### Nuprl Lemma : exp-of-mul

`∀[x,y:ℤ]. ∀[n:ℕ].  ((x * y)^n = (x^n * y^n) ∈ ℤ)`

Proof

Definitions occuring in Statement :  exp: `i^n` nat: `ℕ` uall: `∀[x:A]. B[x]` multiply: `n * m` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uiff: `uiff(P;Q)` guard: `{T}` sq_type: `SQType(T)` or: `P ∨ Q` decidable: `Dec(P)` nat_plus: `ℕ+` less_than': `less_than'(a;b)` le: `A ≤ B` primtailrec: `primtailrec(n;i;b;f)` primrec: `primrec(n;b;c)` exp: `i^n` prop: `ℙ` and: `P ∧ Q` top: `Top` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` satisfiable_int_formula: `satisfiable_int_formula(fmla)` not: `¬A` uimplies: `b supposing a` ge: `i ≥ j ` false: `False` implies: `P `` Q` nat: `ℕ` member: `t ∈ T` uall: `∀[x:A]. B[x]`
Lemmas referenced :  exp_step false_wf int_term_value_mul_lemma int_formula_prop_eq_lemma itermMultiply_wf intformeq_wf multiply-is-int-iff decidable__equal_int int_subtype_base subtype_base_sq int_formula_prop_not_lemma intformnot_wf decidable__lt istype-nat subtract-1-ge-0 istype-le exp_wf2 istype-less_than ge_wf int_formula_prop_wf int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma istype-void int_formula_prop_and_lemma istype-int intformless_wf itermVar_wf itermConstant_wf intformle_wf intformand_wf full-omega-unsat nat_properties
Rules used in proof :  productElimination baseClosed closedConclusion baseApply promote_hyp pointwiseFunctionality equalitySymmetry equalityTransitivity intEquality cumulativity instantiate unionElimination because_Cache Error :isectIsTypeImplies,  Error :dependent_set_memberEquality_alt,  multiplyEquality Error :inhabitedIsType,  Error :functionIsTypeImplies,  axiomEquality Error :universeIsType,  independent_pairFormation sqequalRule voidElimination Error :isect_memberEquality_alt,  dependent_functionElimination int_eqEquality Error :lambdaEquality_alt,  Error :dependent_pairFormation_alt,  independent_functionElimination approximateComputation independent_isectElimination natural_numberEquality Error :lambdaFormation_alt,  intWeakElimination rename setElimination hypothesis hypothesisEquality thin isectElimination sqequalHypSubstitution extract_by_obid cut introduction Error :isect_memberFormation_alt,  sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution

Latex:
\mforall{}[x,y:\mBbbZ{}].  \mforall{}[n:\mBbbN{}].    ((x  *  y)\^{}n  =  (x\^{}n  *  y\^{}n))

Date html generated: 2019_06_20-PM-02_26_26
Last ObjectModification: 2019_06_19-PM-00_11_31

Theory : num_thy_1

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