Nuprl Lemma : mul_positive
∀a,b:ℤ. (0 < a ⇒ 0 < b ⇒ 0 < a * b)
Proof
Definitions occuring in Statement :
less_than: a < b,
all: ∀x:A. B[x],
implies: P ⇒ Q,
multiply: n * m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
prop: ℙ,
squash: ↓T,
member: t ∈ T,
less_than': less_than'(a;b),
cand: A c∧ B,
and: P ∧ Q,
less_than: a < b,
implies: P ⇒ Q,
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
top: Top,
true: True,
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
Lemmas referenced :
less_than_wf,
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,
bool_wf,
iff_transitivity,
assert_wf,
bnot_wf,
not_wf,
iff_weakening_uiff,
assert_of_bnot,
istype-assert
Rules used in proof :
intEquality,
isectElimination,
extract_by_obid,
sqequalHypSubstitution,
baseClosed,
thin,
imageMemberEquality,
sqequalRule,
introduction,
hypothesisEquality,
multiplyEquality,
natural_numberEquality,
hypothesis,
independent_pairFormation,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
multiplyPositive,
Error :inhabitedIsType,
Error :lambdaFormation_alt,
unionElimination,
equalityElimination,
because_Cache,
productElimination,
independent_isectElimination,
lessCases,
Error :isect_memberFormation_alt,
axiomSqEquality,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
voidElimination,
imageElimination,
independent_functionElimination,
Error :dependent_pairFormation_alt,
equalityTransitivity,
equalitySymmetry,
Error :equalityIsType4,
baseApply,
closedConclusion,
applyEquality,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
Error :functionIsType,
Error :equalityIsType1
Latex:
\mforall{}a,b:\mBbbZ{}. (0 < a {}\mRightarrow{} 0 < b {}\mRightarrow{} 0 < a * b)
Date html generated:
2019_06_20-AM-11_23_15
Last ObjectModification:
2018_10_16-PM-00_58_40
Theory : arithmetic
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