Nuprl Lemma : left-convex3
∀e:OrientedPlane. ∀b,p,q,x,z:Point. (q leftof pb ⇒ Colinear(p;b;z) ⇒ z_q_x ⇒ x leftof pb)
Proof
Definitions occuring in Statement :
oriented-plane: OrientedPlane,
geo-colinear: Colinear(a;b;c),
geo-left: a leftof bc,
geo-between: a_b_c,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
and: P ∧ Q,
oriented-plane: OrientedPlane,
or: P ∨ Q,
euclidean-plane: EuclideanPlane,
false: False,
cand: A c∧ B,
not: ¬A,
geo-colinear: Colinear(a;b;c),
geo-lsep: a # bc,
subtract: n - m,
cons: [a / b],
select: L[n],
true: True,
squash: ↓T,
less_than: a < b,
less_than': less_than'(a;b),
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
top: Top,
l_all: (∀x∈L.P[x]),
geo-colinear-set: geo-colinear-set(e; L),
basic-geometry-: BasicGeometry-,
exists: ∃x:A. B[x],
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q
Lemmas referenced :
geo-point_wf,
geo-left_wf,
geo-colinear_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
oriented-plane_wf,
subtype_rel_transitivity,
oriented-plane-subtype,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-between_wf,
left-implies-sep,
euclidean-plane-axioms,
geo-sep_wf,
geo-sep-or,
not_wf,
geo-between-symmetry,
colinear-lsep',
colinear-lsep,
geo-between-sep,
geo-sep-sym,
lsep-implies-sep,
lelt_wf,
false_wf,
length_of_nil_lemma,
length_of_cons_lemma,
geo-left-axioms_wf,
basic-geo-axioms_wf,
subtype_rel_self,
geo-between-implies-colinear,
geo-colinear-is-colinear-set,
not-left-and-right,
lsep-opposite-iff,
geo-between-exchange3,
oriented-colinear-trans,
not-lsep-iff-colinear
Rules used in proof :
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
hypothesis,
applyEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_functionElimination,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_set_memberEquality,
rename,
setElimination,
productEquality,
voidElimination,
independent_pairFormation,
inrFormation,
baseClosed,
imageMemberEquality,
natural_numberEquality,
voidEquality,
isect_memberEquality,
cumulativity,
setEquality,
inlFormation
Latex:
\mforall{}e:OrientedPlane. \mforall{}b,p,q,x,z:Point. (q leftof pb {}\mRightarrow{} Colinear(p;b;z) {}\mRightarrow{} z\_q\_x {}\mRightarrow{} x leftof pb)
Date html generated:
2017_10_02-PM-04_48_01
Last ObjectModification:
2017_08_07-PM-00_12_38
Theory : euclidean!plane!geometry
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