Nuprl Lemma : length-rneq-real-vec-sep
∀n:ℕ. ∀x,v:ℝ^n. (||v|| ≠ ||x|| ⇒ x ≠ v)
Proof
Definitions occuring in Statement :
real-vec-sep: a ≠ b,
real-vec-norm: ||x||,
real-vec: ℝ^n,
rneq: x ≠ y,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
real-vec-sep: a ≠ b,
guard: {T},
subtype_rel: A ⊆r B,
prop: ℙ,
uimplies: b supposing a
Lemmas referenced :
rneq-iff-rabs,
real-vec-norm_wf,
real-vec-dist-lower-bound,
rless_transitivity1,
int-to-real_wf,
rabs_wf,
rsub_wf,
real-vec-dist_wf,
real_wf,
rleq_wf,
rneq_wf,
real-vec_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
productElimination,
independent_functionElimination,
natural_numberEquality,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
sqequalRule,
independent_isectElimination
Latex:
\mforall{}n:\mBbbN{}. \mforall{}x,v:\mBbbR{}\^{}n. (||v|| \mneq{} ||x|| {}\mRightarrow{} x \mneq{} v)
Date html generated:
2017_10_03-AM-11_02_23
Last ObjectModification:
2017_06_19-PM-06_55_43
Theory : reals
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