Nuprl Lemma : real-vec-dist-lower-bound
∀[n:ℕ]. ∀[x,y:ℝ^n]. (|||y|| - ||x||| ≤ d(x;y))
Proof
Definitions occuring in Statement :
real-vec-dist: d(x;y),
real-vec-norm: ||x||,
real-vec: ℝ^n,
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
nat: ℕ,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
real-vec-dist: d(x;y),
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
nat: ℕ,
uimplies: b supposing a,
req-vec: req-vec(n;x;y),
real-vec-sub: X - Y,
rleq: x ≤ y,
rnonneg: rnonneg(x),
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
req_int_terms: t1 ≡ t2,
false: False,
implies: P ⇒ Q,
not: ¬A,
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x,y:\mBbbR{}\^{}n]. (|||y|| - ||x||| \mleq{} d(x;y))
Date html generated:
2020_05_20-PM-00_41_55
Last ObjectModification:
2019_12_14-PM-03_04_23
Theory : reals
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