Nuprl Lemma : rv-circle-circle-lemma3'
∀a,b,c,d:ℝ^2. ∀p:{p:ℝ^2| ab=ap} . ∀q:{q:ℝ^2| cd=cq} . ∀x:{x:ℝ^2| cp=cx ∧ (¬(c ≠ x ∧ x ≠ d ∧ (¬c-x-d)))} .
∀y:{y:ℝ^2| aq=ay ∧ (¬(a ≠ y ∧ y ≠ b ∧ (¬a-y-b)))} .
(a ≠ c
⇒ (∃u,v:{p:ℝ^2| ab=ap ∧ cd=cp} . (((d(a;y) < d(a;b)) ∧ (d(c;x) < d(c;d))) ⇒ (r2-left(u;c;a) ∧ r2-left(v;a;c)))))
Proof
Definitions occuring in Statement :
r2-left: r2-left(p;q;r),
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,
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,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
false: False,
prop: ℙ,
rv-congruent: ab=cd,
subtype_rel: A ⊆r B,
sq_stable: SqStable(P),
squash: ↓T,
real-vec-dist: d(x;y),
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
rev_implies: P ⇐ Q,
stable: Stable{P},
or: P ∨ Q,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
nat_plus: ℕ+,
sq_exists: ∃x:A [B[x]],
rless: x < y,
real-vec-sep: a ≠ b,
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,
less_than: a < b,
r2-left: r2-left(p;q;r),
r2-det: |pqr|
Latex:
\mforall{}a,b,c,d:\mBbbR{}\^{}2. \mforall{}p:\{p:\mBbbR{}\^{}2| ab=ap\} . \mforall{}q:\{q:\mBbbR{}\^{}2| cd=cq\} . \mforall{}x:\{x:\mBbbR{}\^{}2|
cp=cx \mwedge{} (\mneg{}(c \mneq{} x \mwedge{} x \mneq{} d \mwedge{} (\mneg{}c-x-d)))\} .
\mforall{}y:\{y:\mBbbR{}\^{}2| 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{}\^{}2| ab=ap \mwedge{} cd=cp\}
(((d(a;y) < d(a;b)) \mwedge{} (d(c;x) < d(c;d))) {}\mRightarrow{} (r2-left(u;c;a) \mwedge{} r2-left(v;a;c)))))
Date html generated:
2020_05_20-PM-01_01_04
Last ObjectModification:
2020_01_06-PM-00_31_47
Theory : reals
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