Nuprl Lemma : rv-circle-circle-lemma3
∀n:{2...}. ∀a,b,c,d:ℝ^n. ∀p:{p:ℝ^n| ab=ap} . ∀q:{q:ℝ^n| cd=cq} . ∀x:{x:ℝ^n| cp=cx ∧ (¬(c ≠ x ∧ x ≠ d ∧ (¬c-x-d)))} .
∀y:{y:ℝ^n| aq=ay ∧ (¬(a ≠ y ∧ y ≠ b ∧ (¬a-y-b)))} .
(a ≠ c ⇒ (∃u,v:{p:ℝ^n| ab=ap ∧ cd=cp} . (((d(a;y) < d(a;b)) ∧ (d(c;x) < d(c;d))) ⇒ u ≠ v)))
Proof
Definitions occuring in Statement :
rv-between: 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,
int_upper: {i...},
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: 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,
nat: ℕ,
int_upper: {i...},
real-vec-sep: a ≠ b,
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
rv-congruent: ab=cd,
sq_stable: SqStable(P),
squash: ↓T,
rev_implies: P ⇐ Q,
guard: {T},
iff: P ⇐⇒ Q,
real-vec-dist: d(x;y),
stable: Stable{P},
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
rge: x ≥ y,
req_int_terms: t1 ≡ t2,
cand: A c∧ B,
rv-between: a-b-c,
true: True,
lelt: i ≤ j < k,
int_seg: {i..j-},
real-vec: ℝ^n,
req-vec: req-vec(n;x;y),
real-vec-sub: X - Y,
real-vec-add: X + Y
Latex:
\mforall{}n:\{2...\}. \mforall{}a,b,c,d:\mBbbR{}\^{}n. \mforall{}p:\{p:\mBbbR{}\^{}n| ab=ap\} . \mforall{}q:\{q:\mBbbR{}\^{}n| cd=cq\} . \mforall{}x:\{x:\mBbbR{}\^{}n|
cp=cx
\mwedge{} (\mneg{}(c \mneq{} x \mwedge{} x \mneq{} d \mwedge{} (\mneg{}c-x-d)))\} \000C.
\mforall{}y:\{y:\mBbbR{}\^{}n| aq=ay \mwedge{} (\mneg{}(a \mneq{} y \mwedge{} y \mneq{} b \mwedge{} (\mneg{}a-y-b)))\} .
(a \mneq{} c {}\mRightarrow{} (\mexists{}u,v:\{p:\mBbbR{}\^{}n| ab=ap \mwedge{} cd=cp\} . (((d(a;y) < d(a;b)) \mwedge{} (d(c;x) < d(c;d))) {}\mRightarrow{} u \mneq{} v)))
Date html generated:
2020_05_20-PM-00_59_09
Last ObjectModification:
2020_01_06-PM-00_51_21
Theory : reals
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