Nuprl Lemma : plane-sep-imp-Opasch_left
∀e:EuclideanPlane. ∀a,b:Point. ∀c:{c:Point| B(abc)} . ∀x:Point. ∀y:{y:Point| b-x-y} .
(x leftof ab ⇒ (∃p:Point [(B(axp) ∧ B(cpy))]))
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-strict-between: a-b-c,
geo-between: B(abc),
geo-left: a leftof bc,
geo-point: Point,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
euclidean-plane: EuclideanPlane,
uall: ∀[x:A]. B[x],
sq_stable: SqStable(P),
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
or: P ∨ Q,
and: P ∧ Q,
oriented-plane: OrientedPlane,
basic-geometry-: BasicGeometry-,
exists: ∃x:A. B[x],
sq_exists: ∃x:A [B[x]],
cand: A c∧ B,
not: ¬A,
false: False,
stable: Stable{P},
geo-eq: a ≡ b,
iff: P ⇐⇒ Q,
basic-geometry: BasicGeometry,
geo-lsep: a # bc,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
select: L[n],
cons: [a / b],
subtract: n - m,
geo-strict-between: a-b-c,
geo-colinear: Colinear(a;b;c),
rev_implies: P ⇐ Q
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b:Point. \mforall{}c:\{c:Point| B(abc)\} . \mforall{}x:Point. \mforall{}y:\{y:Point| b-x-y\} .
(x leftof ab {}\mRightarrow{} (\mexists{}p:Point [(B(axp) \mwedge{} B(cpy))]))
Date html generated:
2020_05_20-AM-10_01_17
Last ObjectModification:
2019_12_03-AM-09_53_01
Theory : euclidean!plane!geometry
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