Nuprl Lemma : p-adics-subtype
∀[p:ℕ+]. (p-adics(p) ⊆r (ℕ+ ⟶ ℤ))
Proof
Definitions occuring in Statement :
p-adics: p-adics(p),
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
p-adics: p-adics(p),
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtract: n - m,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
int_seg: {i..j-},
so_apply: x[s]
Lemmas referenced :
subtype_rel_set,
nat_plus_wf,
int_seg_wf,
exp_wf2,
nat_plus_subtype_nat,
all_wf,
eqmod_wf,
decidable__lt,
false_wf,
not-lt-2,
less-iff-le,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
less_than_wf,
subtype_rel_dep_function
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
functionEquality,
hypothesis,
natural_numberEquality,
hypothesisEquality,
applyEquality,
setElimination,
rename,
intEquality,
lambdaEquality,
because_Cache,
functionExtensionality,
dependent_set_memberEquality,
addEquality,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
lambdaFormation,
voidElimination,
productElimination,
independent_functionElimination,
independent_isectElimination,
minusEquality,
axiomEquality
Latex:
\mforall{}[p:\mBbbN{}\msupplus{}]. (p-adics(p) \msubseteq{}r (\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}))
Date html generated:
2018_05_21-PM-03_18_33
Last ObjectModification:
2018_05_19-AM-08_09_24
Theory : rings_1
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