### Nuprl Lemma : p-digit_wf

`∀[p:ℕ+]. ∀[a:p-adics(p)]. ∀[n:ℕ+].  (p-digit(p;a;n) ∈ ℕp)`

Proof

Definitions occuring in Statement :  p-digit: `p-digit(p;a;n)` p-adics: `p-adics(p)` int_seg: `{i..j-}` nat_plus: `ℕ+` uall: `∀[x:A]. B[x]` member: `t ∈ T` natural_number: `\$n`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` p-adics: `p-adics(p)` all: `∀x:A. B[x]` nat_plus: `ℕ+` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` sq_type: `SQType(T)` implies: `P `` Q` guard: `{T}` so_apply: `x[s]` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` prop: `ℙ` and: `P ∧ Q` true: `True` less_than': `less_than'(a;b)` squash: `↓T` less_than: `a < b` btrue: `tt` ifthenelse: `if b then t else f fi ` subtract: `n - m` eq_int: `(i =z j)` p-digit: `p-digit(p;a;n)` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` eqmod: `a ≡ b mod m` divides: `b | a` assert: `↑b` bnot: `¬bb` bfalse: `ff` top: `Top` uiff: `uiff(P;Q)` it: `⋅` unit: `Unit` bool: `𝔹` nat: `ℕ` nequal: `a ≠ b ∈ T ` int_nzero: `ℤ-o` int_seg: `{i..j-}` lelt: `i ≤ j < k`
Lemmas referenced :  decidable__equal_int subtype_base_sq int_subtype_base p-adics_wf nat_plus_wf int_seg_wf subtype_rel_self exp1 exp-positive set_subtype_base less_than_wf subtract_wf nat_plus_properties decidable__lt full-omega-unsat intformand_wf intformnot_wf intformless_wf itermConstant_wf itermSubtract_wf itermVar_wf intformeq_wf istype-int int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_subtract_lemma int_term_value_var_lemma int_formula_prop_eq_lemma int_formula_prop_wf istype-less_than subtract-add-cancel neg_assert_of_eq_int assert-bnot bool_subtype_base bool_cases_sqequal equal_wf eqff_to_assert assert_of_eq_int eqtt_to_assert bool_wf eq_int_wf le_wf int_formula_prop_le_lemma intformle_wf decidable__le exp-fastexp equal-wf-base nequal_wf subtype_rel_sets exp_wf3 div-cancel2 p-adic-bounds eqmod_wf exp_wf2 istype-le itermAdd_wf int_term_value_add_lemma false_wf int_term_value_mul_lemma itermMultiply_wf multiply-is-int-iff exp_wf_nat_plus mul_cancel_in_le mul_cancel_in_lt mul-commutes exp_step int_seg_properties
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut sqequalHypSubstitution setElimination thin rename extract_by_obid dependent_functionElimination hypothesisEquality hypothesis natural_numberEquality unionElimination instantiate isectElimination cumulativity intEquality independent_isectElimination because_Cache independent_functionElimination sqequalRule axiomEquality equalityTransitivity equalitySymmetry inhabitedIsType isect_memberEquality_alt isectIsTypeImplies universeIsType lambdaEquality baseClosed imageMemberEquality independent_pairFormation dependent_set_memberEquality applyEquality dependent_set_memberEquality_alt approximateComputation dependent_pairFormation_alt lambdaEquality_alt int_eqEquality Error :memTop,  voidElimination productElimination promote_hyp voidEquality isect_memberEquality dependent_pairFormation equalityElimination lambdaFormation setEquality functionIsType addEquality productIsType closedConclusion baseApply pointwiseFunctionality multiplyEquality applyLambdaEquality

Latex:
\mforall{}[p:\mBbbN{}\msupplus{}].  \mforall{}[a:p-adics(p)].  \mforall{}[n:\mBbbN{}\msupplus{}].    (p-digit(p;a;n)  \mmember{}  \mBbbN{}p)

Date html generated: 2020_05_19-PM-10_08_18
Last ObjectModification: 2020_01_08-PM-06_00_08

Theory : rings_1

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