Nuprl Lemma : circle-circle-continuity2
∀e:EuclideanPlane. ∀a,b,c,d:Point.
((¬(a = c ∈ Point))
⇒ (∃p,q,x,z:Point. ((a_x_b ∧ a_b_z ∧ ap=ax ∧ aq=az ∧ cp=cd ∧ cq=cd) ∧ (¬(x = z ∈ Point))))
⇒ (∃z1,z2:Point. (az1=ab ∧ az2=ab ∧ cz1=cd ∧ cz2=cd ∧ (¬(z1 = z2 ∈ Point)))))
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
eu-between-eq: a_b_c,
eu-congruent: ab=cd,
eu-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
uimplies: b supposing a,
not: ¬A,
false: False,
prop: ℙ,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
circle-circle-continuity,
equal_wf,
eu-point_wf,
eu-congruent_wf,
not_wf,
exists_wf,
eu-between-eq_wf,
euclidean-plane_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
lemma_by_obid,
dependent_functionElimination,
hypothesisEquality,
independent_isectElimination,
hypothesis,
dependent_pairFormation,
independent_functionElimination,
independent_pairFormation,
equalitySymmetry,
voidElimination,
isectElimination,
setElimination,
rename,
productEquality,
because_Cache,
sqequalRule,
lambdaEquality
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,d:Point.
((\mneg{}(a = c))
{}\mRightarrow{} (\mexists{}p,q,x,z:Point. ((a\_x\_b \mwedge{} a\_b\_z \mwedge{} ap=ax \mwedge{} aq=az \mwedge{} cp=cd \mwedge{} cq=cd) \mwedge{} (\mneg{}(x = z))))
{}\mRightarrow{} (\mexists{}z1,z2:Point. (az1=ab \mwedge{} az2=ab \mwedge{} cz1=cd \mwedge{} cz2=cd \mwedge{} (\mneg{}(z1 = z2)))))
Date html generated:
2016_05_18-AM-06_41_41
Last ObjectModification:
2015_12_28-AM-09_23_48
Theory : euclidean!geometry
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