Nuprl Lemma : implies-geo-between_functionality
∀e:EuclideanPlaneStructure
(BasicGeometryAxioms(e) ⇒ (∀a1,a2,b1,b2,c1,c2:Point. (a1 ≡ a2 ⇒ b1 ≡ b2 ⇒ c1 ≡ c2 ⇒ (B(a1b1c1) ⇐⇒ B(a2b2c2)))))
Proof
Definitions occuring in Statement :
euclidean-plane-structure: EuclideanPlaneStructure,
basic-geo-axioms: BasicGeometryAxioms(g),
geo-eq: a ≡ b,
geo-between: B(abc),
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,
member: t ∈ T,
and: P ∧ Q,
subtype_rel: A ⊆r B,
basic-geo-axioms: BasicGeometryAxioms(g),
cand: A c∧ B,
geo-eq: a ≡ b,
not: ¬A,
false: False,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
geo-between: B(abc),
geo-lsep: a # bc,
or: P ∨ Q,
geo-ge: ab ≥ cd,
geo-sep: a # b,
geo-gt-prim: ab>cd,
record-select: r.x,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Latex:
\mforall{}e:EuclideanPlaneStructure
(BasicGeometryAxioms(e)
{}\mRightarrow{} (\mforall{}a1,a2,b1,b2,c1,c2:Point. (a1 \mequiv{} a2 {}\mRightarrow{} b1 \mequiv{} b2 {}\mRightarrow{} c1 \mequiv{} c2 {}\mRightarrow{} (B(a1b1c1) \mLeftarrow{}{}\mRightarrow{} B(a2b2c2)))))
Date html generated:
2020_05_20-AM-09_43_35
Last ObjectModification:
2020_01_27-PM-03_32_29
Theory : euclidean!plane!geometry
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