### 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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