Nuprl Lemma : Euler-Fermat
Euler's Generalization Of Fermat's
(we also have a different, combinatorial, proof of
fermat-little, fermat-little2)⋅
∀n:{2...}. ∀a:ℕ+. (CoPrime(n,a) ⇒ (a^totient(n) ≡ 1 mod n))
This theorem is one of freek's list of 100 theorems
Proof
Definitions occuring in Statement :
totient: totient(n),
eqmod: a ≡ b mod m,
coprime: CoPrime(a,b),
exp: i^n,
int_upper: {i...},
nat_plus: ℕ+,
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
bag-product: Πx ∈ b. f[x],
bag-map: bag-map(f;bs),
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
nat: ℕ,
int_upper: {i...},
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
residue-mul: (ai mod n),
modulus: a mod n,
remainder: n rem m,
residue: residue(n),
int_seg: {i..j-},
cand: A c∧ B,
assoc: Assoc(T;op),
infix_ap: x f y,
comm: Comm(T;op),
squash: ↓T,
totient: totient(n),
bag-summation: Σ(x∈b). f[x],
bag-accum: bag-accum(v,x.f[v; x];init;bs),
lelt: i ≤ j < k,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
guard: {T},
sq_stable: SqStable(P),
ge: i ≥ j ,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cons: [a / b],
colength: colength(L),
nil: [],
it: ⋅,
sq_type: SQType(T),
less_than: a < b,
true: True,
coprime: CoPrime(a,b),
gcd_p: GCD(a;b;y)
Lemmas referenced :
bag-summation-map,
residues-mod_wf,
upper_subtype_nat,
istype-false,
subtype_rel_list,
residue_wf,
nat_plus_properties,
int_upper_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_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_wf,
istype-le,
top_wf,
coprime_wf,
nat_plus_wf,
istype-int_upper,
assoc_wf,
comm_wf,
bag_wf,
residues-equal-bags,
list-subtype-bag,
equal-wf-T-base,
subtype_rel_bag,
set_subtype_base,
le_wf,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
itermMultiply_wf,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
bag-product_wf,
squash_wf,
subtype_rel_sets,
lelt_wf,
istype-less_than,
list_induction,
all_wf,
eqmod_wf,
list_accum_wf,
exp_wf2,
length_wf_nat,
residue-mul_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
list_wf,
list_accum_nil_lemma,
length_of_nil_lemma,
exp0_lemma,
mul-commutes,
one-mul,
eqmod_refl,
list_accum_cons_lemma,
length_of_cons_lemma,
length_wf,
add_nat_wf,
istype-void,
nat_properties,
sq_stable__coprime,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
false_wf,
eqmod_functionality_wrt_eqmod,
eqmod_weakening,
eqmod_inversion,
ge_wf,
set_wf,
list-cases,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
subtract-1-ge-0,
subtype_base_sq,
spread_cons_lemma,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
true_wf,
istype-universe,
istype-nat,
multiply_functionality_wrt_eqmod,
mod-eqmod,
mul-associates,
mul-swap,
exp_add,
subtype_rel_self,
iff_weakening_equal,
exp1,
totient_wf,
eqmod_cancellation,
one_divs_any,
divides_wf,
coprime_symmetry,
coprime_prod
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalHypSubstitution,
sqequalRule,
introduction,
extract_by_obid,
isectElimination,
thin,
Error :memTop,
hypothesisEquality,
applyEquality,
natural_numberEquality,
independent_isectElimination,
independent_pairFormation,
hypothesis,
dependent_functionElimination,
dependent_set_memberEquality_alt,
setElimination,
rename,
because_Cache,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
universeIsType,
voidElimination,
intEquality,
multiplyEquality,
inhabitedIsType,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
baseApply,
closedConclusion,
baseClosed,
isect_memberFormation_alt,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
imageElimination,
productElimination,
equalityTransitivity,
productEquality,
functionIsType,
universeEquality,
imageMemberEquality,
setEquality,
setIsType,
productIsType,
equalityIstype,
addEquality,
pointwiseFunctionality,
promote_hyp,
intWeakElimination,
axiomSqEquality,
functionIsTypeImplies,
hypothesis_subsumption,
instantiate,
sqequalBase,
cumulativity,
functionEquality
Latex:
\mforall{}n:\{2...\}. \mforall{}a:\mBbbN{}\msupplus{}. (CoPrime(n,a) {}\mRightarrow{} (a\^{}totient(n) \mequiv{} 1 mod n))
Date html generated:
2020_05_20-AM-08_20_12
Last ObjectModification:
2020_01_01-PM-02_21_03
Theory : general
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