Nuprl Lemma : mkpadic-equiv
∀[p:{2...}]. ∀[n:ℕ]. ∀[a:p-adics(p)]. ((a/p^n) ∈ {x:basic-padic(p)| bpa-equiv(p;<n, a>;x)} )
Proof
Definitions occuring in Statement :
mkpadic: (a/p^n),
bpa-equiv: bpa-equiv(p;x;y),
basic-padic: basic-padic(p),
p-adics: p-adics(p),
int_upper: {i...},
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
pair: <a, b>,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
mkpadic: (a/p^n),
all: ∀x:A. B[x],
basic-padic: basic-padic(p),
nat_plus: ℕ+,
int_upper: {i...},
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
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
Lemmas referenced :
bpa-norm-equiv,
mkpadic_wf,
decidable__lt,
false_wf,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
less_than_wf,
padic_subtype_basic-padic,
bpa-equiv_wf,
p-adics_wf,
nat_wf,
int_upper_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_pairEquality,
hypothesis,
dependent_set_memberEquality,
isectElimination,
setElimination,
rename,
productElimination,
natural_numberEquality,
unionElimination,
independent_pairFormation,
lambdaFormation,
voidElimination,
independent_functionElimination,
independent_isectElimination,
applyEquality,
because_Cache,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality
Latex:
\mforall{}[p:\{2...\}]. \mforall{}[n:\mBbbN{}]. \mforall{}[a:p-adics(p)]. ((a/p\^{}n) \mmember{} \{x:basic-padic(p)| bpa-equiv(p;<n, a>x)\} )
Date html generated:
2018_05_21-PM-03_26_39
Last ObjectModification:
2018_05_19-AM-08_23_45
Theory : rings_1
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