Nuprl Lemma : pos_mul_arg_bounds
∀a,b:ℤ. ((a * b) > 0 ⇐⇒ ((a > 0) ∧ (b > 0)) ∨ (a < 0 ∧ b < 0))
Proof
Definitions occuring in Statement :
less_than: a < b,
gt: i > j,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
multiply: n * m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
gt: i > j,
or: P ∨ Q,
cand: A c∧ B,
decidable: Dec(P),
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
nat_plus: ℕ+,
squash: ↓T,
label: ...$L... t,
true: True,
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
gt_wf,
or_wf,
less_than_wf,
int_trichot,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermMultiply_wf,
itermMinus_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_mul_lemma,
int_term_value_minus_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
mul_cancel_in_lt,
mul_bounds_1b,
squash_wf,
true_wf,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
multiplyEquality,
hypothesisEquality,
natural_numberEquality,
hypothesis,
productEquality,
intEquality,
dependent_functionElimination,
unionElimination,
inrFormation,
minusEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
computeAll,
dependent_set_memberEquality,
because_Cache,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality,
inlFormation,
productElimination,
applyEquality,
imageElimination,
equalityTransitivity,
imageMemberEquality,
baseClosed,
universeEquality,
independent_functionElimination
Latex:
\mforall{}a,b:\mBbbZ{}. ((a * b) > 0 \mLeftarrow{}{}\mRightarrow{} ((a > 0) \mwedge{} (b > 0)) \mvee{} (a < 0 \mwedge{} b < 0))
Date html generated:
2016_10_21-AM-09_58_35
Last ObjectModification:
2016_07_12-AM-05_12_30
Theory : int_2
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