Nuprl Lemma : geo-colinear-cong-tri-exists
∀[e:BasicGeometry]. ∀[a,b,c,a',c':Point].
(Colinear(a;b;c) ⇒ ac ≅ a'c' ⇒ (¬¬(∃b':Point. (Cong3(abc,a'b'c') ∧ Colinear(a';b';c')))))
Proof
Definitions occuring in Statement :
geo-cong-tri: Cong3(abc,a'b'c'),
basic-geometry: BasicGeometry,
geo-colinear: Colinear(a;b;c),
geo-congruent: ab ≅ cd,
geo-point: Point,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
not: ¬A,
false: False,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
basic-geometry-: BasicGeometry-,
prop: ℙ,
exists: ∃x:A. B[x],
and: P ∧ Q,
geo-cong-tri: Cong3(abc,a'b'c'),
guard: {T},
uimplies: b supposing a,
cand: A c∧ B,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
stable: Stable{P},
geo-eq: a ≡ b,
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,
squash: ↓T,
true: True
Latex:
\mforall{}[e:BasicGeometry]. \mforall{}[a,b,c,a',c':Point].
(Colinear(a;b;c) {}\mRightarrow{} ac \mcong{} a'c' {}\mRightarrow{} (\mneg{}\mneg{}(\mexists{}b':Point. (Cong3(abc,a'b'c') \mwedge{} Colinear(a';b';c')))))
Date html generated:
2020_05_20-AM-09_54_34
Last ObjectModification:
2020_01_13-PM-03_29_41
Theory : euclidean!plane!geometry
Home
Index