Nuprl Lemma : p-unitize-unit
∀p:ℕ+. ∀a:p-units(p). ∀n:ℕ+. (p-unitize(p;a;n) = <0, a> ∈ (k:ℕ × {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,
nat_plus: ℕ+,
nat: ℕ,
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
pair: <a, b>,
product: x:A × B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
p-units: p-units(p),
p-unitize: p-unitize(p;a;n),
member: t ∈ T,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
top: Top,
p-adics: p-adics(p),
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
ge: i ≥ j ,
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
Lemmas referenced :
false_wf,
le_wf,
exp0_lemma,
not_wf,
equal-wf-T-base,
less_than_wf,
equal_wf,
p-adics_wf,
p-mul_wf,
p-int_wf,
p-units_wf,
exp_wf2,
nat_plus_wf,
subtype_base_sq,
nat_wf,
set_subtype_base,
int_subtype_base,
decidable__equal_nat,
greatest-p-zero_wf,
greatest-p-zero-property,
nat_plus_subtype_nat,
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,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
nat_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
decidable__equal_int,
p-adics-equal,
modulus_wf_int_mod,
exp_wf_nat_plus,
int-subtype-int_mod,
int_seg_wf,
one-mul,
p-adic-property,
eqmod_functionality_wrt_eqmod,
eqmod_transitivity,
mod-eqmod,
multiply_functionality_wrt_eqmod,
eqmod_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
sqequalRule,
dependent_pairEquality,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
intEquality,
applyEquality,
imageMemberEquality,
baseClosed,
because_Cache,
setEquality,
instantiate,
cumulativity,
independent_isectElimination,
lambdaEquality,
unionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
productElimination,
addEquality,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
applyLambdaEquality,
multiplyEquality
Latex:
\mforall{}p:\mBbbN{}\msupplus{}. \mforall{}a:p-units(p). \mforall{}n:\mBbbN{}\msupplus{}. (p-unitize(p;a;n) = ɘ, a>)
Date html generated:
2018_05_21-PM-03_22_36
Last ObjectModification:
2018_05_19-AM-08_22_27
Theory : rings_1
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