Nuprl Lemma : interior-implies-lt-angle
∀e:EuclideanPlane. ∀a,b,c,x,y,z:Point.
(x # yz ⇒ c leftof ba ⇒ (∃f:Point. ((f leftof ba ∧ f leftof cb) ∧ abf ≅a xyz)) ⇒ xyz < abc)
Proof
Definitions occuring in Statement :
geo-lt-angle: abc < xyz,
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
geo-left: a leftof bc,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
and: P ∧ Q,
geo-lt-angle: abc < xyz,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
cand: A c∧ B,
not: ¬A,
false: False,
geo-out: out(p ab),
iff: P ⇐⇒ Q,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
top: Top,
int_seg: {i..j-},
rev_implies: P ⇐ Q,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
select: L[n],
cons: [a / b],
subtract: n - m,
basic-geometry-: BasicGeometry-,
geo-eq: a ≡ b,
geo-strict-between: a-b-c,
oriented-plane: OrientedPlane
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,x,y,z:Point.
(x \# yz {}\mRightarrow{} c leftof ba {}\mRightarrow{} (\mexists{}f:Point. ((f leftof ba \mwedge{} f leftof cb) \mwedge{} abf \mcong{}\msuba{} xyz)) {}\mRightarrow{} xyz < abc)
Date html generated:
2020_05_20-AM-10_06_06
Last ObjectModification:
2020_01_13-PM-04_02_44
Theory : euclidean!plane!geometry
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