Nuprl Lemma : geo-colinear-swap
∀e:EuclideanPlane. ∀a,b,c:Point. (Colinear(a;b;c) ⇒ Colinear(b;a;c))
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-colinear: Colinear(a;b;c),
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
geo-colinear: Colinear(a;b;c),
not: ¬A,
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
prop: ℙ,
false: False,
guard: {T}
Lemmas referenced :
geo-between-symmetry,
geo-between_wf,
istype-void,
geo-colinear_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
introduction,
independent_functionElimination,
thin,
productElimination,
cut,
hypothesis,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
isectElimination,
independent_isectElimination,
universeIsType,
applyEquality,
because_Cache,
sqequalRule,
independent_pairFormation,
voidElimination,
productIsType,
functionIsType,
instantiate,
inhabitedIsType
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c:Point. (Colinear(a;b;c) {}\mRightarrow{} Colinear(b;a;c))
Date html generated:
2019_10_16-PM-01_14_27
Last ObjectModification:
2018_11_12-PM-03_10_39
Theory : euclidean!plane!geometry
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