Nuprl Lemma : rabs-bounds
∀[x:ℝ]. ((-(|x|) ≤ x) ∧ (x ≤ |x|))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rabs: |x|,
rminus: -(x),
real: ℝ,
uall: ∀[x:A]. B[x],
and: P ∧ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
real: ℝ,
prop: ℙ,
top: Top,
cand: A c∧ B,
uimplies: b supposing a,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
uiff: uiff(P;Q)
Lemmas referenced :
rleq-rmax,
rminus_wf,
less_than'_wf,
rsub_wf,
rabs_wf,
real_wf,
nat_plus_wf,
rleq-implies-rleq,
real_term_polynomial,
itermSubtract_wf,
itermVar_wf,
itermMinus_wf,
int-to-real_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma,
req-iff-rsub-is-0,
rabs-as-rmax
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
productElimination,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
voidElimination,
applyEquality,
setElimination,
rename,
minusEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidEquality,
because_Cache,
independent_isectElimination,
computeAll,
int_eqEquality,
intEquality,
independent_pairFormation
Latex:
\mforall{}[x:\mBbbR{}]. ((-(|x|) \mleq{} x) \mwedge{} (x \mleq{} |x|))
Date html generated:
2017_10_03-AM-08_37_26
Last ObjectModification:
2017_07_28-AM-07_30_08
Theory : reals
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