### Nuprl Lemma : fermat-little2

`∀p:ℕ. (prime(p) `` (∀x:ℕ. x^p - 1 ≡ 1 mod p supposing ¬(p | x)))`

Proof

Definitions occuring in Statement :  eqmod: `a ≡ b mod m` prime: `prime(a)` divides: `b | a` exp: `i^n` nat: `ℕ` uimplies: `b supposing a` all: `∀x:A. B[x]` not: `¬A` implies: `P `` Q` subtract: `n - m` natural_number: `\$n`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` uimplies: `b supposing a` member: `t ∈ T` not: `¬A` false: `False` uall: `∀[x:A]. B[x]` nat: `ℕ` prop: `ℙ` prime: `prime(a)` and: `P ∧ Q` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` coprime: `CoPrime(a,b)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` sq_type: `SQType(T)` guard: `{T}` le: `A ≤ B` less_than': `less_than'(a;b)` exp: `i^n` squash: `↓T` true: `True`
Lemmas referenced :  int_term_value_add_lemma itermAdd_wf decidable__equal_int true_wf squash_wf primrec1_lemma false_wf exp_add mul-one int_subtype_base subtype_base_sq coprime_iff_ndivides gcd_p_sym le_wf int_formula_prop_wf int_formula_prop_eq_lemma int_term_value_var_lemma int_term_value_subtract_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma intformeq_wf itermVar_wf itermSubtract_wf itermConstant_wf intformle_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__le nat_properties subtract_wf prime_wf nat_wf not_wf exp_wf2 eqmod_cancellation fermat-little divides_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation isect_memberFormation cut introduction sqequalRule sqequalHypSubstitution lambdaEquality dependent_functionElimination thin hypothesisEquality voidElimination lemma_by_obid isectElimination setElimination rename hypothesis independent_functionElimination natural_numberEquality productElimination dependent_set_memberEquality unionElimination independent_isectElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidEquality independent_pairFormation computeAll instantiate because_Cache equalityTransitivity equalitySymmetry cumulativity applyEquality imageElimination addEquality imageMemberEquality baseClosed

Latex:
\mforall{}p:\mBbbN{}.  (prime(p)  {}\mRightarrow{}  (\mforall{}x:\mBbbN{}.  x\^{}p  -  1  \mequiv{}  1  mod  p  supposing  \mneg{}(p  |  x)))

Date html generated: 2016_05_14-PM-09_29_43
Last ObjectModification: 2016_01_14-PM-11_31_35

Theory : num_thy_1

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