Nuprl Lemma : absval-divides
∀a,b:ℤ. (|a| | b ⇐⇒ a | b)
Proof
Definitions occuring in Statement :
divides: b | a,
absval: |i|,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
less_than: a < b,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
not: ¬A,
false: False,
iff: P ⇐⇒ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b
Lemmas referenced :
absval_unfold,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-top,
istype-void,
divides_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
istype-less_than,
divides_invar_1,
minus-minus,
istype-int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
minusEquality,
natural_numberEquality,
Error :inhabitedIsType,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
because_Cache,
lessCases,
Error :isect_memberFormation_alt,
axiomSqEquality,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
independent_pairFormation,
voidElimination,
imageMemberEquality,
baseClosed,
imageElimination,
independent_functionElimination,
Error :universeIsType,
Error :dependent_pairFormation_alt,
Error :equalityIstype,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity
Latex:
\mforall{}a,b:\mBbbZ{}. (|a| | b \mLeftarrow{}{}\mRightarrow{} a | b)
Date html generated:
2019_06_20-PM-02_20_02
Last ObjectModification:
2019_01_10-PM-01_19_34
Theory : num_thy_1
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