Nuprl Lemma : qabs-non-zero
∀[q:ℚ]. uiff(0 < |q|;¬(q = 0 ∈ ℚ))
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,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
qabs: |r|,
callbyvalueall: callbyvalueall,
evalall: evalall(t),
ifthenelse: if b then t else f fi ,
qpositive: qpositive(r),
btrue: tt,
lt_int: i <z j,
bfalse: ff,
qmul: r * s,
qless: r < s,
grp_lt: a < b,
set_lt: a <p b,
assert: ↑b,
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),
bor: p ∨bq,
qsub: r - s,
qadd: r + s,
qeq: qeq(r;s),
eq_int: (i =z j),
bnot: ¬bb,
all: ∀x:A. B[x],
or: P ∨ Q,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
equal-wf-T-base,
rationals_wf,
qless_wf,
int-subtype-rationals,
qabs_wf,
qless_witness,
not_wf,
zero-qle-qabs,
qle-iff,
qabs-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
thin,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
baseClosed,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
because_Cache,
natural_numberEquality,
applyEquality,
productElimination,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality,
independent_isectElimination,
unionElimination
Latex:
\mforall{}[q:\mBbbQ{}]. uiff(0 < |q|;\mneg{}(q = 0))
Date html generated:
2016_10_26-AM-06_32_06
Last ObjectModification:
2016_07_12-AM-07_51_11
Theory : rationals
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