`∀p:ℕ+. ∀a:p-adics(p). ∀n:ℕ+. ∀m:{n...}.  ((a m) ≡ (a n) mod p^n)`

Proof

Definitions occuring in Statement :  p-adics: `p-adics(p)` eqmod: `a ≡ b mod m` exp: `i^n` int_upper: `{i...}` nat_plus: `ℕ+` all: `∀x:A. B[x]` apply: `f a`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` uall: `∀[x:A]. B[x]` subtype_rel: `A ⊆r B` nat_plus: `ℕ+` p-adics: `p-adics(p)` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` top: `Top` and: `P ∧ Q` int_seg: `{i..j-}` nat: `ℕ` guard: `{T}` lelt: `i ≤ j < k` so_lambda: `λ2x.t[x]` so_apply: `x[s]` ge: `i ≥ j ` int_upper: `{i...}` subtract: `n - m` sq_type: `SQType(T)` sq_stable: `SqStable(P)` squash: `↓T` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` p-reduce: `i mod(p^n)`
Lemmas referenced :  eqmod_wf exp_wf2 nat_plus_subtype_nat subtract_wf nat_plus_properties decidable__lt full-omega-unsat intformand_wf intformnot_wf intformless_wf itermConstant_wf itermAdd_wf itermVar_wf itermSubtract_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_term_value_subtract_lemma int_formula_prop_wf less_than_wf int_seg_wf int_seg_properties decidable__le intformle_wf int_formula_prop_le_lemma le_wf set_wf primrec-wf2 nat_properties nat_wf int_upper_properties minus-one-mul add-swap add-mul-special zero-mul add-zero subtype_base_sq int_subtype_base int_upper_wf nat_plus_wf p-adics_wf eqmod_weakening sq_stable__eqmod decidable__equal_int intformeq_wf int_formula_prop_eq_lemma equal_wf modulus-equal-iff-eqmod exp_wf_nat_plus p-reduce-eqmod-exp eqmod_inversion eqmod_transitivity
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut thin rename setElimination introduction extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality applyEquality hypothesis sqequalRule because_Cache dependent_set_memberEquality addEquality natural_numberEquality dependent_functionElimination unionElimination independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation equalityTransitivity equalitySymmetry applyLambdaEquality productElimination hyp_replacement instantiate functionExtensionality cumulativity imageMemberEquality baseClosed imageElimination

Latex:
\mforall{}p:\mBbbN{}\msupplus{}.  \mforall{}a:p-adics(p).  \mforall{}n:\mBbbN{}\msupplus{}.  \mforall{}m:\{n...\}.    ((a  m)  \mequiv{}  (a  n)  mod  p\^{}n)

Date html generated: 2018_05_21-PM-03_19_37
Last ObjectModification: 2018_05_19-AM-08_12_17

Theory : rings_1

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