Nuprl Lemma : prime-mul
∀x,y:ℤ. (prime(x * y) ⇒ ((x ~ 1) ∨ (y ~ 1)))
Proof
Definitions occuring in Statement :
prime: prime(a),
assoced: a ~ b,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: 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,
and: P ∧ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
divides: b | a,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
or: P ∨ Q,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
not: ¬A,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
decidable: Dec(P)
Lemmas referenced :
prime_elim,
prime_wf,
equal-wf-base,
int_subtype_base,
assoced_wf,
assoced_elim,
mul_cancel_in_eq,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
itermMultiply_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
nequal_wf,
decidable__equal_int,
itermMinus_wf,
int_term_value_minus_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
multiplyEquality,
hypothesisEquality,
independent_functionElimination,
hypothesis,
productElimination,
isectElimination,
intEquality,
dependent_pairFormation,
sqequalRule,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
because_Cache,
unionElimination,
inlFormation,
natural_numberEquality,
inrFormation,
dependent_set_memberEquality,
independent_isectElimination,
approximateComputation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
minusEquality
Latex:
\mforall{}x,y:\mBbbZ{}. (prime(x * y) {}\mRightarrow{} ((x \msim{} 1) \mvee{} (y \msim{} 1)))
Date html generated:
2019_06_20-PM-02_23_08
Last ObjectModification:
2018_09_22-PM-05_25_11
Theory : num_thy_1
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