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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