Nuprl Lemma : geo-between-same-side2-or-strong
∀e:BasicGeometry. ∀A,B,C,d:Point. ((A ≠ B ∧ C ≠ d) ⇒ A_B_C ⇒ A_B_d ⇒ (B_C_d ∨ B_d_C))
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
or: P ∨ Q,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
basic-geometry-: BasicGeometry-
Lemmas referenced :
geo-between-same-side-or,
geo-between_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-sep_wf,
geo-point_wf,
geo-between-symmetry,
geo-between-inner-trans,
geo-between-exchange3,
subtype_rel_self,
basic-geometry-_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
productElimination,
universeIsType,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
productIsType,
inhabitedIsType,
unionElimination,
inlFormation_alt,
inrFormation_alt
Latex:
\mforall{}e:BasicGeometry. \mforall{}A,B,C,d:Point. ((A \mneq{} B \mwedge{} C \mneq{} d) {}\mRightarrow{} A\_B\_C {}\mRightarrow{} A\_B\_d {}\mRightarrow{} (B\_C\_d \mvee{} B\_d\_C))
Date html generated:
2019_10_16-PM-01_18_13
Last ObjectModification:
2019_07_24-PM-09_31_18
Theory : euclidean!plane!geometry
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