Nuprl Lemma : Prop22-inequality-implies-triangle
∀e:EuclideanPlane. ∀a,b,c:Point. (|ac| < |ab| + |bc| ⇒ |ab| < |ac| + |bc| ⇒ |bc| < |ac| + |ab| ⇒ a # bc)
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-lsep: a # bc,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
exists: ∃x:A. B[x],
and: P ∧ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
squash: ↓T,
cand: A c∧ B,
oriented-plane: OrientedPlane,
or: P ∨ Q,
false: False,
not: ¬A,
geo-eq: a ≡ b,
stable: Stable{P},
geo-lsep: a # bc,
uiff: uiff(P;Q),
geo-strict-between: a-b-c,
basic-geometry-: BasicGeometry-
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c:Point.
(|ac| < |ab| + |bc| {}\mRightarrow{} |ab| < |ac| + |bc| {}\mRightarrow{} |bc| < |ac| + |ab| {}\mRightarrow{} a \# bc)
Date html generated:
2020_05_20-AM-10_40_10
Last ObjectModification:
2020_01_13-PM-05_11_14
Theory : euclidean!plane!geometry
Home
Index