Nuprl Lemma : pa-inv_wf
∀p:{p:{2...}| prime(p)} . ∀x:{x:padic(p)| pa-sep(p;x;0(p))} .
(pa-inv(p;x) ∈ {y:padic(p)| pa-mul(p;x;y) = 1(p) ∈ padic(p)} )
Proof
Definitions occuring in Statement :
pa-inv: pa-inv(p;x),
pa-sep: pa-sep(p;x;y),
pa-mul: pa-mul(p;x;y),
pa-int: k(p),
padic: padic(p),
prime: prime(a),
int_upper: {i...},
all: ∀x:A. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
padic: padic(p),
pa-int: k(p),
pa-sep: pa-sep(p;x;y),
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
int_upper: {i...},
subtype_rel: A ⊆r B,
prop: ℙ,
not: ¬A,
false: False,
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
btrue: tt,
p-sep: p-sep(x;y),
exists: ∃x:A. B[x],
p-int: k(p),
p-reduce: i mod(p^n),
nat_plus: ℕ+,
le: A ≤ B,
and: P ∧ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
less_than': less_than'(a;b),
true: True,
int_seg: {i..j-},
lelt: i ≤ j < k,
pa-inv: pa-inv(p;x),
p-adics: p-adics(p),
subtract: n - m,
sq_stable: SqStable(P),
squash: ↓T,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
has-value: (a)↓,
bnot: ¬bb,
ge: i ≥ j ,
less_than: a < b,
p-units: p-units(p),
basic-padic: basic-padic(p),
bool: 𝔹,
unit: Unit,
it: ⋅,
bfalse: ff,
assert: ↑b,
pa-mul: pa-mul(p;x;y),
bpa-mul: bpa-mul(p;x;y),
bpa-equiv: bpa-equiv(p;x;y)
Lemmas referenced :
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
padic_wf,
pa-sep_wf,
padic_subtype_basic-padic,
pa-int_wf,
int_upper_wf,
prime_wf,
modulus_base,
exp_wf_nat_plus,
nat_plus_subtype_nat,
decidable__lt,
istype-false,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
less_than_wf,
exp-positive,
le_wf,
exp_wf2,
mu_wf,
bnot_wf,
eq_int_wf,
condition-implies-le,
minus-add,
minus-one-mul,
minus-one-mul-top,
add-associates,
add-zero,
nat_wf,
subtract_wf,
nat_plus_properties,
sq_stable_from_decidable,
decidable__prime,
upper_subtype_nat,
int_upper_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
iff_transitivity,
assert_wf,
add-swap,
not_wf,
equal-wf-T-base,
iff_weakening_uiff,
assert_of_bnot,
assert_of_eq_int,
set_subtype_base,
lelt_wf,
subtract-add-cancel,
int_seg_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
mu-property,
value-type-has-value,
set-value-type,
int-value-type,
p-unitize_wf,
assert_elim,
bfalse_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
bool_wf,
eq_int_eq_true,
subtype_rel_self,
iff_weakening_equal,
btrue_neq_bfalse,
nat_properties,
itermAdd_wf,
int_term_value_add_lemma,
p-inv_wf,
false_wf,
int_seg_wf,
p-unit-iff,
p-adics_wf,
p-mul-comm,
p-mul_wf,
pa-mul_wf,
int_seg_subtype_nat,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
ifthenelse_wf,
p-units_wf,
bpa-norm-padic,
bpa-equiv-iff-norm2,
bpa-mul_wf,
subtype_rel_product,
exp0_lemma,
p-int_wf,
p-mul-1,
nat_plus_wf,
p-1-mul,
p-mul-assoc,
fastexp_wf,
p-adic-property,
exp-fastexp,
eqmod_wf,
less-iff-le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
setElimination,
thin,
rename,
cut,
sqequalHypSubstitution,
productElimination,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
natural_numberEquality,
unionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
because_Cache,
independent_functionElimination,
setIsType,
universeIsType,
applyEquality,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
voidElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
lambdaEquality_alt,
addEquality,
minusEquality,
dependent_pairFormation_alt,
imageMemberEquality,
baseClosed,
imageElimination,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
equalityIsType4,
applyLambdaEquality,
inhabitedIsType,
equalityIsType1,
isectIsType,
callbyvalueReduce,
universeEquality,
equalityIsType3,
lambdaFormation,
lambdaEquality,
dependent_set_memberEquality,
baseApply,
closedConclusion,
independent_pairEquality,
dependent_pairEquality_alt,
equalityElimination,
equalityIsType2,
promote_hyp,
functionIsType
Latex:
\mforall{}p:\{p:\{2...\}| prime(p)\} . \mforall{}x:\{x:padic(p)| pa-sep(p;x;0(p))\} .
(pa-inv(p;x) \mmember{} \{y:padic(p)| pa-mul(p;x;y) = 1(p)\} )
Date html generated:
2019_10_15-AM-10_36_44
Last ObjectModification:
2018_10_08-PM-05_26_06
Theory : rings_1
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