Nuprl Lemma : eu-proper-extend-exists
∀e:EuclideanPlane. ∀q:Point. ∀a:{a:Point| ¬(q = a ∈ Point)} . ∀b:Point. ∀c:{c:Point| ¬(b = c ∈ Point)} .
∃x:Point. (q-a-x ∧ ax=bc)
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
eu-between: a-b-c,
eu-congruent: ab=cd,
eu-point: Point,
all: ∀x:A. B[x],
exists: ∃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],
member: t ∈ T,
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
prop: ℙ,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
so_lambda: λ2x.t[x],
so_apply: x[s],
stable: Stable{P},
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
false: False,
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
eu-extend-exists,
eu-between_wf,
eu-congruent_wf,
set_wf,
eu-point_wf,
not_wf,
equal_wf,
euclidean-plane_wf,
stable__eu-between,
eu-between-eq-def,
sq_stable__eu-between,
eu-congruence-identity-sym
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
setElimination,
rename,
productElimination,
dependent_pairFormation,
independent_pairFormation,
productEquality,
isectElimination,
because_Cache,
sqequalRule,
lambdaEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
imageMemberEquality,
baseClosed,
imageElimination,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality,
equalityTransitivity
Latex:
\mforall{}e:EuclideanPlane. \mforall{}q:Point. \mforall{}a:\{a:Point| \mneg{}(q = a)\} . \mforall{}b:Point. \mforall{}c:\{c:Point| \mneg{}(b = c)\} .
\mexists{}x:Point. (q-a-x \mwedge{} ax=bc)
Date html generated:
2016_10_26-AM-07_41_46
Last ObjectModification:
2016_07_12-AM-08_08_02
Theory : euclidean!geometry
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