Nuprl Lemma : geo-ge_functionality
∀e:EuclideanPlane. ∀a1,a2,b1,b2,c1,c2,d1,d2:Point.
(a1 ≡ a2 ⇒ b1 ≡ b2 ⇒ c1 ≡ c2 ⇒ d1 ≡ d2 ⇒ (a1b1 ≥ c1d1 ⇐⇒ a2b2 ≥ c2d2))
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-ge: ab ≥ cd,
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,
geo-ge: ab ≥ cd,
member: t ∈ T,
euclidean-plane: EuclideanPlane,
sq_stable: SqStable(P),
squash: ↓T,
and: P ∧ Q,
iff: P ⇐⇒ Q,
not: ¬A,
false: False,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
rev_implies: P ⇐ Q,
guard: {T},
uimplies: b supposing a,
basic-geo-axioms: BasicGeometryAxioms(g),
cand: A c∧ B,
geo-congruent: ab ≅ cd,
geo-length-sep: ab # cd),
or: P ∨ Q
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a1,a2,b1,b2,c1,c2,d1,d2:Point.
(a1 \mequiv{} a2 {}\mRightarrow{} b1 \mequiv{} b2 {}\mRightarrow{} c1 \mequiv{} c2 {}\mRightarrow{} d1 \mequiv{} d2 {}\mRightarrow{} (a1b1 \mgeq{} c1d1 \mLeftarrow{}{}\mRightarrow{} a2b2 \mgeq{} c2d2))
Date html generated:
2020_05_20-AM-09_46_41
Last ObjectModification:
2020_01_27-PM-10_23_29
Theory : euclidean!plane!geometry
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