Nuprl Lemma : eu-extend-property
∀e:EuclideanPlane
∀[q:Point]. ∀[a:{a:Point| ¬(q = a ∈ Point)} ]. ∀[b,c:Point]. (q_a_(extend qa by bc) ∧ a(extend qa by bc)=bc)
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
eu-extend: (extend ab by cd),
eu-between-eq: a_b_c,
eu-congruent: ab=cd,
eu-point: Point,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
and: P ∧ Q,
set: {x:A| B[x]} ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
and: P ∧ Q,
cand: A c∧ B,
euclidean-plane: EuclideanPlane,
member: t ∈ T,
euclidean-axioms: euclidean-axioms(e),
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T}
Lemmas referenced :
euclidean-plane_wf,
equal_wf,
not_wf,
set_wf,
eu-point_wf,
sq_stable__eu-congruent,
eu-extend_wf,
sq_stable__eu-between-eq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
lemma_by_obid,
dependent_functionElimination,
productElimination,
hypothesisEquality,
isectElimination,
hypothesis,
independent_functionElimination,
introduction,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
because_Cache,
lambdaEquality
Latex:
\mforall{}e:EuclideanPlane
\mforall{}[q:Point]. \mforall{}[a:\{a:Point| \mneg{}(q = a)\} ]. \mforall{}[b,c:Point].
(q\_a\_(extend qa by bc) \mwedge{} a(extend qa by bc)=bc)
Date html generated:
2016_05_18-AM-06_33_38
Last ObjectModification:
2016_01_16-PM-10_31_52
Theory : euclidean!geometry
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