Nuprl Lemma : rabs-positive-iff
∀x:ℝ. (x ≠ r0 ⇐⇒ r0 < |x|)
Proof
Definitions occuring in Statement :
rneq: x ≠ y,
rless: x < y,
rabs: |x|,
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,
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
uimplies: b supposing a,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
uiff: uiff(P;Q)
Lemmas referenced :
rneq-iff-rabs,
int-to-real_wf,
rneq_wf,
iff_wf,
rless_wf,
rabs_wf,
rsub_wf,
real_wf,
rmul_wf,
rless_functionality,
req_weakening,
rabs_functionality,
req_transitivity,
real_term_polynomial,
itermSubtract_wf,
itermVar_wf,
itermConstant_wf,
itermMultiply_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_mul_lemma,
req-iff-rsub-is-0,
rmul-identity1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
addLevel,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
impliesFunctionality,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
isectElimination,
natural_numberEquality,
hypothesis,
independent_functionElimination,
because_Cache,
independent_isectElimination,
sqequalRule,
computeAll,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}x:\mBbbR{}. (x \mneq{} r0 \mLeftarrow{}{}\mRightarrow{} r0 < |x|)
Date html generated:
2017_10_03-AM-08_31_26
Last ObjectModification:
2017_07_28-AM-07_27_04
Theory : reals
Home
Index