### Nuprl Lemma : p-unitize-unit

`∀p:ℕ+. ∀a:p-units(p). ∀n:ℕ+.  (p-unitize(p;a;n) = <0, a> ∈ (k:ℕ × {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` nat_plus: `ℕ+` nat: `ℕ` all: `∀x:A. B[x]` set: `{x:A| B[x]} ` pair: `<a, b>` product: `x:A × B[x]` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` p-units: `p-units(p)` p-unitize: `p-unitize(p;a;n)` member: `t ∈ T` nat: `ℕ` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` implies: `P `` Q` prop: `ℙ` uall: `∀[x:A]. B[x]` top: `Top` p-adics: `p-adics(p)` nat_plus: `ℕ+` less_than: `a < b` squash: `↓T` true: `True` subtype_rel: `A ⊆r B` uimplies: `b supposing a` so_lambda: `λ2x.t[x]` so_apply: `x[s]` decidable: `Dec(P)` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` ge: `i ≥ j ` 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`
Lemmas referenced :  false_wf le_wf exp0_lemma not_wf equal-wf-T-base less_than_wf equal_wf p-adics_wf p-mul_wf p-int_wf p-units_wf exp_wf2 nat_plus_wf subtype_base_sq nat_wf set_subtype_base int_subtype_base decidable__equal_nat greatest-p-zero_wf greatest-p-zero-property nat_plus_subtype_nat 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 decidable__le intformle_wf int_formula_prop_le_lemma nat_properties intformeq_wf int_formula_prop_eq_lemma decidable__equal_int p-adics-equal modulus_wf_int_mod exp_wf_nat_plus int-subtype-int_mod int_seg_wf one-mul p-adic-property eqmod_functionality_wrt_eqmod eqmod_transitivity mod-eqmod multiply_functionality_wrt_eqmod eqmod_weakening
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut sqequalHypSubstitution setElimination thin rename sqequalRule dependent_pairEquality dependent_set_memberEquality natural_numberEquality independent_pairFormation hypothesis introduction extract_by_obid isectElimination hypothesisEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality intEquality applyEquality imageMemberEquality baseClosed because_Cache setEquality instantiate cumulativity independent_isectElimination lambdaEquality unionElimination equalityTransitivity equalitySymmetry independent_functionElimination productElimination addEquality approximateComputation dependent_pairFormation int_eqEquality applyLambdaEquality multiplyEquality

Latex:
\mforall{}p:\mBbbN{}\msupplus{}.  \mforall{}a:p-units(p).  \mforall{}n:\mBbbN{}\msupplus{}.    (p-unitize(p;a;n)  =  ɘ,  a>)

Date html generated: 2018_05_21-PM-03_22_36
Last ObjectModification: 2018_05_19-AM-08_22_27

Theory : rings_1

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