Nuprl Lemma : prime_divs_prod
∀p:ℤ. (prime(p) ⇒ (∀a1,a2:ℤ. ((p | (a1 * a2)) ⇒ ((p | a1) ∨ (p | a2)))))
Proof
Definitions occuring in Statement :
prime: prime(a),
divides: b | a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
multiply: n * m,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
prime: prime(a),
and: P ∧ Q
Lemmas referenced :
divides_wf,
prime_wf,
istype-int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
Error :universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
multiplyEquality,
hypothesis,
Error :inhabitedIsType,
productElimination,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}p:\mBbbZ{}. (prime(p) {}\mRightarrow{} (\mforall{}a1,a2:\mBbbZ{}. ((p | (a1 * a2)) {}\mRightarrow{} ((p | a1) \mvee{} (p | a2)))))
Date html generated:
2019_06_20-PM-02_23_58
Last ObjectModification:
2018_10_03-AM-00_12_59
Theory : num_thy_1
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