Nuprl Lemma : colinear-implies-congruent_or_midpoint
∀e:BasicGeometry. ∀M,A,B:Point. (Colinear(A;M;B) ⇒ MA ≅ MB ⇒ (¬¬(A ≡ B ∨ A=M=B)))
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-midpoint: a=m=b,
geo-colinear: Colinear(a;b;c),
geo-eq: a ≡ b,
geo-congruent: ab ≅ cd,
geo-point: Point,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
false: False,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
stable: Stable{P},
or: P ∨ Q,
geo-eq: a ≡ b
Lemmas referenced :
not_wf,
or_wf,
geo-eq_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
basic-geometry-subtype,
subtype_rel_transitivity,
basic-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-midpoint_wf,
geo-congruent_wf,
geo-colinear_wf,
geo-point_wf,
stable__false,
false_wf,
geo-sep_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
colinear-implies-midpoint,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
functionEquality,
unionElimination,
inlFormation,
dependent_functionElimination,
inrFormation
Latex:
\mforall{}e:BasicGeometry. \mforall{}M,A,B:Point. (Colinear(A;M;B) {}\mRightarrow{} MA \mcong{} MB {}\mRightarrow{} (\mneg{}\mneg{}(A \mequiv{} B \mvee{} A=M=B)))
Date html generated:
2018_05_22-PM-00_01_53
Last ObjectModification:
2018_03_27-PM-02_13_08
Theory : euclidean!plane!geometry
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