Nuprl Lemma : geo-lt-implies-gt-strong-1
∀g:EuclideanPlane. ∀a,b,c,d:Point. (|cd| < |ab| ⇒ (∃w:Point. (B(awb) ∧ aw ≅ cd ∧ w # b)))
Proof
Definitions occuring in Statement :
geo-lt: p < q,
geo-length: |s|,
geo-mk-seg: ab,
euclidean-plane: EuclideanPlane,
geo-congruent: ab ≅ cd,
geo-between: B(abc),
geo-sep: a # b,
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,
member: t ∈ T,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
and: P ∧ Q,
exists: ∃x:A. B[x],
or: P ∨ Q,
cand: A c∧ B,
geo-lt: p < q,
rev_implies: P ⇐ Q,
true: True,
squash: ↓T,
uiff: uiff(P;Q),
basic-geometry-: BasicGeometry-,
iff: P ⇐⇒ Q,
l_member: (x ∈ l),
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
false: False,
select: L[n],
cons: [a / b],
subtract: n - m,
less_than: a < b,
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,
geo-strict-between: a-b-c,
geo-eq: a ≡ b
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,d:Point. (|cd| < |ab| {}\mRightarrow{} (\mexists{}w:Point. (B(awb) \mwedge{} aw \mcong{} cd \mwedge{} w \# b)))
Date html generated:
2020_05_20-AM-10_00_01
Last ObjectModification:
2020_01_13-PM-03_43_08
Theory : euclidean!plane!geometry
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