Nuprl Lemma : mul_positive_iff
∀a,b:ℤ. (0 < a * b
⇐⇒ (0 < a ∧ 0 < b) ∨ (a < 0 ∧ b < 0))
Proof
Definitions occuring in Statement :
less_than: a < b
,
all: ∀x:A. B[x]
,
iff: P
⇐⇒ Q
,
or: P ∨ Q
,
and: P ∧ Q
,
multiply: n * m
,
natural_number: $n
,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x]
,
iff: P
⇐⇒ Q
,
and: P ∧ Q
,
implies: P
⇒ Q
,
member: t ∈ T
,
prop: ℙ
,
uall: ∀[x:A]. B[x]
,
rev_implies: P
⇐ Q
,
decidable: Dec(P)
,
or: P ∨ Q
,
cand: A c∧ B
,
not: ¬A
,
false: False
,
nat_plus: ℕ+
,
uimplies: b supposing a
,
uiff: uiff(P;Q)
,
subtract: n - m
,
subtype_rel: A ⊆r B
,
top: Top
,
le: A ≤ B
,
less_than': less_than'(a;b)
,
true: True
,
sq_type: SQType(T)
,
guard: {T}
Lemmas referenced :
less_than_wf,
or_wf,
decidable__lt,
less-trichotomy,
mul_preserves_lt,
less-iff-le,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
mul-commutes,
zero-mul,
add_functionality_wrt_le,
add-associates,
add-zero,
add-commutes,
le-add-cancel,
subtype_base_sq,
int_subtype_base,
minus-zero,
mul_positive,
add_functionality_wrt_lt,
le_reflexive,
add-inverse,
mul-associates,
mul-swap,
one-mul
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
multiplyEquality,
hypothesisEquality,
hypothesis,
productEquality,
intEquality,
dependent_functionElimination,
unionElimination,
because_Cache,
inlFormation,
independent_functionElimination,
voidElimination,
dependent_set_memberEquality,
independent_isectElimination,
productElimination,
addEquality,
sqequalRule,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidEquality,
minusEquality,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
inrFormation
Latex:
\mforall{}a,b:\mBbbZ{}. (0 < a * b \mLeftarrow{}{}\mRightarrow{} (0 < a \mwedge{} 0 < b) \mvee{} (a < 0 \mwedge{} b < 0))
Date html generated:
2019_06_20-AM-11_23_27
Last ObjectModification:
2018_08_20-AM-11_00_40
Theory : arithmetic
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