Nuprl Lemma : rabs-difference-bound-iff
∀x,y,z:ℝ. (|x - y| < z ⇐⇒ ((y - z) < x) ∧ (x < (y + z)))
Proof
Definitions occuring in Statement :
rless: x < y,
rabs: |x|,
rsub: x - y,
radd: a + b,
real: ℝ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
top: Top,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
uimplies: b supposing a,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
uiff: uiff(P;Q),
prop: ℙ,
cand: A c∧ B
Lemmas referenced :
rabs-as-rmax,
rmax_strict_lb,
rsub_wf,
rminus_wf,
rless-implies-rless,
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,
radd_wf,
itermAdd_wf,
real_term_value_add_lemma,
rless_wf,
rmax_wf,
real_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalTransitivity,
computationStep,
isectElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
lambdaFormation,
independent_pairFormation,
dependent_functionElimination,
hypothesisEquality,
productElimination,
independent_functionElimination,
independent_isectElimination,
natural_numberEquality,
computeAll,
lambdaEquality,
int_eqEquality,
intEquality,
because_Cache,
productEquality
Latex:
\mforall{}x,y,z:\mBbbR{}. (|x - y| < z \mLeftarrow{}{}\mRightarrow{} ((y - z) < x) \mwedge{} (x < (y + z)))
Date html generated:
2017_10_03-AM-08_39_15
Last ObjectModification:
2017_07_28-AM-07_30_41
Theory : reals
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