Nuprl Lemma : p-adic-ring_wf
∀[p:{2...}]. (ℤ(p) ∈ CRng)
Proof
Definitions occuring in Statement :
p-adic-ring: ℤ(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),
rng_sig: RngSig,
int_upper: {i...},
nat_plus: ℕ+,
le: A ≤ B,
and: P ∧ Q,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
true: True,
rng_car: |r|,
pi1: fst(t),
rng_plus: +r,
pi2: snd(t),
rng_zero: 0,
rng_minus: -r,
rng_times: *,
rng_one: 1,
ring_p: IsRing(T;plus;zero;neg;times;one),
bilinear: BiLinear(T;pl;tm),
monoid_p: IsMonoid(T;op;id),
group_p: IsGroup(T;op;id;inv),
infix_ap: x f y,
ident: Ident(T;op;id),
assoc: Assoc(T;op),
inverse: Inverse(T;op;id;inv),
cand: A c∧ B,
rev_uimplies: rev_uimplies(P;Q),
p-add: x + y,
p-reduce: i mod(p^n),
p-int: k(p),
p-minus: -(x),
p-mul: x * y,
comm: Comm(T;op),
p-adics: p-adics(p),
int_seg: {i..j-},
guard: {T},
nat: ℕ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
lelt: i ≤ j < k
Lemmas referenced :
p-adics_wf,
bfalse_wf,
p-add_wf,
decidable__lt,
false_wf,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
less_than_wf,
p-int_wf,
p-minus_wf,
p-mul_wf,
it_wf,
unit_wf2,
bool_wf,
p-adics-equal,
nat_plus_wf,
ring_p_wf,
rng_car_wf,
rng_plus_wf,
rng_zero_wf,
rng_minus_wf,
rng_times_wf,
rng_one_wf,
comm_wf,
int_upper_wf,
exp_wf2,
nat_plus_subtype_nat,
modulus_wf_int_mod,
exp_wf_nat_plus,
int_seg_wf,
int-subtype-int_mod,
eqmod_weakening,
nat_plus_properties,
int_upper_properties,
decidable__equal_int,
int_seg_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,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
add-zero,
p-adic-property,
itermMinus_wf,
int_term_value_minus_lemma,
itermMultiply_wf,
int_term_value_mul_lemma,
mul-one,
eqmod_functionality_wrt_eqmod,
eqmod_transitivity,
mod-eqmod,
add_functionality_wrt_eqmod,
multiply_functionality_wrt_eqmod
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
dependent_set_memberEquality,
sqequalRule,
dependent_pairEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
lambdaEquality,
productElimination,
dependent_functionElimination,
natural_numberEquality,
hypothesisEquality,
unionElimination,
independent_pairFormation,
lambdaFormation,
voidElimination,
independent_functionElimination,
independent_isectElimination,
applyEquality,
inrEquality,
functionEquality,
unionEquality,
productEquality,
isect_memberEquality,
axiomEquality,
independent_pairEquality,
equalityTransitivity,
equalitySymmetry,
addEquality,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidEquality,
applyLambdaEquality,
minusEquality,
multiplyEquality
Latex:
\mforall{}[p:\{2...\}]. (\mBbbZ{}(p) \mmember{} CRng)
Date html generated:
2018_05_21-PM-03_20_48
Last ObjectModification:
2018_05_19-AM-08_14_17
Theory : rings_1
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