Nuprl Lemma : prime-mult
∀n:{2...}. ∀x:ℤ. (prime(n * x) ⇒ (prime(n) ∧ (x ~ 1)))
Proof
Definitions occuring in Statement :
prime: prime(a),
assoced: a ~ b,
int_upper: {i...},
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
multiply: n * m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
int_upper: {i...},
or: P ∨ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
guard: {T},
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
rev_implies: P ⇐ Q
Lemmas referenced :
prime-mul,
prime_wf,
int_upper_wf,
assoced_elim,
int_upper_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
assoced-prime,
mul-one,
assoced_functionality_wrt_assoced,
multiply_functionality_wrt_assoced,
assoced_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
hypothesisEquality,
independent_functionElimination,
unionElimination,
isectElimination,
multiplyEquality,
intEquality,
natural_numberEquality,
productElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
minusEquality
Latex:
\mforall{}n:\{2...\}. \mforall{}x:\mBbbZ{}. (prime(n * x) {}\mRightarrow{} (prime(n) \mwedge{} (x \msim{} 1)))
Date html generated:
2019_06_20-PM-02_23_11
Last ObjectModification:
2018_09_22-PM-05_54_49
Theory : num_thy_1
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