Nuprl Lemma : residue-mul-inverse
∀n:{2...}. ∀a:ℕ.
(CoPrime(n,a)
⇒ (∃b:ℤ
(CoPrime(n,b)
∧ (∀i:residue(n)
((((ba mod n) = 1 ∈ residue(n)) ∧ ((ab mod n) = 1 ∈ residue(n)))
∧ ((b(ai mod n) mod n) = i ∈ residue(n))
∧ ((a(bi mod n) mod n) = i ∈ residue(n)))))))
Proof
Definitions occuring in Statement :
residue-mul: (ai mod n),
residue: residue(n),
coprime: CoPrime(a,b),
int_upper: {i...},
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
residue: residue(n),
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
uall: ∀[x:A]. B[x],
nat: ℕ,
guard: {T},
int_upper: {i...},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
coprime: CoPrime(a,b),
gcd_p: GCD(a;b;y),
cand: A c∧ B,
residue-mul: (ai mod n),
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
modulus: a mod n,
sq_type: SQType(T),
nat_plus: ℕ+,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
true: True,
eqmod: a ≡ b mod m,
divides: b | a,
squash: ↓T,
sq_stable: SqStable(P)
Lemmas referenced :
istype-false,
nat_properties,
int_upper_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
le_wf,
less_than_wf,
one_divs_any,
divides_wf,
coprime_wf,
nat_wf,
int_upper_wf,
coprime_bezout_id,
mul-commutes,
set_subtype_base,
int_subtype_base,
modulus_wf,
subtype_rel_sets,
nequal_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
one-rem,
subtype_base_sq,
modulus-equal-iff-eqmod,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
decidable__equal_int,
itermSubtract_wf,
itermMultiply_wf,
itermMinus_wf,
itermAdd_wf,
int_term_value_subtract_lemma,
int_term_value_mul_lemma,
int_term_value_minus_lemma,
int_term_value_add_lemma,
modulus_base,
subtract_wf,
residue-mul-assoc,
less_than_transitivity2,
one-mul,
mod_bounds_1,
mod_bounds,
equal_wf,
squash_wf,
true_wf,
istype-universe,
residue_wf,
upper_subtype_nat,
subtype_rel_self,
iff_weakening_equal,
residue-mul_wf,
sq_stable__coprime,
int_seg_wf,
lelt_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
sqequalRule,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
universeIsType,
productIsType,
because_Cache,
productElimination,
equalityIsType4,
inhabitedIsType,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
equalityTransitivity,
equalitySymmetry,
multiplyEquality,
setIsType,
instantiate,
cumulativity,
minusEquality,
promote_hyp,
imageElimination,
universeEquality,
imageMemberEquality,
functionIsType,
equalityIsType1
Latex:
\mforall{}n:\{2...\}. \mforall{}a:\mBbbN{}.
(CoPrime(n,a)
{}\mRightarrow{} (\mexists{}b:\mBbbZ{}
(CoPrime(n,b)
\mwedge{} (\mforall{}i:residue(n)
((((ba mod n) = 1) \mwedge{} ((ab mod n) = 1))
\mwedge{} ((b(ai mod n) mod n) = i)
\mwedge{} ((a(bi mod n) mod n) = i))))))
Date html generated:
2019_10_15-AM-11_34_26
Last ObjectModification:
2018_10_11-PM-11_14_04
Theory : general
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