Nuprl Lemma : circle-not-colinear
∀e:EuclideanPlane. ∀a,b,c,d:Point. (ab ≅ ac ⇒ ab ≅ ad ⇒ b ≠ c ⇒ c ≠ d ⇒ d ≠ b ⇒ (¬Colinear(b;c;d)))
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],
not: ¬A,
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
false: False,
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
prop: ℙ,
and: P ∧ Q,
cand: A c∧ B,
basic-geometry: BasicGeometry,
uiff: uiff(P;Q),
uimplies: b supposing a,
guard: {T}
Lemmas referenced :
no-three-colinear-on-circle,
geo-sep-sym,
geo-sep_wf,
geo-colinear_wf,
geo-congruent-iff-length,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-congruent_wf,
geo-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
dependent_functionElimination,
independent_functionElimination,
hypothesis,
dependent_set_memberEquality,
because_Cache,
applyEquality,
sqequalRule,
independent_pairFormation,
productEquality,
productElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
voidElimination,
instantiate
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,d:Point.
(ab \mcong{} ac {}\mRightarrow{} ab \mcong{} ad {}\mRightarrow{} b \mneq{} c {}\mRightarrow{} c \mneq{} d {}\mRightarrow{} d \mneq{} b {}\mRightarrow{} (\mneg{}Colinear(b;c;d)))
Date html generated:
2018_05_22-PM-00_12_26
Last ObjectModification:
2018_03_30-PM-09_23_36
Theory : euclidean!plane!geometry
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