### Nuprl Lemma : prime-factors

`∀n:{2...}. (∃factors:{m:{2...}| prime(m)}  List [(n = Π(factors)  ∈ ℤ)])`

Proof

Definitions occuring in Statement :  mul-list: `Π(ns) ` prime: `prime(a)` list: `T List` int_upper: `{i...}` all: `∀x:A. B[x]` sq_exists: `∃x:A [B[x]]` set: `{x:A| B[x]} ` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  sq_exists: `∃x:A [B[x]]` all: `∀x:A. B[x]` member: `t ∈ T` uall: `∀[x:A]. B[x]` nat: `ℕ` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` implies: `P `` Q` prop: `ℙ` uimplies: `b supposing a` subtype_rel: `A ⊆r B` cand: `A c∧ B` prime: `prime(a)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` guard: `{T}` int_upper: `{i...}` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` sq_type: `SQType(T)` mul-list: `Π(ns) `
Lemmas referenced :  factorit_wf le_wf false_wf int_upper_wf nil_wf nat_wf prime_wf less_than_wf subtype_base_sq set_subtype_base int_subtype_base nat_properties int_upper_properties decidable__equal_int satisfiable-full-omega-tt intformand_wf intformnot_wf intformeq_wf itermVar_wf itermConstant_wf intformless_wf intformle_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_less_lemma int_formula_prop_le_lemma int_formula_prop_wf decidable__le null_nil_lemma btrue_wf member-implies-null-eq-bfalse btrue_neq_bfalse set_wf all_wf l_member_wf subtype_rel_list not_wf divides_wf assoced_weakening reduce_nil_lemma list_wf equal_wf reduce_wf mul-list_wf subtype_rel_list_set itermMultiply_wf int_term_value_mul_lemma
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep lambdaFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin because_Cache dependent_set_memberEquality natural_numberEquality independent_pairFormation hypothesis hypothesisEquality independent_isectElimination applyEquality setEquality productEquality setElimination rename productElimination independent_functionElimination instantiate cumulativity intEquality lambdaEquality dependent_functionElimination unionElimination dependent_pairFormation int_eqEquality isect_memberEquality voidElimination voidEquality computeAll equalityTransitivity equalitySymmetry functionEquality multiplyEquality

Latex:
\mforall{}n:\{2...\}.  (\mexists{}factors:\{m:\{2...\}|  prime(m)\}    List  [(n  =  \mPi{}(factors)  )])

Date html generated: 2018_05_21-PM-06_57_49
Last ObjectModification: 2017_07_26-PM-05_00_01

Theory : general

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