Nuprl Lemma : qmul-positive
∀a,b:ℚ. ((0 < a ∧ 0 < b) ∨ (0 < -(a) ∧ 0 < -(b)) ⇐⇒ 0 < a * b)
Proof
Definitions occuring in Statement :
qless: r < s,
qmul: r * s,
rationals: ℚ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
or: P ∨ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
squash: ↓T,
true: True,
guard: {T},
cand: A c∧ B,
qpositive: qpositive(r),
callbyvalueall: callbyvalueall,
evalall: evalall(t),
lt_int: i <z j,
ifthenelse: if b then t else f fi ,
btrue: tt,
assert: ↑b,
bfalse: ff,
false: False,
not: ¬A
Lemmas referenced :
or_wf,
qless_wf,
int-subtype-rationals,
qmul_wf,
rationals_wf,
assert-qpositive,
qmul_positive,
assert_wf,
qpositive_wf,
equal_wf,
squash_wf,
true_wf,
qmul_assoc,
iff_weakening_equal,
qmul_ac_1_qrng,
qinv_inv_q,
q_trichotomy,
assert_functionality_wrt_uiff,
qmul_zero_qrng,
qminus_positive,
uiff_transitivity,
qmul_over_minus_qrng,
qmul_comm_qrng
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
unionElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
productEquality,
natural_numberEquality,
hypothesis,
applyEquality,
sqequalRule,
hypothesisEquality,
because_Cache,
minusEquality,
productElimination,
independent_isectElimination,
addLevel,
levelHypothesis,
promote_hyp,
andLevelFunctionality,
equalitySymmetry,
lambdaEquality,
imageElimination,
equalityTransitivity,
universeEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
hyp_replacement,
applyLambdaEquality,
dependent_functionElimination,
inlFormation,
callbyvalueReduce,
sqleReflexivity,
isintReduceTrue,
voidElimination,
inrFormation
Latex:
\mforall{}a,b:\mBbbQ{}. ((0 < a \mwedge{} 0 < b) \mvee{} (0 < -(a) \mwedge{} 0 < -(b)) \mLeftarrow{}{}\mRightarrow{} 0 < a * b)
Date html generated:
2018_05_21-PM-11_52_19
Last ObjectModification:
2017_07_26-PM-06_45_00
Theory : rationals
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