Nuprl Lemma : left-between-implies-right1
∀g:OrientedPlane. ∀a,b,x,y:Point. (x leftof ab ⇒ x_b_y ⇒ y ≠ b ⇒ y leftof ba)
Proof
Definitions occuring in Statement :
oriented-plane: OrientedPlane,
geo-left: a leftof bc,
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
oriented-plane: OrientedPlane,
guard: {T},
and: P ∧ Q,
cand: A c∧ B,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
less_than: a < b,
squash: ↓T,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
exists: ∃x:A. B[x]
Lemmas referenced :
lsep-opposite-iff,
lsep-all-sym2,
colinear-lsep-cycle,
geo-colinear-is-colinear-set,
geo-between-implies-colinear,
length_of_cons_lemma,
length_of_nil_lemma,
false_wf,
lelt_wf,
lsep-all-sym,
geo-sep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
oriented-plane-subtype,
subtype_rel_transitivity,
oriented-plane_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-between_wf,
geo-left_wf,
geo-point_wf,
geo-colinear-same,
euclidean-plane-subtype-basic,
basic-geometry_wf,
geo-colinear_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
because_Cache,
hypothesis,
productElimination,
isectElimination,
independent_isectElimination,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
applyEquality,
instantiate,
dependent_pairFormation,
productEquality
Latex:
\mforall{}g:OrientedPlane. \mforall{}a,b,x,y:Point. (x leftof ab {}\mRightarrow{} x\_b\_y {}\mRightarrow{} y \mneq{} b {}\mRightarrow{} y leftof ba)
Date html generated:
2018_05_22-AM-11_54_44
Last ObjectModification:
2018_04_17-PM-05_54_20
Theory : euclidean!plane!geometry
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