Nuprl Lemma : circle-circle-continuity1
∀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))
⇒ (∃y:Point. (ay=ab ∧ cy=cd)))
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,
cand: A c∧ B,
prop: ℙ,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
circle-circle-continuity,
and_wf,
eu-congruent_wf,
exists_wf,
eu-point_wf,
eu-between-eq_wf,
not_wf,
equal_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_pairFormation,
isectElimination,
setElimination,
rename,
sqequalRule,
lambdaEquality,
because_Cache,
productEquality
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))
{}\mRightarrow{} (\mexists{}y:Point. (ay=ab \mwedge{} cy=cd)))
Date html generated:
2016_05_18-AM-06_41_44
Last ObjectModification:
2015_12_28-AM-09_23_23
Theory : euclidean!geometry
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