Nuprl Lemma : Euclid-drop-perp-ext
∀e:EuclideanPlane. ∀a:Point. ∀b:{b:Point| a # b} . ∀c:{c:Point| c # ab} . (∃p:Point [(Colinear(a;b;p) ∧ ab ⊥p pc)])
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]],
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
member: t ∈ T,
record-select: r.x,
geo-CCL: CCL(a;b;c;d),
geo-CCR: geo-CCR(g;a;b;c;d),
ifthenelse: if b then t else f fi ,
midpoint-construction: Mid(a;b),
let: let,
Euclid-drop-perp,
Euclid-drop-perp-1,
Euclid-drop-perp-0,
sq_stable__colinear,
sq_stable__from_stable,
stable__geo-perp-in,
colinear-equidistant-points-exist,
geo-CC-2,
geo-congruent-refl,
geo-sep-sym,
geo-sep-or,
stable__and,
stable__all,
symmetric-point-construction,
use-SC,
sq_stable__geo-sep,
geo-between-implies-colinear,
geo-congruent-sep,
geo-congruent-symmetry,
geo-congruent-iff-length,
sq_stable__and,
sq_stable__geo-congruent,
sq_stable__geo-between,
sq_stable__all,
basic-geo-sep-sym,
sq_stable__geo-axioms,
geo-ge-sep,
geo-seg-congruent-iff-length,
sq_stable-geo-axioms-if,
sq_stable__geo-gt-prim,
sq_stable__geo-lsep,
any: any x,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda4,
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s]
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a:Point. \mforall{}b:\{b:Point| a \# b\} . \mforall{}c:\{c:Point| c \# ab\} .
(\mexists{}p:Point [(Colinear(a;b;p) \mwedge{} ab \mbot{}p pc)])
Date html generated:
2020_05_20-AM-10_04_33
Last ObjectModification:
2020_01_27-PM-07_08_00
Theory : euclidean!plane!geometry
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