Nuprl Lemma : qmul-negative
∀a,b:ℚ. ((a < 0 ∧ 0 < b) ∨ (0 < a ∧ b < 0) ⇐⇒ a * b < 0)
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,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
uimplies: b supposing a,
qless: r < s,
grp_lt: a < b,
set_lt: a <p b,
assert: ↑b,
ifthenelse: if b then t else f fi ,
set_blt: a <b b,
band: p ∧b q,
infix_ap: x f y,
set_le: ≤b,
pi2: snd(t),
oset_of_ocmon: g↓oset,
dset_of_mon: g↓set,
grp_le: ≤b,
pi1: fst(t),
qadd_grp: <ℚ+>,
q_le: q_le(r;s),
callbyvalueall: callbyvalueall,
evalall: evalall(t),
bor: p ∨bq,
qpositive: qpositive(r),
qsub: r - s,
qadd: r + s,
qmul: r * s,
btrue: tt,
lt_int: i <z j,
bnot: ¬bb,
bfalse: ff,
qeq: qeq(r;s),
eq_int: (i =z j),
true: True,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
squash: ↓T,
guard: {T},
or: P ∨ Q
Lemmas referenced :
qmul-positive2,
and_wf,
or_wf,
iff_wf,
iff_weakening_equal,
qinv_inv_q,
qmul_zero_qrng,
qmul_over_minus_qrng,
true_wf,
squash_wf,
int-subtype-rationals,
qmul_reverses_qless,
qmul_wf,
qless_wf,
rationals_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
hypothesis,
independent_pairFormation,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
natural_numberEquality,
applyEquality,
because_Cache,
sqequalRule,
minusEquality,
independent_isectElimination,
productElimination,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
universeEquality,
independent_functionElimination,
addLevel,
impliesFunctionality,
unionElimination,
inlFormation,
promote_hyp,
inrFormation,
dependent_functionElimination,
productEquality
Latex:
\mforall{}a,b:\mBbbQ{}. ((a < 0 \mwedge{} 0 < b) \mvee{} (0 < a \mwedge{} b < 0) \mLeftarrow{}{}\mRightarrow{} a * b < 0)
Date html generated:
2016_05_15-PM-11_00_58
Last ObjectModification:
2016_01_16-PM-09_33_10
Theory : rationals
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