Nuprl Lemma : r2-equidistant-implies'
∀a,b:ℝ^2.  (a ≠ b 
⇒ (∀x:ℝ^2. (xa=xb 
⇒ (∃t:ℝ. req-vec(2;x;vec-midpoint(a;b) + t*r2-perp(b - a))))))
Proof
Definitions occuring in Statement : 
vec-midpoint: vec-midpoint(a;b)
, 
r2-perp: r2-perp(x)
, 
real-vec-sep: a ≠ b
, 
rv-congruent: ab=cd
, 
real-vec-mul: a*X
, 
real-vec-sub: X - Y
, 
real-vec-add: X + Y
, 
req-vec: req-vec(n;x;y)
, 
real-vec: ℝ^n
, 
real: ℝ
, 
all: ∀x:A. B[x]
, 
exists: ∃x:A. B[x]
, 
implies: P 
⇒ Q
, 
natural_number: $n
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
member: t ∈ T
, 
implies: P 
⇒ Q
, 
rv-congruent: ab=cd
, 
prop: ℙ
, 
uall: ∀[x:A]. B[x]
, 
nat: ℕ
, 
le: A ≤ B
, 
and: P ∧ Q
, 
less_than': less_than'(a;b)
, 
false: False
, 
not: ¬A
, 
subtype_rel: A ⊆r B
, 
uimplies: b supposing a
, 
uiff: uiff(P;Q)
, 
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced : 
r2-equidistant-implies, 
rv-congruent_wf, 
false_wf, 
le_wf, 
real-vec_wf, 
real-vec-sep_wf, 
real-vec-dist_wf, 
req_functionality, 
real-vec-dist-symmetry
Rules used in proof : 
cut, 
introduction, 
extract_by_obid, 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation, 
hypothesis, 
sqequalHypSubstitution, 
dependent_functionElimination, 
thin, 
hypothesisEquality, 
independent_functionElimination, 
isectElimination, 
dependent_set_memberEquality, 
natural_numberEquality, 
sqequalRule, 
independent_pairFormation, 
because_Cache, 
applyEquality, 
independent_isectElimination, 
productElimination
Latex:
\mforall{}a,b:\mBbbR{}\^{}2.    (a  \mneq{}  b  {}\mRightarrow{}  (\mforall{}x:\mBbbR{}\^{}2.  (xa=xb  {}\mRightarrow{}  (\mexists{}t:\mBbbR{}.  req-vec(2;x;vec-midpoint(a;b)  +  t*r2-perp(b  -  a))))))
Date html generated:
2016_10_28-AM-07_42_59
Last ObjectModification:
2016_09_28-PM-09_43_47
Theory : reals!model!euclidean!geometry
Home
Index