Nuprl Lemma : r2-upper-dimension
∀[a,b,p,q,r:ℝ^2].
  (¬(p ≠ q ∧ q ≠ r ∧ r ≠ p ∧ (¬p-q-r) ∧ (¬q-r-p) ∧ (¬r-p-q))) supposing (ra=rb and qa=qb and pa=pb and a ≠ b)
Proof
Definitions occuring in Statement : 
rv-between: a-b-c
, 
real-vec-sep: a ≠ b
, 
rv-congruent: ab=cd
, 
real-vec: ℝ^n
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
not: ¬A
, 
and: P ∧ Q
, 
natural_number: $n
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
uimplies: b supposing a
, 
not: ¬A
, 
implies: P 
⇒ Q
, 
false: False
, 
all: ∀x:A. B[x]
, 
so_lambda: λ2x.t[x]
, 
and: P ∧ Q
, 
prop: ℙ
, 
so_apply: x[s]
, 
nat: ℕ
, 
le: A ≤ B
, 
less_than': less_than'(a;b)
, 
real-vec-dist: d(x;y)
, 
subtype_rel: A ⊆r B
, 
rless: x < y
, 
sq_exists: ∃x:{A| B[x]}
, 
real-vec-sep: a ≠ b
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
exists: ∃x:A. B[x]
, 
cand: A c∧ B
, 
or: P ∨ Q
, 
less_than: a < b
, 
squash: ↓T
, 
true: True
, 
rneq: x ≠ y
, 
guard: {T}
, 
rsub: x - y
Lemmas referenced : 
r2-equidistant-implies', 
vec-midpoint_wf, 
r2-perp_wf, 
real-vec-sub_wf, 
set_wf, 
real-vec_wf, 
req_wf, 
dot-product_wf, 
int-to-real_wf, 
real-vec-norm_wf, 
equal_wf, 
real-vec-sep_wf, 
false_wf, 
le_wf, 
not_wf, 
rv-between_wf, 
rv-congruent_wf, 
real-vec-dist_wf, 
rless_functionality, 
req_weakening, 
real-vec-dist-symmetry, 
exists_wf, 
real_wf, 
req-vec_wf, 
real-vec-add_wf, 
real-vec-mul_wf, 
real-vec-sep_functionality, 
req-vec_weakening, 
rv-between_functionality, 
rv-between-translation, 
rv-between-vec-mul, 
or_wf, 
rless_wf, 
rless-int, 
rmul_wf, 
rabs_wf, 
rsub_wf, 
radd_wf, 
rminus_wf, 
rabs-positive-iff, 
radd-preserves-rless, 
real-vec-dist-translation2, 
real-vec-dist-vec-mul, 
rmul_functionality, 
rmul-one-both, 
rabs_functionality, 
radd_comm, 
req_transitivity, 
radd_functionality, 
rminus-as-rmul, 
radd-assoc, 
req_inversion, 
rmul-identity1, 
rmul-distrib2, 
radd-int, 
rmul-zero-both, 
radd-zero-both, 
radd-ac, 
radd-rminus-both
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
lambdaFormation, 
thin, 
extract_by_obid, 
sqequalHypSubstitution, 
dependent_functionElimination, 
hypothesisEquality, 
independent_functionElimination, 
hypothesis, 
because_Cache, 
isectElimination, 
sqequalRule, 
lambdaEquality, 
productEquality, 
natural_numberEquality, 
equalityTransitivity, 
equalitySymmetry, 
voidElimination, 
dependent_set_memberEquality, 
independent_pairFormation, 
isect_memberEquality, 
applyEquality, 
independent_isectElimination, 
productElimination, 
setElimination, 
rename, 
promote_hyp, 
addLevel, 
impliesFunctionality, 
andLevelFunctionality, 
impliesLevelFunctionality, 
levelHypothesis, 
imageMemberEquality, 
baseClosed, 
unionElimination, 
minusEquality, 
addEquality, 
inlFormation, 
inrFormation
Latex:
\mforall{}[a,b,p,q,r:\mBbbR{}\^{}2].
    (\mneg{}(p  \mneq{}  q  \mwedge{}  q  \mneq{}  r  \mwedge{}  r  \mneq{}  p  \mwedge{}  (\mneg{}p-q-r)  \mwedge{}  (\mneg{}q-r-p)  \mwedge{}  (\mneg{}r-p-q)))  supposing 
          (ra=rb  and 
          qa=qb  and 
          pa=pb  and 
          a  \mneq{}  b)
Date html generated:
2017_10_03-PM-00_47_13
Last ObjectModification:
2017_07_28-AM-08_47_17
Theory : reals!model!euclidean!geometry
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