Nuprl Lemma : bpa-equiv-iff-norm2
∀p:{2...}. ∀x,y:basic-padic(p). (bpa-norm(p;x) = bpa-norm(p;y) ∈ padic(p) ⇐⇒ bpa-equiv(p;x;y))
Proof
Definitions occuring in Statement :
padic: padic(p),
bpa-norm: bpa-norm(p;x),
bpa-equiv: bpa-equiv(p;x;y),
basic-padic: basic-padic(p),
int_upper: {i...},
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
int_upper: {i...},
nat_plus: ℕ+,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
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-equiv-iff-norm,
equal_wf,
basic-padic_wf,
bpa-norm_wf,
decidable__lt,
false_wf,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
less_than_wf,
bpa-equiv_wf,
equal-padics,
padic_wf,
bpa-norm_wf_padic,
iff_wf,
int_upper_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairFormation,
isectElimination,
setElimination,
rename,
dependent_set_memberEquality,
because_Cache,
natural_numberEquality,
unionElimination,
voidElimination,
independent_functionElimination,
independent_isectElimination,
sqequalRule,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidEquality,
intEquality,
addLevel,
impliesFunctionality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}p:\{2...\}. \mforall{}x,y:basic-padic(p). (bpa-norm(p;x) = bpa-norm(p;y) \mLeftarrow{}{}\mRightarrow{} bpa-equiv(p;x;y))
Date html generated:
2018_05_21-PM-03_26_21
Last ObjectModification:
2018_05_19-AM-08_23_26
Theory : rings_1
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