Nuprl Lemma : absval_cases
∀x:ℤ. ∀[y:ℕ]. uiff(|x| = y ∈ ℤ;(x = y ∈ ℤ) ∨ (x = (-y) ∈ ℤ))
Proof
Definitions occuring in Statement :
absval: |i|,
nat: ℕ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
or: P ∨ Q,
minus: -n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B,
nat: ℕ,
implies: P ⇒ Q,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
not: ¬A,
false: False,
guard: {T},
sq_type: SQType(T),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
equal-wf-base-T,
int_subtype_base,
or_wf,
nat_wf,
absval_unfold2,
decidable__lt,
top_wf,
less_than_wf,
subtype_base_sq,
minus-minus,
equal_wf,
squash_wf,
true_wf,
absval_pos,
iff_weakening_equal,
absval_sym
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
independent_pairFormation,
cut,
introduction,
axiomEquality,
hypothesis,
thin,
rename,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
intEquality,
sqequalRule,
baseApply,
closedConclusion,
baseClosed,
hypothesisEquality,
applyEquality,
setElimination,
because_Cache,
minusEquality,
independent_functionElimination,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
lessCases,
sqequalAxiom,
isect_memberEquality,
voidElimination,
voidEquality,
imageMemberEquality,
imageElimination,
productElimination,
inlFormation,
inrFormation,
instantiate,
cumulativity,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
universeEquality
Latex:
\mforall{}x:\mBbbZ{}. \mforall{}[y:\mBbbN{}]. uiff(|x| = y;(x = y) \mvee{} (x = (-y)))
Date html generated:
2017_04_14-AM-07_17_26
Last ObjectModification:
2017_02_27-PM-02_52_03
Theory : arithmetic
Home
Index