Nuprl Lemma : geo-between_functionality
∀e:EuclideanPlane. ∀a1,a2,b1,b2,c1,c2:Point.  (a1 ≡ a2 
⇒ b1 ≡ b2 
⇒ c1 ≡ c2 
⇒ (a1_b1_c1 
⇐⇒ a2_b2_c2))
Proof
Definitions occuring in Statement : 
euclidean-plane: EuclideanPlane
, 
geo-eq: a ≡ b
, 
geo-between: a_b_c
, 
geo-point: Point
, 
all: ∀x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
implies: P 
⇒ Q
Definitions unfolded in proof : 
squash: ↓T
, 
and: P ∧ Q
, 
sq_stable: SqStable(P)
, 
implies: P 
⇒ Q
, 
euclidean-plane: EuclideanPlane
, 
member: t ∈ T
, 
all: ∀x:A. B[x]
Lemmas referenced : 
sq_stable__geo-axioms, 
euclidean-plane_wf, 
implies-geo-between_functionality
Rules used in proof : 
imageElimination, 
baseClosed, 
imageMemberEquality, 
sqequalRule, 
productElimination, 
independent_functionElimination, 
hypothesisEquality, 
rename, 
setElimination, 
thin, 
dependent_functionElimination, 
sqequalHypSubstitution, 
hypothesis, 
lambdaFormation, 
sqequalReflexivity, 
computationStep, 
sqequalTransitivity, 
sqequalSubstitution, 
extract_by_obid, 
introduction, 
cut
Latex:
\mforall{}e:EuclideanPlane.  \mforall{}a1,a2,b1,b2,c1,c2:Point.
    (a1  \mequiv{}  a2  {}\mRightarrow{}  b1  \mequiv{}  b2  {}\mRightarrow{}  c1  \mequiv{}  c2  {}\mRightarrow{}  (a1\_b1\_c1  \mLeftarrow{}{}\mRightarrow{}  a2\_b2\_c2))
Date html generated:
2018_05_22-AM-11_53_12
Last ObjectModification:
2018_05_21-AM-01_13_12
Theory : euclidean!plane!geometry
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