Nuprl Lemma : geo-tar-opp-side-iff
∀e:BasicGeometry. ∀A,B,P,Q:Point. (geo-tar-opp-side(e;P;Q;A;B) ⇐⇒ P # AB ∧ Q # AB ∧ (P leftof AB ⇐⇒ Q leftof BA))
Proof
Definitions occuring in Statement :
geo-tar-opp-side: geo-tar-opp-side(e;a;b;p;q),
basic-geometry: BasicGeometry,
geo-lsep: a # bc,
geo-left: a leftof bc,
geo-point: Point,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
rev_implies: P ⇐ Q,
geo-tar-opp-side: geo-tar-opp-side(e;a;b;p;q),
exists: ∃x:A. B[x],
geo-lsep: a # bc,
or: P ∨ Q,
cand: A c∧ B,
basic-geometry: BasicGeometry,
not: ¬A,
false: False,
euclidean-plane: EuclideanPlane,
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
geo-left_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-tar-opp-side_wf,
geo-lsep_wf,
geo-point_wf,
left-between,
euclidean-plane-subtype-oriented,
oriented-plane_wf,
geo-between_wf,
geo-between-symmetry,
lsep-all-sym2,
not-lsep-iff-colinear,
lsep-all-sym,
geo-SS_wf,
sq_stable__colinear,
sq_stable__geo-between,
geo-colinear_wf,
not-left-and-right
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
productElimination,
productIsType,
functionIsType,
inhabitedIsType,
unionElimination,
dependent_functionElimination,
independent_functionElimination,
voidElimination,
setElimination,
rename,
dependent_set_memberEquality_alt,
dependent_pairFormation_alt,
imageMemberEquality,
baseClosed,
imageElimination,
equalityIstype,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}e:BasicGeometry. \mforall{}A,B,P,Q:Point.
(geo-tar-opp-side(e;P;Q;A;B) \mLeftarrow{}{}\mRightarrow{} P \# AB \mwedge{} Q \# AB \mwedge{} (P leftof AB \mLeftarrow{}{}\mRightarrow{} Q leftof BA))
Date html generated:
2019_10_16-PM-01_21_27
Last ObjectModification:
2018_12_11-PM-00_19_25
Theory : euclidean!plane!geometry
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