Nuprl Lemma : p-adic-inv-lemma1
∀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
Home
Index