Nuprl Lemma : p-unitize_wf
∀p:ℕ+. ∀a:p-adics(p). ∀n:ℕ+.
((¬((a n) = 0 ∈ ℤ)) ⇒ (p-unitize(p;a;n) ∈ k:ℕn + 1 × {b:p-units(p)| p^k(p) * b = a ∈ p-adics(p)} ))
Proof
Definitions occuring in Statement :
p-unitize: p-unitize(p;a;n),
p-units: p-units(p),
p-int: k(p),
p-mul: x * y,
p-adics: p-adics(p),
exp: i^n,
int_seg: {i..j-},
nat_plus: ℕ+,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
product: x:A × B[x],
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
p-unitize: p-unitize(p;a;n),
uall: ∀[x:A]. B[x],
p-adics: p-adics(p),
subtype_rel: A ⊆r B,
int_seg: {i..j-},
nat_plus: ℕ+,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
prop: ℙ,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
p-units: p-units(p),
less_than: a < b,
squash: ↓T,
true: True,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
p-int: k(p),
p-mul: x * y,
p-reduce: i mod(p^n),
int_upper: {i...},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
has-value: (a)↓,
so_lambda: λ2x.t[x],
so_apply: x[s],
ge: i ≥ j
Lemmas referenced :
decidable__equal_int,
greatest-p-zero_wf,
int_seg_wf,
exp_wf2,
nat_plus_wf,
nat_wf,
not_wf,
equal-wf-T-base,
nat_plus_subtype_nat,
p-adics_wf,
subtype_base_sq,
int_subtype_base,
false_wf,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
lelt_wf,
exp0_lemma,
equal_wf,
p-mul_wf,
p-int_wf,
p-units_wf,
int_seg_subtype_nat,
less_than_wf,
le_wf,
greatest-p-zero-property,
p-adics-equal,
modulus_wf_int_mod,
exp_wf_nat_plus,
int-subtype-int_mod,
one-mul,
p-adic-property,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
eqmod_functionality_wrt_eqmod,
eqmod_transitivity,
mod-eqmod,
multiply_functionality_wrt_eqmod,
eqmod_weakening,
value-type-has-value,
set-value-type,
int-value-type,
le_weakening2,
int_seg_properties,
nat_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
shift-greatest-p-zero-unit,
p-shift-mul
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
setElimination,
rename,
functionExtensionality,
applyEquality,
hypothesisEquality,
hypothesis,
lambdaEquality,
natural_numberEquality,
because_Cache,
sqequalRule,
unionElimination,
intEquality,
baseClosed,
instantiate,
cumulativity,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
dependent_pairEquality,
dependent_set_memberEquality,
independent_pairFormation,
addEquality,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
setEquality,
imageMemberEquality,
productElimination,
multiplyEquality,
callbyvalueReduce,
applyLambdaEquality
Latex:
\mforall{}p:\mBbbN{}\msupplus{}. \mforall{}a:p-adics(p). \mforall{}n:\mBbbN{}\msupplus{}.
((\mneg{}((a n) = 0)) {}\mRightarrow{} (p-unitize(p;a;n) \mmember{} k:\mBbbN{}n + 1 \mtimes{} \{b:p-units(p)| p\^{}k(p) * b = a\} ))
Date html generated:
2018_05_21-PM-03_22_30
Last ObjectModification:
2018_05_19-AM-08_20_53
Theory : rings_1
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