### Nuprl Lemma : p-inv_wf

`∀p:{p:{2...}| prime(p)} . ∀a:p-units(p).  (p-inv(p;a) ∈ {b:p-adics(p)| a * b = 1(p) ∈ p-adics(p)} )`

Proof

Definitions occuring in Statement :  p-inv: `p-inv(p;a)` p-units: `p-units(p)` p-int: `k(p)` p-mul: `x * y` p-adics: `p-adics(p)` prime: `prime(a)` int_upper: `{i...}` all: `∀x:A. B[x]` member: `t ∈ T` set: `{x:A| B[x]} ` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  p-units: `p-units(p)` all: `∀x:A. B[x]` member: `t ∈ T` uall: `∀[x:A]. B[x]` int_upper: `{i...}` not: `¬A` implies: `P `` Q` p-adics: `p-adics(p)` nat_plus: `ℕ+` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` true: `True` and: `P ∧ Q` subtype_rel: `A ⊆r B` int_seg: `{i..j-}` so_lambda: `λ2x.t[x]` nat: `ℕ` le: `A ≤ B` false: `False` so_apply: `x[s]` uimplies: `b supposing a` prop: `ℙ` p-inv: `p-inv(p;a)` has-value: `(a)↓` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` p-adic-inv-lemma sign: `sign(x)` lelt: `i ≤ j < k` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` sq_exists: `∃x:A [B[x]]` sq_stable: `SqStable(P)` eqmod: `a ≡ b mod m` divides: `b | a` ge: `i ≥ j ` int_nzero: `ℤ-o` nequal: `a ≠ b ∈ T ` cand: `A c∧ B` rev_uimplies: `rev_uimplies(P;Q)` p-int: `k(p)` p-mul: `x * y`
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep lambdaFormation_alt cut sqequalHypSubstitution hypothesis setIsType universeIsType introduction extract_by_obid isectElimination thin setElimination rename hypothesisEquality functionIsType equalityIstype applyEquality dependent_set_memberEquality_alt natural_numberEquality independent_pairFormation imageMemberEquality baseClosed intEquality lambdaEquality_alt independent_isectElimination sqequalBase equalitySymmetry functionExtensionality callbyvalueReduce dependent_functionElimination unionElimination approximateComputation independent_functionElimination dependent_pairFormation_alt int_eqEquality isect_memberEquality_alt voidElimination because_Cache instantiate cumulativity inhabitedIsType productElimination equalityElimination equalityTransitivity promote_hyp applyLambdaEquality imageElimination functionEquality setEquality multiplyEquality addEquality universeEquality baseApply closedConclusion inrFormation_alt inlFormation_alt minusEquality

Latex:
\mforall{}p:\{p:\{2...\}|  prime(p)\}  .  \mforall{}a:p-units(p).    (p-inv(p;a)  \mmember{}  \{b:p-adics(p)|  a  *  b  =  1(p)\}  )

Date html generated: 2019_10_15-AM-10_35_07
Last ObjectModification: 2019_06_24-PM-00_57_15

Theory : rings_1

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