### Nuprl Lemma : p-unitize_wf

`∀p:ℕ+. ∀a:p-adics(p). ∀n:ℕ+.`
`  ((¬((a n) = 0 ∈ ℤ)) `` (p-unitize(p;a;n) ∈ k:ℕn + 1 × {b:p-units(p)| p^k(p) * b = a ∈ p-adics(p)} ))`

Proof

Definitions occuring in Statement :  p-unitize: `p-unitize(p;a;n)` p-units: `p-units(p)` p-int: `k(p)` p-mul: `x * y` p-adics: `p-adics(p)` exp: `i^n` int_seg: `{i..j-}` nat_plus: `ℕ+` all: `∀x:A. B[x]` not: `¬A` implies: `P `` Q` member: `t ∈ T` set: `{x:A| B[x]} ` apply: `f a` product: `x:A × B[x]` add: `n + m` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` p-unitize: `p-unitize(p;a;n)` uall: `∀[x:A]. B[x]` p-adics: `p-adics(p)` subtype_rel: `A ⊆r B` int_seg: `{i..j-}` nat_plus: `ℕ+` nat: `ℕ` decidable: `Dec(P)` or: `P ∨ Q` prop: `ℙ` uimplies: `b supposing a` sq_type: `SQType(T)` guard: `{T}` lelt: `i ≤ j < k` and: `P ∧ Q` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` p-units: `p-units(p)` less_than: `a < b` squash: `↓T` true: `True` uiff: `uiff(P;Q)` rev_uimplies: `rev_uimplies(P;Q)` p-int: `k(p)` p-mul: `x * y` p-reduce: `i mod(p^n)` int_upper: `{i...}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` has-value: `(a)↓` so_lambda: `λ2x.t[x]` so_apply: `x[s]` ge: `i ≥ j `
Lemmas referenced :  decidable__equal_int greatest-p-zero_wf int_seg_wf exp_wf2 nat_plus_wf nat_wf not_wf equal-wf-T-base nat_plus_subtype_nat p-adics_wf subtype_base_sq int_subtype_base false_wf nat_plus_properties decidable__lt full-omega-unsat intformand_wf intformnot_wf intformless_wf itermConstant_wf itermAdd_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_add_lemma int_term_value_var_lemma int_formula_prop_wf lelt_wf exp0_lemma equal_wf p-mul_wf p-int_wf p-units_wf int_seg_subtype_nat less_than_wf le_wf greatest-p-zero-property p-adics-equal modulus_wf_int_mod exp_wf_nat_plus int-subtype-int_mod one-mul p-adic-property decidable__le intformle_wf int_formula_prop_le_lemma eqmod_functionality_wrt_eqmod eqmod_transitivity mod-eqmod multiply_functionality_wrt_eqmod eqmod_weakening value-type-has-value set-value-type int-value-type le_weakening2 int_seg_properties nat_properties intformeq_wf int_formula_prop_eq_lemma shift-greatest-p-zero-unit p-shift-mul
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination thin isectElimination setElimination rename functionExtensionality applyEquality hypothesisEquality hypothesis lambdaEquality natural_numberEquality because_Cache sqequalRule unionElimination intEquality baseClosed instantiate cumulativity independent_isectElimination equalityTransitivity equalitySymmetry independent_functionElimination dependent_pairEquality dependent_set_memberEquality independent_pairFormation addEquality approximateComputation dependent_pairFormation int_eqEquality isect_memberEquality voidElimination voidEquality setEquality imageMemberEquality productElimination multiplyEquality callbyvalueReduce applyLambdaEquality

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