Nuprl Lemma : qabs-positive-iff
∀[r:ℚ]. uiff(¬(r = 0 ∈ ℚ);0 < |r|)
Proof
Definitions occuring in Statement :
qabs: |r|,
qless: r < s,
rationals: ℚ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
not: ¬A,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
prop: ℙ,
not: ¬A,
false: False,
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),
qabs: |r|,
qpositive: qpositive(r),
btrue: tt,
lt_int: i <z j,
bfalse: ff,
qmul: r * s,
bor: p ∨bq,
qsub: r - s,
qadd: r + s,
qeq: qeq(r;s),
eq_int: (i =z j),
bnot: ¬bb
Lemmas referenced :
qless_witness,
int-subtype-rationals,
qabs_wf,
not_wf,
equal-wf-T-base,
rationals_wf,
qless_wf,
qabs-positive
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesis,
applyEquality,
sqequalRule,
hypothesisEquality,
independent_functionElimination,
baseClosed,
lambdaFormation,
voidElimination,
because_Cache,
lambdaEquality,
dependent_functionElimination,
productElimination,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
hyp_replacement,
Error :applyLambdaEquality
Latex:
\mforall{}[r:\mBbbQ{}]. uiff(\mneg{}(r = 0);0 < |r|)
Date html generated:
2016_10_25-PM-00_05_11
Last ObjectModification:
2016_07_12-AM-07_49_13
Theory : rationals
Home
Index