Nuprl Lemma : left-convex
∀g:EuclideanPlane. ∀a,b,x,y:Point. (x leftof ab ⇒ (B(bxy) ∨ (B(byx) ∧ y # b)) ⇒ y leftof ab)
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-between: B(abc),
geo-left: a leftof bc,
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 :
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
and: P ∧ Q,
geo-between: B(abc),
geo-colinear: Colinear(a;b;c),
not: ¬A,
euclidean-plane: EuclideanPlane,
cand: A c∧ B,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
select: L[n],
cons: [a / b],
subtract: n - m,
geo-lsep: a # bc,
stable: Stable{P},
geo-eq: a ≡ b,
iff: P ⇐⇒ Q,
l_member: (x ∈ l),
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
squash: ↓T,
true: True,
ge: i ≥ j ,
append: as @ bs,
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
rev_implies: P ⇐ Q
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,x,y:Point. (x leftof ab {}\mRightarrow{} (B(bxy) \mvee{} (B(byx) \mwedge{} y \# b)) {}\mRightarrow{} y leftof ab)
Date html generated:
2020_05_20-AM-09_48_04
Last ObjectModification:
2019_11_13-PM-03_27_59
Theory : euclidean!plane!geometry
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