sym,
assoced_weakening,
gcd_is_divisor_2,
divides_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
intEquality,
productElimination,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
independent_isectElimination
Latex:
\mforall{}x,y:\mBbbZ{}. (x | y \mLeftarrow{}{}\mRightarrow{} gcd(x;y) \msim{} x)
Date html generated:
2018_05_21-PM-01_10_28
Last ObjectModification:
2018_01_28-PM-02_03_50
Theory : num_thy_1
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