Nuprl Lemma : mkpadic_functionality
∀[p:{2...}]. ∀[n,m:ℕ]. ∀[a,b:p-adics(p)]. (a/p^n) = (b/p^m) ∈ padic(p) supposing bpa-equiv(p;<n, a>;<m, b>)
Proof
Definitions occuring in Statement :
mkpadic: (a/p^n),
padic: padic(p),
bpa-equiv: bpa-equiv(p;x;y),
p-adics: p-adics(p),
int_upper: {i...},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
pair: <a, b>,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
basic-padic: basic-padic(p),
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
mkpadic: (a/p^n),
prop: ℙ,
nat_plus: ℕ+,
int_upper: {i...},
nat: ℕ,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
top: Top,
less_than': less_than'(a;b),
true: True,
padic: padic(p),
so_lambda: λ2x.t[x],
so_apply: x[s],
pi2: snd(t),
pi1: fst(t),
sq_type: SQType(T),
guard: {T},
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
btrue: tt,
bool: 𝔹,
unit: Unit,
it: ⋅,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
p-units: p-units(p),
p-adics: p-adics(p),
less_than: a < b,
squash: ↓T,
int_seg: {i..j-}
Lemmas referenced :
bpa-equiv-iff-norm,
bpa-equiv_wf,
decidable__lt,
false_wf,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
less_than_wf,
p-adics_wf,
nat_wf,
int_upper_wf,
mkpadic_wf,
padic_wf,
equal_wf,
basic-padic_wf,
padic_subtype_basic-padic,
pi2_wf,
pi1_wf,
subtype_base_sq,
set_subtype_base,
le_wf,
int_subtype_base,
ifthenelse_wf,
eq_int_wf,
p-units_wf,
decidable__equal_int,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
nat_properties,
int_upper_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
not_wf,
equal-wf-T-base,
int_seg_wf,
exp_wf2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
sqequalRule,
independent_pairEquality,
hypothesis,
productElimination,
independent_functionElimination,
isectElimination,
dependent_set_memberEquality,
setElimination,
rename,
natural_numberEquality,
unionElimination,
independent_pairFormation,
lambdaFormation,
voidElimination,
independent_isectElimination,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidEquality,
intEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
promote_hyp,
instantiate,
cumulativity,
dependent_pairEquality,
universeEquality,
equalityElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[p:\{2...\}]. \mforall{}[n,m:\mBbbN{}]. \mforall{}[a,b:p-adics(p)]. (a/p\^{}n) = (b/p\^{}m) supposing bpa-equiv(p;<n, a><m, b>)
Date html generated:
2018_05_21-PM-03_26_35
Last ObjectModification:
2018_05_19-AM-08_23_49
Theory : rings_1
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