`∀p:{p:{2...}| prime(p)} . ∀a:{a:p-adics(p)| ¬((a 1) = 0 ∈ ℤ)} . ∀n:ℕ+.  (∃c:ℕp^n [((c * (a n)) ≡ 1 mod p^n)])`

Proof

Definitions occuring in Statement :  p-adics: `p-adics(p)` eqmod: `a ≡ b mod m` prime: `prime(a)` exp: `i^n` int_upper: `{i...}` int_seg: `{i..j-}` nat_plus: `ℕ+` all: `∀x:A. B[x]` sq_exists: `∃x:A [B[x]]` not: `¬A` set: `{x:A| B[x]} ` apply: `f a` multiply: `n * m` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` member: `t ∈ T` nat_plus: `ℕ+` int_upper: `{i...}` le: `A ≤ B` and: `P ∧ Q` decidable: `Dec(P)` or: `P ∨ Q` iff: `P `⇐⇒` Q` not: `¬A` rev_implies: `P `` Q` implies: `P `` Q` false: `False` uiff: `uiff(P;Q)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` subtype_rel: `A ⊆r B` top: `Top` less_than': `less_than'(a;b)` true: `True` prop: `ℙ` less_than: `a < b` squash: `↓T` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_stable: `SqStable(P)` guard: `{T}` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` sq_type: `SQType(T)` p-adics: `p-adics(p)` int_seg: `{i..j-}` nat: `ℕ` assert: `↑b` bnot: `¬bb` bfalse: `ff` ifthenelse: `if b then t else f fi ` btrue: `tt` it: `⋅` unit: `Unit` bool: `𝔹` exp: `i^n` eqmod: `a ≡ b mod m` divides: `b | a` lelt: `i ≤ j < k` sq_exists: `∃x:A [B[x]]`
Lemmas referenced :  p-adic-property decidable__lt istype-false not-lt-2 add_functionality_wrt_le add-commutes istype-void zero-add le-add-cancel less_than_wf subtype_rel_sets le_wf sq_stable_from_decidable prime_wf decidable__prime upper_subtype_nat int_upper_properties decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf istype-int int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf subtype_base_sq nat_plus_wf set_subtype_base int_subtype_base exp-positive exp1 p-adics_wf lelt_wf exp_wf2 false_wf int_upper_wf nat_wf primrec-wf2 set_wf int_term_value_subtract_lemma itermSubtract_wf subtract_wf coprime_wf exp0_lemma coprime_bezout_id exists_wf equal-wf-base int_term_value_mul_lemma int_term_value_add_lemma int_formula_prop_eq_lemma itermMultiply_wf itermAdd_wf intformeq_wf decidable__equal_int assert-bnot bool_subtype_base bool_cases_sqequal equal_wf eqff_to_assert assert_of_lt_int eqtt_to_assert bool_wf lt_int_wf primrec-unroll coprime_prod nat_plus_subtype_nat coprime_iff_ndivides divides_wf nat_plus_properties divisor_bound subtype_rel_set int_seg_properties coprime_inversion gcd-reduce-coprime p-reduce_wf eqmod_wf eqmod_functionality_wrt_eqmod multiply_functionality_wrt_eqmod p-reduce-eqmod eqmod_weakening exp_wf_nat_plus itermMinus_wf int_term_value_minus_lemma
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation_alt cut introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination thin dependent_set_memberEquality_alt setElimination rename because_Cache hypothesis productElimination natural_numberEquality hypothesisEquality unionElimination independent_pairFormation voidElimination independent_functionElimination independent_isectElimination isectElimination sqequalRule applyEquality lambdaEquality_alt isect_memberEquality_alt universeIsType intEquality closedConclusion imageMemberEquality baseClosed setIsType imageElimination approximateComputation dependent_pairFormation_alt int_eqEquality instantiate cumulativity equalityTransitivity equalitySymmetry functionIsType equalityIsType4 lambdaEquality dependent_pairFormation dependent_set_memberEquality voidEquality isect_memberEquality lambdaFormation baseApply promote_hyp equalityElimination inhabitedIsType pointwiseFunctionality applyLambdaEquality equalityIsType1 dependent_set_memberFormation_alt multiplyEquality minusEquality

Latex:
\mforall{}p:\{p:\{2...\}|  prime(p)\}  .  \mforall{}a:\{a:p-adics(p)|  \mneg{}((a  1)  =  0)\}  .  \mforall{}n:\mBbbN{}\msupplus{}.
(\mexists{}c:\mBbbN{}p\^{}n  [((c  *  (a  n))  \mequiv{}  1  mod  p\^{}n)])

Date html generated: 2019_10_15-AM-10_34_43
Last ObjectModification: 2018_10_16-AM-00_03_13

Theory : rings_1

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