Nuprl Lemma : absval-minus
∀[x:ℤ]. (|-x| = |x| ∈ ℤ)
Proof
Definitions occuring in Statement :
absval: |i|
,
uall: ∀[x:A]. B[x]
,
minus: -n
,
int: ℤ
,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x]
,
member: t ∈ T
,
all: ∀x:A. B[x]
,
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
,
prop: ℙ
,
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
,
iff: P
⇐⇒ Q
,
rev_implies: P
⇐ Q
,
le: A ≤ B
Lemmas referenced :
absval_unfold2,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
iff_transitivity,
assert_wf,
bnot_wf,
not_wf,
iff_weakening_uiff,
assert_of_bnot,
minus-minus,
add_functionality_wrt_lt,
le_reflexive,
less_than_transitivity2,
le_weakening2,
less_than_irreflexivity,
minus-one-mul,
zero-add,
add-commutes,
add-mul-special,
zero-mul,
not-lt-2,
int_subtype_base,
minus-zero,
add_functionality_wrt_le,
le_antisymmetry
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
minusEquality,
hypothesisEquality,
hypothesis,
natural_numberEquality,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
because_Cache,
lessCases,
sqequalAxiom,
isect_memberEquality,
independent_pairFormation,
voidElimination,
voidEquality,
imageMemberEquality,
baseClosed,
imageElimination,
independent_functionElimination,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
impliesFunctionality,
cumulativity,
multiplyEquality,
intEquality
Latex:
\mforall{}[x:\mBbbZ{}]. (|-x| = |x|)
Date html generated:
2017_04_14-AM-07_16_54
Last ObjectModification:
2017_02_27-PM-02_51_41
Theory : arithmetic
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