Nuprl Lemma : rv-line-circle-3
∀n:ℕ. ∀c,b,d,q:ℝ^n.
(c_b_d
⇒ (d(c;d) < d(c;q))
⇒ (∃u:{u:ℝ^n| cd=cu ∧ q_u_b}
(∃v:ℝ^n [(cd=cv
∧ q_b_v
∧ (b ≠ d ⇒ (u ≠ v ∧ u ≠ b ∧ v ≠ b))
∧ (req-vec(n;b;d)
⇒ ((u ≠ v ⇒ ((req-vec(n;u;b) ∧ (r0 < b - c⋅q - b)) ∨ (req-vec(n;v;b) ∧ (b - c⋅q - b < r0))))
∧ (req-vec(n;u;v) ⇒ ((b - c⋅q - b = r0) ∧ req-vec(n;u;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),
dot-product: x⋅y,
real-vec-sub: X - Y,
req-vec: req-vec(n;x;y),
real-vec: ℝ^n,
rless: x < y,
req: x = y,
int-to-real: r(n),
nat: ℕ,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
guard: {T},
subtype_rel: A ⊆r B,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
prop: ℙ,
real-vec-sep: a ≠ b,
rev_implies: P ⇐ Q,
rge: x ≥ y,
rv-be: a_b_c,
exists: ∃x:A. B[x],
sq_exists: ∃x:A [B[x]],
cand: A c∧ B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
or: P ∨ Q,
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
rless: x < y,
rv-between: a-b-c
Latex:
\mforall{}n:\mBbbN{}. \mforall{}c,b,d,q:\mBbbR{}\^{}n.
(c\_b\_d
{}\mRightarrow{} (d(c;d) < d(c;q))
{}\mRightarrow{} (\mexists{}u:\{u:\mBbbR{}\^{}n| cd=cu \mwedge{} q\_u\_b\}
(\mexists{}v:\mBbbR{}\^{}n [(cd=cv
\mwedge{} q\_b\_v
\mwedge{} (b \mneq{} d {}\mRightarrow{} (u \mneq{} v \mwedge{} u \mneq{} b \mwedge{} v \mneq{} b))
\mwedge{} (req-vec(n;b;d)
{}\mRightarrow{} ((u \mneq{} v
{}\mRightarrow{} ((req-vec(n;u;b) \mwedge{} (r0 < b - c\mcdot{}q - b))
\mvee{} (req-vec(n;v;b) \mwedge{} (b - c\mcdot{}q - b < r0))))
\mwedge{} (req-vec(n;u;v) {}\mRightarrow{} ((b - c\mcdot{}q - b = r0) \mwedge{} req-vec(n;u;b))))))])))
Date html generated:
2020_05_20-PM-00_56_53
Last ObjectModification:
2020_01_06-PM-00_08_16
Theory : reals
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