Nuprl Lemma : pa-mul_wf
∀[p:{2...}]. ∀[x,y:basic-padic(p)]. (pa-mul(p;x;y) ∈ padic(p))
Proof
Definitions occuring in Statement :
pa-mul: pa-mul(p;x;y),
padic: padic(p),
basic-padic: basic-padic(p),
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,
pa-mul: pa-mul(p;x;y),
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
Lemmas referenced :
bpa-norm_wf_padic,
decidable__lt,
false_wf,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
less_than_wf,
bpa-mul_wf,
basic-padic_wf,
int_upper_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality,
setElimination,
rename,
because_Cache,
hypothesis,
productElimination,
dependent_functionElimination,
natural_numberEquality,
hypothesisEquality,
unionElimination,
independent_pairFormation,
lambdaFormation,
voidElimination,
independent_functionElimination,
independent_isectElimination,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidEquality,
intEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[p:\{2...\}]. \mforall{}[x,y:basic-padic(p)]. (pa-mul(p;x;y) \mmember{} padic(p))
Date html generated:
2018_05_21-PM-03_26_55
Last ObjectModification:
2018_05_19-AM-08_23_58
Theory : rings_1
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