Nuprl Lemma : real-vec-sep-implies
∀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],
implies: P ⇒ Q,
apply: f a,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
real-vec-sep: a ≠ b,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
false: False,
prop: ℙ,
real-vec-dist: d(x;y),
real-vec-norm: ||x||,
real-vec-sub: X - Y,
dot-product: x⋅y,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
so_lambda: λ2x.t[x],
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
so_apply: x[s],
less_than: a < b,
squash: ↓T
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,c:\mBbbR{}\^{}n. (a \mneq{} c {}\mRightarrow{} (\mexists{}i:\mBbbN{}n. (r0 < |(a i) - c i|)))
Date html generated:
2020_05_20-PM-00_44_59
Last ObjectModification:
2019_12_14-PM-03_03_49
Theory : reals
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