Nuprl Lemma : rneq-iff-rabs
∀x,y:ℝ. (x ≠ y ⇐⇒ r0 < |x - y|)
Proof
Definitions occuring in Statement :
rneq: x ≠ y,
rless: x < y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
top: Top,
rneq: x ≠ y,
or: P ∨ Q,
guard: {T},
uimplies: b supposing a,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
uiff: uiff(P;Q)
Lemmas referenced :
rneq_wf,
rless_wf,
int-to-real_wf,
rabs_wf,
rsub_wf,
real_wf,
rabs-as-rmax,
rmax_strict_ub,
rminus_wf,
rless-implies-rless,
real_term_polynomial,
itermSubtract_wf,
itermVar_wf,
itermMinus_wf,
itermConstant_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,
rneq-if-rabs
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
natural_numberEquality,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_functionElimination,
productElimination,
independent_functionElimination,
unionElimination,
inrFormation,
independent_isectElimination,
computeAll,
lambdaEquality,
int_eqEquality,
intEquality,
inlFormation,
because_Cache
Latex:
\mforall{}x,y:\mBbbR{}. (x \mneq{} y \mLeftarrow{}{}\mRightarrow{} r0 < |x - y|)
Date html generated:
2017_10_03-AM-08_31_12
Last ObjectModification:
2017_07_28-AM-07_26_59
Theory : reals
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