Nuprl Lemma : gcd_properties
∀a,b:ℤ. (((gcd(a;b) | a) ∧ (gcd(a;b) | b)) ∧ (∀c:ℤ. ((c | a) ⇒ (c | b) ⇒ (c | gcd(a;b)))))
This theorem is one of freek's list of 100 theorems
Proof
Definitions occuring in Statement :
divides: b | a,
gcd: gcd(a;b),
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
cand: A c∧ B
Lemmas referenced :
gcd_is_divisor_1,
gcd_is_divisor_2,
gcd_is_gcd
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
intEquality,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis
Latex:
\mforall{}a,b:\mBbbZ{}. (((gcd(a;b) | a) \mwedge{} (gcd(a;b) | b)) \mwedge{} (\mforall{}c:\mBbbZ{}. ((c | a) {}\mRightarrow{} (c | b) {}\mRightarrow{} (c | gcd(a;b)))))
Date html generated:
2016_05_14-PM-04_18_48
Last ObjectModification:
2015_12_26-PM-08_15_58
Theory : num_thy_1
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