Nuprl Lemma : geo-colinear_functionality
∀e:EuclideanPlane. ∀a1,a2,b1,b2,c1,c2:Point.
(a1 ≡ a2 ⇒ b1 ≡ b2 ⇒ c1 ≡ c2 ⇒ (Colinear(a1;b1;c1) ⇐⇒ Colinear(a2;b2;c2)))
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-colinear: Colinear(a;b;c),
geo-eq: a ≡ b,
geo-point: Point,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
geo-colinear: Colinear(a;b;c),
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
rev_implies: P ⇐ Q,
not: ¬A,
false: False
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a1,a2,b1,b2,c1,c2:Point.
(a1 \mequiv{} a2 {}\mRightarrow{} b1 \mequiv{} b2 {}\mRightarrow{} c1 \mequiv{} c2 {}\mRightarrow{} (Colinear(a1;b1;c1) \mLeftarrow{}{}\mRightarrow{} Colinear(a2;b2;c2)))
Date html generated:
2020_05_20-AM-09_47_45
Last ObjectModification:
2019_11_12-AM-10_47_47
Theory : euclidean!plane!geometry
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