Nuprl Lemma : upper-dimension-axiom
∀g:EuclideanPlane. ∀a,b,c,x,y:Point. (ax ≅ ay ⇒ bx ≅ by ⇒ cx ≅ cy ⇒ x # y ⇒ Colinear(a;b;c))
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-colinear: Colinear(a;b;c),
geo-congruent: ab ≅ cd,
geo-sep: a # b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
geo-colinear: Colinear(a;b;c),
cand: A c∧ B,
and: P ∧ Q,
basic-geo-axioms: BasicGeometryAxioms(g),
squash: ↓T,
implies: P ⇒ Q,
sq_stable: SqStable(P),
member: t ∈ T,
euclidean-plane: EuclideanPlane,
all: ∀x:A. B[x]
Lemmas referenced :
euclidean-plane_wf,
sq_stable__geo-axioms
Rules used in proof :
universeIsType,
productElimination,
imageElimination,
baseClosed,
imageMemberEquality,
sqequalRule,
independent_functionElimination,
hypothesis,
hypothesisEquality,
dependent_functionElimination,
extract_by_obid,
introduction,
rename,
thin,
setElimination,
sqequalHypSubstitution,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,x,y:Point. (ax \mcong{} ay {}\mRightarrow{} bx \mcong{} by {}\mRightarrow{} cx \mcong{} cy {}\mRightarrow{} x \# y {}\mRightarrow{} Colinear(a;b;c))
Date html generated:
2019_10_30-AM-06_18_22
Last ObjectModification:
2019_10_29-PM-02_56_42
Theory : euclidean!plane!geometry
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