Nuprl Lemma : Euclid-Prop21
∀g:EuclideanPlane. ∀a,b,c,d:Point. (I(abc;d) ⇒ {|cd| + |bd| < |ba| + |ac| ∧ bac < bdc})
Proof
Definitions occuring in Statement :
geo-interior-point: I(abc;d),
geo-lt-angle: abc < xyz,
geo-lt: p < q,
geo-add-length: p + q,
geo-length: |s|,
geo-mk-seg: ab,
euclidean-plane: EuclideanPlane,
geo-point: Point,
guard: {T},
all: ∀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,
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
geo-interior-point: I(abc;d),
and: P ∧ Q,
exists: ∃x:A. B[x],
geo-strict-between: a-b-c,
cand: A c∧ B,
basic-geometry-: BasicGeometry-,
euclidean-plane: EuclideanPlane,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
not: ¬A,
false: False,
subtract: n - m,
cons: [a / b],
select: L[n],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
lelt: i ≤ j < k,
int_seg: {i..j-},
top: Top,
l_all: (∀x∈L.P[x]),
geo-colinear-set: geo-colinear-set(e; L),
basic-geometry: BasicGeometry,
oriented-plane: OrientedPlane,
geo-lsep: a # bc,
squash: ↓T,
true: True,
geo-zero-length: 0,
geo-length-type: Length,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,d:Point. (I(abc;d) {}\mRightarrow{} \{|cd| + |bd| < |ba| + |ac| \mwedge{} bac < bdc\})
Date html generated:
2020_05_20-AM-10_39_20
Last ObjectModification:
2020_01_13-PM-04_53_02
Theory : euclidean!plane!geometry
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