Nuprl Lemma : geo-perp-in-iff
∀e:BasicGeometry. ∀a,b,c,d,x:Point.
  (a ≠ b
  
⇒ c ≠ d
  
⇒ (ab  ⊥x cd
     
⇐⇒ Colinear(a;b;x) ∧ Colinear(c;d;x) ∧ (∃u,v:Point. (Colinear(a;b;u) ∧ Colinear(c;d;v) ∧ u ≠ x ∧ v ≠ x ∧ Ruxv))))
Proof
Definitions occuring in Statement : 
geo-perp-in: ab  ⊥x cd
, 
right-angle: Rabc
, 
basic-geometry: BasicGeometry
, 
geo-colinear: Colinear(a;b;c)
, 
geo-sep: a ≠ b
, 
geo-point: Point
, 
all: ∀x:A. B[x]
, 
exists: ∃x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
implies: P 
⇒ Q
, 
and: P ∧ Q
Definitions unfolded in proof : 
geo-perp-in: ab  ⊥x cd
, 
exists: ∃x:A. B[x]
, 
so_apply: x[s]
, 
so_lambda: λ2x.t[x]
, 
uimplies: b supposing a
, 
guard: {T}
, 
subtype_rel: A ⊆r B
, 
basic-geometry: BasicGeometry
, 
rev_implies: P 
⇐ Q
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
member: t ∈ T
, 
and: P ∧ Q
, 
iff: P 
⇐⇒ Q
, 
implies: P 
⇒ Q
, 
all: ∀x:A. B[x]
, 
cand: A c∧ B
, 
or: P ∨ Q
, 
subtract: n - m
, 
cons: [a / b]
, 
select: L[n]
, 
true: True
, 
squash: ↓T
, 
less_than: a < b
, 
not: ¬A
, 
false: False
, 
less_than': less_than'(a;b)
, 
le: A ≤ B
, 
lelt: i ≤ j < k
, 
int_seg: {i..j-}
, 
l_all: (∀x∈L.P[x])
, 
geo-colinear-set: geo-colinear-set(e; L)
, 
so_apply: x[s1;s2;s3]
, 
top: Top
, 
so_lambda: so_lambda(x,y,z.t[x; y; z])
, 
append: as @ bs
Lemmas referenced : 
right-angle_wf, 
geo-sep_wf, 
Error :basic-geo-primitives_wf, 
Error :basic-geo-structure_wf, 
basic-geometry_wf, 
subtype_rel_transitivity, 
basic-geometry-subtype, 
geo-point_wf, 
exists_wf, 
geo-colinear_wf, 
geo-perp-in_wf, 
geo-sep-sym, 
geo-colinear-same, 
geo-sep-or, 
lelt_wf, 
false_wf, 
length_of_nil_lemma, 
length_of_cons_lemma, 
list_ind_nil_lemma, 
list_ind_cons_lemma, 
geo-colinear-is-colinear-set, 
equal_wf, 
l_member_wf, 
cons_member, 
nil_wf, 
cons_wf, 
geo-colinear-append, 
right-angle-colinear, 
right-angle-symmetry
Rules used in proof : 
lambdaEquality, 
sqequalRule, 
independent_isectElimination, 
instantiate, 
applyEquality, 
because_Cache, 
rename, 
setElimination, 
productEquality, 
productElimination, 
hypothesis, 
hypothesisEquality, 
thin, 
isectElimination, 
sqequalHypSubstitution, 
extract_by_obid, 
introduction, 
cut, 
independent_pairFormation, 
lambdaFormation, 
sqequalReflexivity, 
computationStep, 
sqequalTransitivity, 
sqequalSubstitution, 
independent_functionElimination, 
dependent_functionElimination, 
dependent_pairFormation, 
unionElimination, 
inrFormation, 
inlFormation, 
dependent_set_memberEquality, 
baseClosed, 
imageMemberEquality, 
natural_numberEquality, 
voidEquality, 
voidElimination, 
isect_memberEquality
Latex:
\mforall{}e:BasicGeometry.  \mforall{}a,b,c,d,x:Point.
    (a  \mneq{}  b
    {}\mRightarrow{}  c  \mneq{}  d
    {}\mRightarrow{}  (ab    \mbot{}x  cd
          \mLeftarrow{}{}\mRightarrow{}  Colinear(a;b;x)
                  \mwedge{}  Colinear(c;d;x)
                  \mwedge{}  (\mexists{}u,v:Point.  (Colinear(a;b;u)  \mwedge{}  Colinear(c;d;v)  \mwedge{}  u  \mneq{}  x  \mwedge{}  v  \mneq{}  x  \mwedge{}  Ruxv))))
Date html generated:
2017_10_02-PM-06_43_15
Last ObjectModification:
2017_08_05-PM-04_49_13
Theory : euclidean!plane!geometry
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