Nuprl Lemma : padic-ring_wf
∀[p:{2...}]. (padic-ring(p) ∈ CRng)
Proof
Definitions occuring in Statement :
padic-ring: padic-ring(p),
crng: CRng,
int_upper: {i...},
uall: ∀[x:A]. B[x],
member: t ∈ T,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
crng: CRng,
rng: Rng,
p-adic-ring: ℤ(p),
ring_p: IsRing(T;plus;zero;neg;times;one),
rng_car: |r|,
pi1: fst(t),
rng_plus: +r,
pi2: snd(t),
rng_zero: 0,
rng_minus: -r,
rng_times: *,
rng_one: 1,
monoid_p: IsMonoid(T;op;id),
group_p: IsGroup(T;op;id;inv),
bilinear: BiLinear(T;pl;tm),
ident: Ident(T;op;id),
assoc: Assoc(T;op),
inverse: Inverse(T;op;id;inv),
infix_ap: x f y,
comm: Comm(T;op),
and: P ∧ Q,
padic-ring: padic-ring(p),
prop: ℙ,
rng_sig: RngSig,
int_upper: {i...},
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
pa-mul: pa-mul(p;x;y),
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
pa-add: pa-add(p;x;y),
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
squash: ↓T,
true: True,
guard: {T},
basic-padic: basic-padic(p),
bpa-add: bpa-add(p;x;y),
bpa-equiv: bpa-equiv(p;x;y),
nat: ℕ,
ge: i ≥ j ,
has-value: (a)↓,
top: Top,
int_seg: {i..j-},
p-adics: p-adics(p),
less_than': less_than'(a;b),
le: A ≤ B,
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
sq_type: SQType(T),
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
so_apply: x[s],
so_lambda: λ2x.t[x],
pa-int: k(p),
subtract: n - m,
pa-minus: pa-minus(p;x),
bpa-minus: bpa-minus(p;x),
bpa-mul: bpa-mul(p;x;y),
lelt: i ≤ j < k,
padic: padic(p),
istype: istype(T),
p-units: p-units(p)
Lemmas referenced :
p-adic-ring_wf,
crng_properties,
rng_properties,
ring_p_wf,
rng_car_wf,
rng_plus_wf,
rng_zero_wf,
rng_minus_wf,
rng_times_wf,
rng_one_wf,
comm_wf,
istype-int_upper,
padic_wf,
bfalse_wf,
pa-add_wf,
padic_subtype_basic-padic,
pa-int_wf,
pa-minus_wf,
pa-mul_wf,
it_wf,
unit_wf2,
bool_wf,
bpa-norm-equiv,
bpa-mul_wf,
int_upper_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
istype-less_than,
basic-padic_wf,
bpa-add_wf,
bpa-equiv-iff-norm2,
equal_wf,
squash_wf,
true_wf,
istype-universe,
bpa-equiv_inversion,
pa-add_functionality,
bpa-equiv_weakening,
imax_ub,
nat_properties,
decidable__le,
istype-le,
imax_wf,
value-type-has-value,
int-value-type,
fastexp_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
le_wf,
subtype_rel_self,
iff_weakening_equal,
p-adics_wf,
p-int_wf,
exp_wf2,
p-mul_wf,
p-add_wf,
itermAdd_wf,
int_term_value_add_lemma,
p-distrib,
nat_plus_wf,
p-mul-assoc,
exp-fastexp,
p-mul-int,
exp_add,
p-add-assoc,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
istype-nat,
bpa-norm-padic,
istype-void,
eqmod_wf,
nat_plus_properties,
p-adic-property,
int_term_value_mul_lemma,
itermMultiply_wf,
mul-one,
exp0_lemma,
set-value-type,
nat_wf,
assert_wf,
iff_weakening_uiff,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
eqff_to_assert,
subtype_base_sq,
assert_of_le_int,
eqtt_to_assert,
le_int_wf,
set_subtype_base,
int_subtype_base,
imax_unfold,
minus-one-mul,
add-mul-special,
zero-mul,
add-zero,
minus-zero,
bpa-minus_wf,
p-minus_wf,
false_wf,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
less_than_wf,
pa-mul_functionality,
all_wf,
nat_plus_subtype_nat,
less-iff-le,
condition-implies-le,
minus-add,
minus-one-mul-top,
add-associates,
int_seg_wf,
int_seg_properties,
p-1-mul,
bpa-norm_wf_padic,
p-mul-1,
ifthenelse_wf,
add_functionality_wrt_eq,
eq_int_wf,
assert_of_eq_int,
neg_assert_of_eq_int
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
setElimination,
rename,
sqequalRule,
dependent_set_memberEquality_alt,
productElimination,
universeIsType,
because_Cache,
natural_numberEquality,
dependent_pairEquality,
lambdaEquality,
applyEquality,
inrEquality,
functionEquality,
unionEquality,
productEquality,
lambdaFormation_alt,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
inhabitedIsType,
independent_pairEquality,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
hyp_replacement,
imageElimination,
instantiate,
universeEquality,
imageMemberEquality,
baseClosed,
inlFormation_alt,
inrFormation_alt,
callbyvalueReduce,
intEquality,
productIsType,
equalityIstype,
addEquality,
multiplyEquality,
functionIsType,
promote_hyp,
cumulativity,
equalityElimination,
sqequalIntensionalEquality,
isect_memberFormation,
dependent_set_memberEquality,
lambdaFormation,
isect_memberEquality,
voidEquality,
dependent_pairFormation,
minusEquality
Latex:
\mforall{}[p:\{2...\}]. (padic-ring(p) \mmember{} CRng)
Date html generated:
2020_05_19-PM-10_09_14
Last ObjectModification:
2020_01_08-PM-06_53_09
Theory : rings_1
Home
Index