Nuprl Lemma : Euclid-drop-perp-0
∀e:EuclideanPlane. ∀a:Point. ∀b:{b:Point| a # b} . ∀c:Point.
∃x:Point. (∃p:Point [(Colinear(p;x;c) ∧ ab ⊥p px ∧ x # ab ∧ x # c)])
Proof
Definitions occuring in Statement :
geo-perp-in: ab ⊥x cd,
euclidean-plane: EuclideanPlane,
geo-colinear: Colinear(a;b;c),
geo-lsep: a # bc,
geo-sep: a # b,
geo-point: Point,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
exists: ∃x:A. B[x],
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
euclidean-plane: EuclideanPlane,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
uall: ∀[x:A]. B[x],
geo-gt-prim: ab>cd,
record-select: r.x,
geo-sep: a # b,
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T},
oriented-plane: OrientedPlane,
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],
less_than: a < b,
true: True,
uimplies: b supposing a,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
subtract: n - m,
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,
sq_exists: ∃x:A [B[x]],
basic-geometry: BasicGeometry,
geo-midpoint: a=m=b,
basic-geometry-: BasicGeometry-,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
geo-eq: a ≡ b,
geo-perp-in: ab ⊥x cd
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a:Point. \mforall{}b:\{b:Point| a \# b\} . \mforall{}c:Point.
\mexists{}x:Point. (\mexists{}p:Point [(Colinear(p;x;c) \mwedge{} ab \mbot{}p px \mwedge{} x \# ab \mwedge{} x \# c)])
Date html generated:
2020_05_20-AM-10_04_10
Last ObjectModification:
2019_12_28-AM-08_24_36
Theory : euclidean!plane!geometry
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