Nuprl Lemma : p5-triangles
∀e:HeytingGeometry. ∀a,b,c:Point. (a # bc ⇒ ab ≅ cb ⇒ (bac ≅a bca ∧ (∀p,q:Point. ((b-a-p ∧ b-c-q) ⇒ pac ≅a qca))))
Proof
Definitions occuring in Statement :
geo-triangle: a # bc,
heyting-geometry: HeytingGeometry,
geo-cong-angle: abc ≅a xyz,
geo-strict-between: a-b-c,
geo-congruent: ab ≅ cd,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
geo-cong-angle: abc ≅a xyz,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
heyting-geometry: HeytingGeometry,
uiff: uiff(P;Q),
exists: ∃x:A. B[x],
euclidean-plane: EuclideanPlane,
basic-geometry-: BasicGeometry-,
squash: ↓T,
true: True
Latex:
\mforall{}e:HeytingGeometry. \mforall{}a,b,c:Point.
(a \# bc {}\mRightarrow{} ab \mcong{} cb {}\mRightarrow{} (bac \mcong{}\msuba{} bca \mwedge{} (\mforall{}p,q:Point. ((b-a-p \mwedge{} b-c-q) {}\mRightarrow{} pac \mcong{}\msuba{} qca))))
Date html generated:
2020_05_20-AM-10_33_46
Last ObjectModification:
2020_01_27-PM-09_59_22
Theory : euclidean!plane!geometry
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