Nuprl Lemma : left-convex2
∀g:EuclideanPlane. ∀a,b,x,y:Point. (x leftof ab ⇒ (a_x_y ∨ (a_y_x ∧ y ≠ a)) ⇒ y leftof ab)
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
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,
or: P ∨ Q,
and: P ∧ Q
Definitions unfolded in proof :
and: P ∧ Q,
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],
or: P ∨ Q,
geo-lsep: a # bc,
subtract: n - m,
cons: [a / b],
select: L[n],
true: True,
squash: ↓T,
less_than: a < b,
not: ¬A,
false: False,
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),
cand: A c∧ B
Lemmas referenced :
geo-point_wf,
geo-left_wf,
geo-sep_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-between_wf,
or_wf,
left-symmetry,
lelt_wf,
false_wf,
length_of_nil_lemma,
length_of_cons_lemma,
geo-between-implies-colinear,
geo-colinear-is-colinear-set,
left-implies-sep,
geo-between-sep,
geo-sep-sym,
colinear-lsep,
lsep-symmetry,
not-left-and-right,
left-convex
Rules used in proof :
because_Cache,
productEquality,
sqequalRule,
independent_isectElimination,
instantiate,
hypothesis,
applyEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_functionElimination,
dependent_functionElimination,
inlFormation,
unionElimination,
baseClosed,
imageMemberEquality,
independent_pairFormation,
natural_numberEquality,
dependent_set_memberEquality,
voidEquality,
voidElimination,
isect_memberEquality,
productElimination,
inrFormation
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,x,y:Point. (x leftof ab {}\mRightarrow{} (a\_x\_y \mvee{} (a\_y\_x \mwedge{} y \mneq{} a)) {}\mRightarrow{} y leftof ab)
Date html generated:
2017_10_02-PM-04_40_41
Last ObjectModification:
2017_08_08-PM-03_52_22
Theory : euclidean!plane!geometry
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