Nuprl Lemma : p-unit-iff
∀p:{p:{2...}| prime(p)} . ∀a:p-adics(p). (¬((a 1) = 0 ∈ ℤ) ⇐⇒ ∃b:p-adics(p). (a * b = 1(p) ∈ p-adics(p)))
Proof
Definitions occuring in Statement :
p-int: k(p),
p-mul: x * y,
p-adics: p-adics(p),
prime: prime(a),
int_upper: {i...},
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
set: {x:A| B[x]} ,
apply: f a,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
p-adics: p-adics(p),
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
nat: ℕ,
le: A ≤ B,
false: False,
not: ¬A,
int_upper: {i...},
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
so_apply: x[s],
exists: ∃x:A. B[x],
p-units: p-units(p),
guard: {T},
sq_stable: SqStable(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
subtract: n - m,
p-int: k(p),
p-mul: x * y,
p-reduce: i mod(p^n),
sq_type: SQType(T),
lelt: i ≤ j < k
Lemmas referenced :
not_wf,
equal-wf-T-base,
less_than_wf,
int_seg_wf,
exp_wf2,
false_wf,
le_wf,
exists_wf,
p-adics_wf,
equal_wf,
p-mul_wf,
decidable__lt,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
p-int_wf,
set_wf,
int_upper_wf,
prime_wf,
p-inv_wf,
p-adic-property,
nat_plus_properties,
sq_stable_from_decidable,
decidable__prime,
int_upper_properties,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
and_wf,
nat_plus_wf,
all_wf,
eqmod_wf,
less-iff-le,
condition-implies-le,
minus-add,
minus-one-mul,
minus-one-mul-top,
add-associates,
add-zero,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
exp-positive,
exp1,
zero-mul,
intformand_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
lelt_wf,
modulus_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
applyEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
lambdaEquality,
independent_functionElimination,
voidElimination,
because_Cache,
productElimination,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
addEquality,
imageElimination,
approximateComputation,
int_eqEquality,
isect_memberEquality,
voidEquality,
functionExtensionality,
addLevel,
levelHypothesis,
equalitySymmetry,
equalityTransitivity,
applyLambdaEquality,
minusEquality,
universeEquality,
instantiate,
cumulativity,
promote_hyp
Latex:
\mforall{}p:\{p:\{2...\}| prime(p)\} . \mforall{}a:p-adics(p). (\mneg{}((a 1) = 0) \mLeftarrow{}{}\mRightarrow{} \mexists{}b:p-adics(p). (a * b = 1(p)))
Date html generated:
2018_05_21-PM-03_23_24
Last ObjectModification:
2018_05_19-AM-08_21_50
Theory : rings_1
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