### Nuprl Lemma : prime-power-divides-product

`∀p:ℕ. (prime(p) `` (∀n:ℕ+. ∀x,y:ℤ.  ((¬(p | x)) `` (p^n | (x * y)) `` (p^n | y))))`

Proof

Definitions occuring in Statement :  prime: `prime(a)` divides: `b | a` exp: `i^n` nat_plus: `ℕ+` nat: `ℕ` all: `∀x:A. B[x]` not: `¬A` implies: `P `` Q` multiply: `n * m` int: `ℤ`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` uall: `∀[x:A]. B[x]` member: `t ∈ T` nat_plus: `ℕ+` not: `¬A` nat: `ℕ` prop: `ℙ` false: `False` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` and: `P ∧ Q` so_lambda: `λ2x.t[x]` so_apply: `x[s]` exp: `i^n` prime: `prime(a)` divides: `b | a` subtract: `n - m` subtype_rel: `A ⊆r B` sq_type: `SQType(T)` guard: `{T}` int_nzero: `ℤ-o` nequal: `a ≠ b ∈ T ` squash: `↓T` true: `True` iff: `P `⇐⇒` Q` uiff: `uiff(P;Q)`
Lemmas referenced :  nat_plus_properties divides_wf istype-void exp_wf2 nat_properties decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf istype-int int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf istype-le primrec-wf-nat-plus not_wf nat_plus_wf prime_wf istype-nat primrec1_lemma mul-one itermAdd_wf int_term_value_add_lemma mul-swap mul-commutes add-commutes exp_step decidable__lt istype-less_than add-associates add-swap zero-add int_subtype_base set_subtype_base le_wf subtype_base_sq mul_cancel_in_eq exp_wf3 nequal_wf equal_wf squash_wf true_wf istype-universe subtype_rel_self iff_weakening_equal decidable__equal_int multiply-is-int-iff intformeq_wf itermMultiply_wf int_formula_prop_eq_lemma int_term_value_mul_lemma false_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation_alt cut thin rename introduction extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination sqequalRule functionIsType inhabitedIsType because_Cache universeIsType dependent_set_memberEquality_alt dependent_functionElimination natural_numberEquality unionElimination independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation_alt lambdaEquality_alt int_eqEquality Error :memTop,  independent_pairFormation voidElimination multiplyEquality functionEquality intEquality productElimination addEquality equalityIstype baseApply closedConclusion baseClosed applyEquality sqequalBase equalitySymmetry instantiate cumulativity equalityTransitivity hyp_replacement imageElimination universeEquality imageMemberEquality pointwiseFunctionality promote_hyp

Latex:
\mforall{}p:\mBbbN{}.  (prime(p)  {}\mRightarrow{}  (\mforall{}n:\mBbbN{}\msupplus{}.  \mforall{}x,y:\mBbbZ{}.    ((\mneg{}(p  |  x))  {}\mRightarrow{}  (p\^{}n  |  (x  *  y))  {}\mRightarrow{}  (p\^{}n  |  y))))

Date html generated: 2020_05_19-PM-10_02_05
Last ObjectModification: 2020_01_04-PM-08_10_12

Theory : num_thy_1

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