Nuprl Lemma : int-ring-hom-p-adic-ring
∀[p:{2...}]. (λk.k(p) ∈ RingHom(ℤ-rng;ℤ(p)))
Proof
Definitions occuring in Statement :
p-adic-ring: ℤ(p),
p-int: k(p),
int_ring: ℤ-rng,
ring_hom: RingHom(R;S),
int_upper: {i...},
uall: ∀[x:A]. B[x],
member: t ∈ T,
lambda: λx.A[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
ring_hom: RingHom(R;S),
nat_plus: ℕ+,
int_upper: {i...},
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,
top: Top,
less_than': less_than'(a;b),
true: True,
int_ring: ℤ-rng,
rng_car: |r|,
pi1: fst(t),
p-adics: p-adics(p),
p-adic-ring: ℤ(p),
guard: {T},
integ_dom: IntegDom{i},
crng: CRng,
rng: Rng,
cand: A c∧ B,
rng_plus: +r,
pi2: snd(t),
fun_thru_2op: FunThru2op(A;B;opa;opb;f),
infix_ap: x f y,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
subtract: n - m,
int_seg: {i..j-},
nat: ℕ,
lelt: i ≤ j < k,
so_apply: x[s],
rng_times: *,
rng_one: 1
Lemmas referenced :
p-int_wf,
decidable__lt,
false_wf,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
less_than_wf,
subtype_rel_self,
rng_car_wf,
p-adic-ring_wf,
rng_subtype_rng_sig,
crng_subtype_rng,
subtype_rel_transitivity,
crng_wf,
rng_wf,
rng_sig_wf,
int_ring_wf,
integ_dom_wf,
p-adic-property,
nat_plus_properties,
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,
le_wf,
nat_plus_wf,
p-add-int,
all_wf,
eqmod_wf,
exp_wf2,
nat_plus_subtype_nat,
less-iff-le,
condition-implies-le,
minus-add,
minus-one-mul,
minus-one-mul-top,
add-associates,
add-zero,
int_seg_wf,
int_seg_properties,
intformand_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
p-mul-int,
fun_thru_2op_wf,
rng_plus_wf,
rng_times_wf,
equal_wf,
rng_one_wf,
int_upper_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
dependent_set_memberEquality,
lambdaEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
productElimination,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_pairFormation,
lambdaFormation,
voidElimination,
independent_functionElimination,
independent_isectElimination,
sqequalRule,
applyEquality,
isect_memberEquality,
voidEquality,
intEquality,
instantiate,
because_Cache,
addEquality,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
equalitySymmetry,
minusEquality,
equalityTransitivity,
applyLambdaEquality,
axiomEquality,
multiplyEquality,
productEquality
Latex:
\mforall{}[p:\{2...\}]. (\mlambda{}k.k(p) \mmember{} RingHom(\mBbbZ{}-rng;\mBbbZ{}(p)))
Date html generated:
2019_10_15-AM-10_35_12
Last ObjectModification:
2018_08_22-AM-09_39_22
Theory : rings_1
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