Nuprl Lemma : Euclid-Prop22
∀e:EuclideanPlane. ∀a1,a2,b1,b2,c1,c2:Point.
(|a1a2| < |b1b2| + |c1c2|
⇒ |b1b2| < |a1a2| + |c1c2|
⇒ |c1c2| < |a1a2| + |b1b2|
⇒ (∃a,b,c:Point. (((a # bc ∧ ab ≅ a1a2) ∧ bc ≅ b1b2) ∧ ca ≅ c1c2)))
Proof
Definitions occuring in Statement :
geo-lt: p < q,
geo-add-length: p + q,
geo-length: |s|,
geo-mk-seg: ab,
euclidean-plane: EuclideanPlane,
geo-congruent: ab ≅ cd,
geo-lsep: a # bc,
geo-point: Point,
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,
basic-geometry: BasicGeometry,
squash: ↓T,
uall: ∀[x:A]. B[x],
prop: ℙ,
euclidean-plane: EuclideanPlane,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
exists: ∃x:A. B[x],
uiff: uiff(P;Q),
basic-geometry-: BasicGeometry-,
geo-strict-between: a-b-c,
or: P ∨ Q,
l_member: (x ∈ l),
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
false: False,
select: L[n],
cons: [a / b],
subtract: n - m,
less_than: a < b,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
append: as @ bs,
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
geo-eq: a ≡ b
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a1,a2,b1,b2,c1,c2:Point.
(|a1a2| < |b1b2| + |c1c2|
{}\mRightarrow{} |b1b2| < |a1a2| + |c1c2|
{}\mRightarrow{} |c1c2| < |a1a2| + |b1b2|
{}\mRightarrow{} (\mexists{}a,b,c:Point. (((a \# bc \mwedge{} ab \mcong{} a1a2) \mwedge{} bc \mcong{} b1b2) \mwedge{} ca \mcong{} c1c2)))
Date html generated:
2020_05_20-AM-10_39_49
Last ObjectModification:
2020_01_28-AM-10_06_49
Theory : euclidean!plane!geometry
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