Nuprl Lemma : rn-metric-sep
∀n:ℕ. ∀x,y:ℝ^n. (r0 < mdist(rn-metric(n);x;y) ⇐⇒ ∃i:ℕn. x i ≠ y i)
Proof
Definitions occuring in Statement :
rn-metric: rn-metric(n),
real-vec: ℝ^n,
mdist: mdist(d;x;y),
rneq: x ≠ y,
rless: 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 :
all: ∀x:A. B[x],
rn-metric: rn-metric(n),
mdist: mdist(d;x;y),
real-vec-sep: a ≠ b,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
nat: ℕ,
real-vec: ℝ^n
Lemmas referenced :
real-vec-sep-iff-rneq,
real-vec-sep_wf,
int_seg_wf,
rneq_wf,
real-vec_wf,
istype-nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalRule,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_functionElimination,
hypothesis,
universeIsType,
isectElimination,
productIsType,
natural_numberEquality,
setElimination,
rename,
applyEquality,
because_Cache
Latex:
\mforall{}n:\mBbbN{}. \mforall{}x,y:\mBbbR{}\^{}n. (r0 < mdist(rn-metric(n);x;y) \mLeftarrow{}{}\mRightarrow{} \mexists{}i:\mBbbN{}n. x i \mneq{} y i)
Date html generated:
2019_10_30-AM-08_43_16
Last ObjectModification:
2019_10_02-AM-10_26_17
Theory : reals
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