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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