Nuprl Lemma : line-circle-continuity
∀e:EuclideanPlane. ∀a,b,p:Point. ∀q:{q:Point| ¬(q = p ∈ Point)} .
((∃x:{x:Point| a_x_b ∧ (¬(x = b ∈ Point))} . ∃y:{y:Point| a_b_y} . (ap=ax ∧ aq=ay))
⇒ (∃y,z:Point. (ay=ab ∧ az=ab ∧ z_p_q ∧ p_y_q ∧ (¬(y = z ∈ 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,
set: {x:A| B[x]} ,
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,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
euclidean-plane: EuclideanPlane,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
prop: ℙ,
cand: A c∧ B,
not: ¬A,
false: False,
top: Top,
euclidean-axioms: euclidean-axioms(e),
sq_stable: SqStable(P),
squash: ↓T,
let: let,
pi1: fst(t),
pi2: snd(t)
Lemmas referenced :
sq_stable__not,
sq_stable__eu-between-eq,
sq_stable__eu-congruent,
sq_stable__and,
squash_wf,
euclidean-plane_wf,
set_wf,
exists_wf,
pi2_wf,
top_wf,
subtype_rel_product,
pi1_wf_top,
eu-congruent_wf,
equal_wf,
not_wf,
eu-between-eq_wf,
and_wf,
eu-point_wf,
subtype_rel_sets,
eu-line-circle_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
cut,
lemma_by_obid,
isectElimination,
because_Cache,
hypothesisEquality,
applyEquality,
setElimination,
rename,
hypothesis,
sqequalRule,
lambdaEquality,
independent_isectElimination,
setEquality,
dependent_set_memberEquality,
independent_pairFormation,
independent_functionElimination,
voidElimination,
productEquality,
isect_memberEquality,
voidEquality,
equalityEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
introduction,
imageMemberEquality,
baseClosed,
imageElimination,
isectEquality
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,p:Point. \mforall{}q:\{q:Point| \mneg{}(q = p)\} .
((\mexists{}x:\{x:Point| a\_x\_b \mwedge{} (\mneg{}(x = b))\} . \mexists{}y:\{y:Point| a\_b\_y\} . (ap=ax \mwedge{} aq=ay))
{}\mRightarrow{} (\mexists{}y,z:Point. (ay=ab \mwedge{} az=ab \mwedge{} z\_p\_q \mwedge{} p\_y\_q \mwedge{} (\mneg{}(y = z)))))
Date html generated:
2016_05_18-AM-06_41_29
Last ObjectModification:
2016_01_16-PM-10_30_26
Theory : euclidean!geometry
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