Nuprl Lemma : small-reciprocal-rneq-zero
∀x:ℝ. (x ≠ r0 ⇒ (∃k:ℕ+. ((r1/r(k)) < |x|)))
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rneq: x ≠ y,
rless: x < y,
rabs: |x|,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
small-reciprocal-real-ext,
rabs_wf,
rless_wf,
int-to-real_wf,
rneq_wf,
real_wf,
rpositive-rless,
rabs-positive
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
dependent_set_memberEquality,
isectElimination,
hypothesisEquality,
hypothesis,
natural_numberEquality,
productElimination,
independent_functionElimination
Latex:
\mforall{}x:\mBbbR{}. (x \mneq{} r0 {}\mRightarrow{} (\mexists{}k:\mBbbN{}\msupplus{}. ((r1/r(k)) < |x|)))
Date html generated:
2016_05_18-AM-07_34_37
Last ObjectModification:
2015_12_28-AM-00_55_49
Theory : reals
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