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
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