Nuprl Lemma : geo-colinear-left-out3
∀e:EuclideanPlane. ∀a,b,c,a',c',x,x':Point.
(Colinear(b;x;x') ⇒ a-x-c ⇒ a'-x'-c' ⇒ out(b aa') ⇒ out(b cc') ⇒ b leftof ac ⇒ b leftof a'c' ⇒ out(b xx'))
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
euclidean-plane: EuclideanPlane,
geo-colinear: Colinear(a;b;c),
geo-strict-between: a-b-c,
geo-left: a leftof 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,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
guard: {T},
uimplies: b supposing a,
basic-geometry-: BasicGeometry-,
geo-out: out(p ab),
and: P ∧ Q,
cand: A c∧ B,
not: ¬A,
geo-eq: a ≡ b,
prop: ℙ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
false: False,
basic-geometry: BasicGeometry,
geo-strict-between: a-b-c,
geo-sep: a # b,
subtract: n - m,
cons: [a / b],
select: L[n],
true: True,
squash: ↓T,
less_than: a < b,
less_than': less_than'(a;b),
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
top: Top,
l_all: (∀x∈L.P[x]),
geo-colinear-set: geo-colinear-set(e; L)
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,a',c',x,x':Point.
(Colinear(b;x;x')
{}\mRightarrow{} a-x-c
{}\mRightarrow{} a'-x'-c'
{}\mRightarrow{} out(b aa')
{}\mRightarrow{} out(b cc')
{}\mRightarrow{} b leftof ac
{}\mRightarrow{} b leftof a'c'
{}\mRightarrow{} out(b xx'))
Date html generated:
2020_05_20-AM-09_59_23
Last ObjectModification:
2020_01_13-PM-03_43_57
Theory : euclidean!plane!geometry
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