Nuprl Lemma : eu-line-circle_wf
∀[e:EuclideanStructure]. ∀[a,b:Point]. ∀[x:{x:Point| a_x_b} ]. ∀[y:{y:Point| a_b_y} ]. ∀[p:{p:Point| ap=ax} ].
∀[q:{q:Point| aq=ay ∧ (¬(q = p ∈ Point))} ].
(intersect pq (at radius xy) with Oab ∈ Point × Point)
Proof
Definitions occuring in Statement :
eu-line-circle: intersect pq (at radius xy) with Oab ,
eu-between-eq: a_b_c,
eu-congruent: ab=cd,
eu-point: Point,
euclidean-structure: EuclideanStructure,
uall: ∀[x:A]. B[x],
not: ¬A,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
product: x:A × B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
eu-line-circle: intersect pq (at radius xy) with Oab ,
and: P ∧ Q,
eu-point: Point,
eu-congruent: ab=cd,
eu-between-eq: a_b_c,
euclidean-structure: EuclideanStructure,
record+: record+,
record-select: r.x,
subtype_rel: A ⊆r B,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
btrue: tt,
guard: {T},
prop: ℙ,
spreadn: spread3,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
cand: A c∧ B
Lemmas referenced :
subtype_rel_self,
not_wf,
equal_wf,
uall_wf,
iff_wf,
and_wf,
isect_wf,
set_wf,
eu-point_wf,
eu-congruent_wf,
eu-between-eq_wf,
euclidean-structure_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
setElimination,
thin,
rename,
sqequalRule,
sqequalHypSubstitution,
productElimination,
dependentIntersectionElimination,
dependentIntersectionEqElimination,
hypothesis,
applyEquality,
tokenEquality,
instantiate,
lemma_by_obid,
isectElimination,
universeEquality,
functionEquality,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
cumulativity,
hypothesisEquality,
because_Cache,
setEquality,
productEquality,
lambdaFormation,
dependent_set_memberEquality,
independent_pairFormation,
axiomEquality,
isect_memberEquality
Latex:
\mforall{}[e:EuclideanStructure]. \mforall{}[a,b:Point]. \mforall{}[x:\{x:Point| a\_x\_b\} ]. \mforall{}[y:\{y:Point| a\_b\_y\} ]. \mforall{}[p:\{p:Point|\000C
ap=ax\} ].
\mforall{}[q:\{q:Point| aq=ay \mwedge{} (\mneg{}(q = p))\} ].
(intersect pq (at radius xy) with Oab \mmember{} Point \mtimes{} Point)
Date html generated:
2016_05_18-AM-06_33_23
Last ObjectModification:
2015_12_28-AM-09_28_18
Theory : euclidean!geometry
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