Nuprl Lemma : rabs-rless-iff
∀x,z:ℝ. (|x| < z ⇐⇒ (-(z) < x) ∧ (x < z))
Proof
Definitions occuring in Statement :
rless: x < y,
rabs: |x|,
rminus: -(x),
real: ℝ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
and: P ∧ Q,
rsub: x - y,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
uimplies: b supposing a,
rev_implies: P ⇐ Q
Lemmas referenced :
rabs-difference-bound-iff,
int-to-real_wf,
real_wf,
rabs_wf,
radd_wf,
rminus_wf,
rless_wf,
rless_functionality,
rabs_functionality,
radd_functionality,
rminus-zero,
req_weakening,
radd_comm,
radd-zero-both
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
natural_numberEquality,
hypothesis,
because_Cache,
productEquality,
sqequalRule,
productElimination,
independent_pairFormation,
independent_functionElimination,
independent_isectElimination,
promote_hyp
Latex:
\mforall{}x,z:\mBbbR{}. (|x| < z \mLeftarrow{}{}\mRightarrow{} (-(z) < x) \mwedge{} (x < z))
Date html generated:
2016_10_26-AM-09_10_23
Last ObjectModification:
2016_09_01-PM-01_37_12
Theory : reals
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