Nuprl Lemma : interior-angles-unique
∀e:EuclideanPlane. ∀a,b,c,d,b',c',p:Point.
(a # bc ⇒ out(b dc) ⇒ out(a bb') ⇒ out(a cc') ⇒ B(b'pc') ⇒ p # c' ⇒ bad < bac ⇒ bap ≅a bad ⇒ out(a dp))
Proof
Definitions occuring in Statement :
geo-lt-angle: abc < xyz,
geo-out: out(p ab),
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-between: B(abc),
geo-lsep: a # bc,
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,
uall: ∀[x:A]. B[x],
basic-geometry: BasicGeometry,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
geo-out: out(p ab),
and: P ∧ Q,
cand: A c∧ B,
or: P ∨ Q,
not: ¬A,
false: False,
geo-cong-angle: abc ≅a xyz,
stable: Stable{P},
geo-eq: a ≡ b,
iff: P ⇐⇒ Q,
geo-colinear: Colinear(a;b;c),
oriented-plane: OrientedPlane,
exists: ∃x:A. B[x],
rev_implies: P ⇐ Q,
append: as @ bs,
so_lambda: so_lambda3,
top: Top,
so_apply: x[s1;s2;s3],
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
select: L[n],
cons: [a / b],
subtract: n - m,
basic-geometry-: BasicGeometry-,
geo-lsep: a # bc,
geo-strict-between: a-b-c
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,d,b',c',p:Point.
(a \# bc
{}\mRightarrow{} out(b dc)
{}\mRightarrow{} out(a bb')
{}\mRightarrow{} out(a cc')
{}\mRightarrow{} B(b'pc')
{}\mRightarrow{} p \# c'
{}\mRightarrow{} bad < bac
{}\mRightarrow{} bap \mcong{}\msuba{} bad
{}\mRightarrow{} out(a dp))
Date html generated:
2020_05_20-AM-10_38_23
Last ObjectModification:
2020_01_13-PM-04_52_05
Theory : euclidean!plane!geometry
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