Nuprl Lemma : real-vec-sep-iff
∀n:ℕ. ∀a,c:ℝ^n. (a ≠ c ⇐⇒ ∃i:ℕn. (r0 < |(a i) - c i|))
Proof
Definitions occuring in Statement :
real-vec-sep: a ≠ b,
real-vec: ℝ^n,
rless: x < y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
natural_number: $n
Definitions unfolded in proof :
so_apply: x[s],
real-vec: ℝ^n,
so_lambda: λ2x.t[x],
nat: ℕ,
real-vec-dist: d(x;y),
real-vec-sep: a ≠ b,
rev_implies: P ⇐ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
real-vec-sub: X - Y
Lemmas referenced :
nat_wf,
real-vec_wf,
rsub_wf,
rabs_wf,
int-to-real_wf,
rless_wf,
int_seg_wf,
exists_wf,
real-vec-sub_wf,
real-vec-norm-positive-iff,
real-vec-sep_wf,
real-vec-sep-implies,
rneq_wf,
rabs-positive-iff,
rneq-symmetry
Rules used in proof :
because_Cache,
applyEquality,
lambdaEquality,
sqequalRule,
rename,
setElimination,
natural_numberEquality,
productElimination,
isectElimination,
hypothesis,
independent_functionElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
dependent_pairFormation
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,c:\mBbbR{}\^{}n. (a \mneq{} c \mLeftarrow{}{}\mRightarrow{} \mexists{}i:\mBbbN{}n. (r0 < |(a i) - c i|))
Date html generated:
2017_10_03-AM-11_00_41
Last ObjectModification:
2017_06_16-AM-11_48_28
Theory : reals
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