Nuprl Lemma : geo-between-same-side-or
∀e:BasicGeometry. ∀A,B,C,d:Point. ((A # B ∧ C # d) ⇒ B(ABC) ⇒ B(ABd) ⇒ (B(ACd) ∨ B(AdC)))
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-between: B(abc),
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 :
geo-eq: a ≡ b,
stable: Stable{P},
uiff: uiff(P;Q),
exists: ∃x:A. B[x],
basic-geometry-: BasicGeometry-,
euclidean-plane: EuclideanPlane,
iff: P ⇐⇒ Q,
or: P ∨ Q,
false: False,
not: ¬A,
basic-geometry: BasicGeometry,
prop: ℙ,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
geo-between-inner-trans,
geo-between-exchange3,
geo-between-exchange4,
stable__geo-between,
double-negation-hyp-elim,
geo-construction-unicity,
geo-strict-between-sep2,
geo-between-symmetry,
geo-strict-between-implies-between,
geo-congruent-iff-length,
geo-proper-extend-exists,
subtype_rel_self,
geo-strict-between-sep1,
geo-sep-or,
geo-between-same-side,
istype-void,
not_over_or,
not_wf,
iff_weakening_uiff,
geo-strict-between-sep3,
geo-sep-sym,
geo-between-sep,
geo-point_wf,
geo-sep_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-between_wf
Rules used in proof :
inlFormation_alt,
inrFormation_alt,
equalitySymmetry,
unionElimination,
dependent_set_memberEquality_alt,
rename,
setElimination,
unionIsType,
functionIsType,
voidElimination,
productEquality,
unionEquality,
independent_functionElimination,
dependent_functionElimination,
inhabitedIsType,
productIsType,
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
hypothesis,
applyEquality,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
cut,
universeIsType,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:BasicGeometry. \mforall{}A,B,C,d:Point. ((A \# B \mwedge{} C \# d) {}\mRightarrow{} B(ABC) {}\mRightarrow{} B(ABd) {}\mRightarrow{} (B(ACd) \mvee{} B(AdC)))
Date html generated:
2019_10_29-AM-09_14_35
Last ObjectModification:
2019_10_18-PM-03_17_42
Theory : euclidean!plane!geometry
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