Nuprl Lemma : qabs-of-non-positive
∀[q:ℚ]. |q| ~ -(q) supposing q ≤ 0
Proof
Definitions occuring in Statement :
qabs: |r|,
qle: r ≤ s,
qmul: r * s,
rationals: ℚ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
minus: -n,
natural_number: $n,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
qabs: |r|,
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
prop: ℙ,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
guard: {T},
false: False
Lemmas referenced :
valueall-type-has-valueall,
rationals_wf,
rationals-valueall-type,
evalall-reduce,
qle_wf,
int-subtype-rationals,
qpositive_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
qless_wf,
bnot_wf,
not_wf,
uiff_transitivity,
eqtt_to_assert,
assert-qpositive,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
equal_wf,
qless_transitivity_2_qorder,
qless_irreflexivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
independent_isectElimination,
hypothesisEquality,
callbyvalueReduce,
sqequalAxiom,
natural_numberEquality,
applyEquality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
baseClosed,
lambdaFormation,
unionElimination,
equalityElimination,
independent_functionElimination,
productElimination,
independent_pairFormation,
impliesFunctionality,
dependent_functionElimination,
voidElimination
Latex:
\mforall{}[q:\mBbbQ{}]. |q| \msim{} -(q) supposing q \mleq{} 0
Date html generated:
2018_05_21-PM-11_52_55
Last ObjectModification:
2017_07_26-PM-06_45_17
Theory : rationals
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