Nuprl Lemma : divides-prime
∀p,q:ℤ. (prime(q) ⇒ (p | q) ⇒ ((p ~ q) ∨ (p ~ 1) ∨ (p = 0 ∈ ℤ)))
Proof
Definitions occuring in Statement :
prime: prime(a),
assoced: a ~ b,
divides: b | a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
atomic: atomic(a),
and: P ∧ Q,
divides: b | a,
exists: ∃x:A. B[x],
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
top: Top,
guard: {T},
sq_type: SQType(T),
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
reducible: reducible(a),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
int_nzero_wf,
exists_wf,
not_wf,
and_wf,
nequal_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
itermVar_wf,
itermConstant_wf,
itermMultiply_wf,
intformeq_wf,
intformnot_wf,
satisfiable-full-omega-tt,
int_subtype_base,
subtype_base_sq,
decidable__equal_int,
assoced_transitivity,
assoced_inversion,
equal_wf,
assoced_wf,
or_wf,
one-mul,
mul-commutes,
assoced_weakening,
multiply_functionality_wrt_assoced,
assoced_functionality_wrt_assoced,
decidable__assoced,
prime_wf,
divides_wf,
prime_imp_atomic
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
productElimination,
intEquality,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
equalityTransitivity,
equalitySymmetry,
multiplyEquality,
because_Cache,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
inlFormation,
instantiate,
cumulativity,
promote_hyp,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
computeAll,
inrFormation,
dependent_set_memberEquality,
independent_pairFormation,
setElimination,
rename
Latex:
\mforall{}p,q:\mBbbZ{}. (prime(q) {}\mRightarrow{} (p | q) {}\mRightarrow{} ((p \msim{} q) \mvee{} (p \msim{} 1) \mvee{} (p = 0)))
Date html generated:
2016_05_14-PM-04_27_05
Last ObjectModification:
2016_01_14-PM-11_36_30
Theory : num_thy_1
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