Nuprl Lemma : absval_mul
∀[x,y:ℤ]. (|x * y| = (|x| * |y|) ∈ ℤ)
Proof
Definitions occuring in Statement :
absval: |i|,
uall: ∀[x:A]. B[x],
multiply: n * m,
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,
bfalse: ff,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
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,
prop: ℙ,
le: A ≤ B,
nat: ℕ,
subtract: n - m,
nat_plus: ℕ+,
decidable: Dec(P),
cand: A c∧ B
Lemmas referenced :
absval_unfold2,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-top,
istype-void,
eqff_to_assert,
int_subtype_base,
bool_subtype_base,
bool_cases_sqequal,
subtype_base_sq,
iff_transitivity,
assert_wf,
bnot_wf,
not_wf,
less_than_wf,
iff_weakening_uiff,
assert_of_bnot,
istype-less_than,
istype-assert,
not-lt-2,
minus-one-mul,
mul-associates,
istype-int,
minus-one-mul-top,
mul-swap,
one-mul,
bool_wf,
less_than_irreflexivity,
less_than_transitivity1,
le_wf,
le_weakening2,
mul_preserves_le,
mul-commutes,
zero-mul,
le-add-cancel,
zero-add,
add-associates,
add_functionality_wrt_le,
add-commutes,
add-zero,
minus-zero,
minus-add,
condition-implies-le,
less-iff-le,
mul_preserves_lt,
decidable__int_equal,
decidable__lt,
istype-false,
not-equal-2,
add_functionality_wrt_lt,
le_reflexive,
add-mul-special
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
multiplyEquality,
hypothesisEquality,
hypothesis,
natural_numberEquality,
Error :inhabitedIsType,
Error :lambdaFormation_alt,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
because_Cache,
lessCases,
axiomSqEquality,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
independent_pairFormation,
voidElimination,
imageMemberEquality,
baseClosed,
imageElimination,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :equalityIsType4,
baseApply,
closedConclusion,
applyEquality,
promote_hyp,
dependent_functionElimination,
instantiate,
Error :functionIsType,
Error :universeIsType,
Error :equalityIsType1,
minusEquality,
Error :lambdaEquality_alt,
cumulativity,
axiomEquality,
voidEquality,
isect_memberEquality,
dependent_set_memberEquality,
intEquality,
lambdaEquality,
addEquality
Latex:
\mforall{}[x,y:\mBbbZ{}]. (|x * y| = (|x| * |y|))
Date html generated:
2019_06_20-AM-11_24_38
Last ObjectModification:
2018_10_27-AM-11_38_11
Theory : arithmetic
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