Nuprl Lemma : qabs-qle-zero
∀[r:ℚ]. uiff(|r| ≤ 0;r = 0 ∈ ℚ)
Proof
Definitions occuring in Statement :
qabs: |r|,
qle: r ≤ s,
rationals: ℚ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
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,
prop: ℙ,
implies: P ⇒ Q,
guard: {T},
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,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A
Lemmas referenced :
zero-qle-qabs,
qle_wf,
qabs_wf,
int-subtype-rationals,
qle_witness,
rationals_wf,
qabs-zero,
qle_antisymmetry,
qle-int,
istype-false
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_pairFormation,
universeIsType,
natural_numberEquality,
applyEquality,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
equalityIstype,
inhabitedIsType,
baseClosed,
sqequalBase,
productElimination,
independent_isectElimination,
closedConclusion,
lambdaFormation_alt,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[r:\mBbbQ{}]. uiff(|r| \mleq{} 0;r = 0)
Date html generated:
2019_10_16-PM-00_31_18
Last ObjectModification:
2018_11_26-PM-03_09_30
Theory : rationals
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