Nuprl Lemma : shift-greatest-p-zero-unit
∀p:ℕ+. ∀a:p-adics(p). ∀n:ℕ+.
((¬((a n) = 0 ∈ ℤ)) ⇒ 0 < greatest-p-zero(n;a) ⇒ (p-shift(p;a;greatest-p-zero(n;a)) ∈ p-units(p)))
Proof
Definitions occuring in Statement :
greatest-p-zero: greatest-p-zero(n;a),
p-shift: p-shift(p;a;k),
p-units: p-units(p),
p-adics: p-adics(p),
nat_plus: ℕ+,
less_than: a < b,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
member: t ∈ T,
apply: f a,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
p-units: p-units(p),
and: P ∧ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
int_seg: {i..j-},
p-adics: p-adics(p),
subtype_rel: A ⊆r B,
lelt: i ≤ j < k,
nat_plus: ℕ+,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
label: ...$L... t,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
p-shift: p-shift(p;a;k),
uiff: uiff(P;Q),
int_nzero: ℤ-o,
so_lambda: λ2x.t[x],
so_apply: x[s],
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
int_upper: {i...},
eqmod: a ≡ b mod m,
divides: b | a
Lemmas referenced :
greatest-p-zero-property,
p-shift_wf,
greatest-p-zero_wf,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
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,
le_wf,
decidable__lt,
itermAdd_wf,
int_term_value_add_lemma,
lelt_wf,
not_wf,
equal-wf-T-base,
less_than_wf,
int_seg_wf,
exp_wf2,
false_wf,
nat_plus_wf,
p-adics_wf,
nat_wf,
equal_wf,
squash_wf,
true_wf,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
subtype_rel_self,
iff_weakening_equal,
not-lt-2,
less-iff-le,
add_functionality_wrt_le,
add-associates,
add-zero,
add-commutes,
zero-add,
le-add-cancel,
exp-positive,
div_rem_sum,
exp_wf3,
subtype_rel_sets,
nequal_wf,
equal-wf-base,
int_subtype_base,
subtype_base_sq,
int_seg_properties,
add-is-int-iff,
itermMultiply_wf,
int_term_value_mul_lemma,
rem_bounds_1,
int_seg_subtype_nat,
nat_plus_subtype_nat,
p-adic-property,
set_subtype_base,
eqmod_wf,
mul_preserves_lt,
less_than_transitivity2,
itermSubtract_wf,
int_term_value_subtract_lemma,
mul_preserves_le,
exp_wf4
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
because_Cache,
dependent_set_memberEquality,
productElimination,
isectElimination,
independent_isectElimination,
hypothesis,
setElimination,
rename,
applyEquality,
sqequalRule,
independent_pairFormation,
natural_numberEquality,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
addEquality,
imageMemberEquality,
baseClosed,
functionExtensionality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
equalityUniverse,
levelHypothesis,
instantiate,
applyLambdaEquality,
setEquality,
cumulativity,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
divideEquality
Latex:
\mforall{}p:\mBbbN{}\msupplus{}. \mforall{}a:p-adics(p). \mforall{}n:\mBbbN{}\msupplus{}.
((\mneg{}((a n) = 0)) {}\mRightarrow{} 0 < greatest-p-zero(n;a) {}\mRightarrow{} (p-shift(p;a;greatest-p-zero(n;a)) \mmember{} p-units(p)))
Date html generated:
2018_05_21-PM-03_22_20
Last ObjectModification:
2018_05_19-AM-08_20_09
Theory : rings_1
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