Nuprl Lemma : qmul-non-neg
∀a,b:ℚ. ((a = 0 ∈ ℚ) ∨ (b = 0 ∈ ℚ) ∨ (0 < a ∧ 0 < b) ∨ (0 < -(a) ∧ 0 < -(b)) ⇐⇒ 0 ≤ (a * b))
Proof
Definitions occuring in Statement :
qle: r ≤ s,
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,
equal: s = t ∈ T
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],
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
or: P ∨ Q,
true: True,
qle: r ≤ s,
grp_leq: a ≤ b,
assert: ↑b,
ifthenelse: if b then t else f fi ,
infix_ap: x f y,
grp_le: ≤b,
pi1: fst(t),
pi2: snd(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,
bfalse: ff,
qeq: qeq(r;s),
eq_int: (i =z j),
squash: ↓T,
uimplies: b supposing a,
guard: {T},
decidable: Dec(P),
cand: A c∧ B,
false: False,
not: ¬A
Lemmas referenced :
or_wf,
equal-wf-T-base,
rationals_wf,
qless_wf,
int-subtype-rationals,
qmul_wf,
qle_wf,
squash_wf,
true_wf,
qmul_zero_qrng,
iff_weakening_equal,
qmul-positive,
qle_weakening_lt_qorder,
decidable__equal_rationals,
qless_trichot_qorder,
qless_transitivity_2_qorder,
qless_irreflexivity,
qmul-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
hypothesisEquality,
baseClosed,
because_Cache,
productEquality,
natural_numberEquality,
applyEquality,
sqequalRule,
minusEquality,
unionElimination,
hyp_replacement,
equalitySymmetry,
Error :applyLambdaEquality,
equalityTransitivity,
lambdaEquality,
imageElimination,
productElimination,
imageMemberEquality,
universeEquality,
independent_isectElimination,
independent_functionElimination,
dependent_functionElimination,
inlFormation,
inrFormation,
voidElimination
Latex:
\mforall{}a,b:\mBbbQ{}. ((a = 0) \mvee{} (b = 0) \mvee{} (0 < a \mwedge{} 0 < b) \mvee{} (0 < -(a) \mwedge{} 0 < -(b)) \mLeftarrow{}{}\mRightarrow{} 0 \mleq{} (a * b))
Date html generated:
2016_10_25-PM-00_07_19
Last ObjectModification:
2016_07_12-AM-07_50_28
Theory : rationals
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