Nuprl Lemma : Euclid-Prop7'
∀e:EuclideanPlane. ∀a,b,c,d:Point. (a # b ⇒ ac ≅ ad ⇒ bc ≅ bd ⇒ (¬c leftof ba ⇐⇒ ¬d leftof ba) ⇒ c ≡ d)
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-eq: a ≡ b,
geo-congruent: ab ≅ cd,
geo-left: a leftof bc,
geo-sep: a # b,
geo-point: Point,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
geo-eq: a ≡ b,
not: ¬A,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
sq_exists: ∃x:A [B[x]],
false: False,
geo-midpoint: a=m=b,
basic-geometry: BasicGeometry,
uiff: uiff(P;Q),
uimplies: b supposing a,
guard: {T},
rev_implies: P ⇐ Q,
or: P ∨ Q,
stable: Stable{P},
oriented-plane: OrientedPlane,
geo-colinear: Colinear(a;b;c),
cand: A c∧ B,
geo-lsep: a # bc,
exists: ∃x:A. B[x],
l_member: (x ∈ l),
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
top: Top,
select: L[n],
cons: [a / b],
subtract: n - m,
less_than: a < b,
squash: ↓T,
true: True,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
append: as @ bs,
so_lambda: so_lambda3,
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
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,d:Point.
(a \# b {}\mRightarrow{} ac \mcong{} ad {}\mRightarrow{} bc \mcong{} bd {}\mRightarrow{} (\mneg{}c leftof ba \mLeftarrow{}{}\mRightarrow{} \mneg{}d leftof ba) {}\mRightarrow{} c \mequiv{} d)
Date html generated:
2020_05_20-AM-10_05_50
Last ObjectModification:
2019_12_03-AM-09_52_00
Theory : euclidean!plane!geometry
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