### Nuprl Lemma : rng_nexp-int

`∀[n:ℕ]. ∀[a:ℤ].  ((a ↑ℤ-rng n) = a^n ∈ ℤ)`

Proof

Definitions occuring in Statement :  rng_nexp: `e ↑r n` int_ring: `ℤ-rng` exp: `i^n` nat: `ℕ` uall: `∀[x:A]. B[x]` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  exp: `i^n` rng_nexp: `e ↑r n` mon_nat_op: `n ⋅ e` mul_mon_of_rng: `r↓xmn` grp_op: `*` pi2: `snd(t)` pi1: `fst(t)` grp_id: `e` int_ring: `ℤ-rng` rng_times: `*` rng_one: `1` nat_op: `n x(op;id) e` uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` all: `∀x:A. B[x]` top: `Top` and: `P ∧ Q` prop: `ℙ` itop: `Π(op,id) lb ≤ i < ub. E[i]` ycomb: `Y` lt_int: `i <z j` infix_ap: `x f y` subtract: `n - m` ifthenelse: `if b then t else f fi ` bfalse: `ff` decidable: `Dec(P)` or: `P ∨ Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` subtype_rel: `A ⊆r B` uiff: `uiff(P;Q)` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` so_lambda: `λ2x.t[x]` so_apply: `x[s]` squash: `↓T` nequal: `a ≠ b ∈ T ` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_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 ge_wf less_than_wf primrec0_lemma decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma primrec-unroll lt_int_wf bool_wf uiff_transitivity equal-wf-base int_subtype_base assert_wf eqtt_to_assert assert_of_lt_int eq_int_wf assert_of_eq_int intformeq_wf int_formula_prop_eq_lemma eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int mul-commutes itop_wf int_seg_wf squash_wf true_wf primrec_wf le_wf iff_weakening_equal le_int_wf bnot_wf assert_functionality_wrt_uiff bnot_of_lt_int assert_of_le_int nat_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename intWeakElimination lambdaFormation natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll independent_functionElimination axiomEquality unionElimination equalityElimination baseApply closedConclusion baseClosed applyEquality because_Cache equalityTransitivity equalitySymmetry productElimination promote_hyp instantiate cumulativity multiplyEquality imageElimination universeEquality dependent_set_memberEquality imageMemberEquality

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[a:\mBbbZ{}].    ((a  \muparrow{}\mBbbZ{}-rng  n)  =  a\^{}n)

Date html generated: 2017_10_01-AM-08_18_52
Last ObjectModification: 2017_02_28-PM-02_03_52

Theory : rings_1

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