Nuprl Lemma : rv-gt-dist
∀n:ℕ. ∀a,b,q:ℝ^n. ((d(a;q) < d(a;b)) ⇒ (∃w:ℝ^n. (a_w_b ∧ aw=aq ∧ w ≠ b)))
Proof
Definitions occuring in Statement :
rv-be: a_b_c,
real-vec-sep: a ≠ b,
rv-congruent: ab=cd,
real-vec-dist: d(x;y),
real-vec: ℝ^n,
rless: x < y,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T},
uimplies: b supposing a,
rneq: x ≠ y,
or: P ∨ Q,
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
real-vec-sep: a ≠ b,
rless: x < y,
sq_exists: ∃x:A [B[x]],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
top: Top,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
real-vec-be: real-vec-be(n;a;b;c),
squash: ↓T,
true: True,
real-vec-mul: a*X,
real-vec-add: X + Y,
real-vec-sub: X - Y,
req-vec: req-vec(n;x;y),
nat: ℕ,
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
rv-be: a_b_c,
rv-congruent: ab=cd,
real-vec-dist: d(x;y),
rat_term_to_real: rat_term_to_real(f;t),
rtermVar: rtermVar(var),
rat_term_ind: rat_term_ind,
pi1: fst(t),
rtermMultiply: left "*" right,
rtermDivide: num "/" denom,
pi2: snd(t)
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b,q:\mBbbR{}\^{}n. ((d(a;q) < d(a;b)) {}\mRightarrow{} (\mexists{}w:\mBbbR{}\^{}n. (a\_w\_b \mwedge{} aw=aq \mwedge{} w \mneq{} b)))
Date html generated:
2020_05_20-PM-00_56_23
Last ObjectModification:
2019_12_11-PM-03_24_21
Theory : reals
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