Nuprl Lemma : eu-inner-pasch-ex
∀e:EuclideanPlane. ∀a,b:Point. ∀c:{c:Point| ¬Colinear(a;b;c)} . ∀p:{p:Point| a-p-c} . ∀q:{q:Point| b_q_c} .
∃x:Point. (p-x-b ∧ q-x-a ∧ (x = eu-inner-pasch(e;a;b;c;p;q) ∈ Point))
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
eu-inner-pasch: eu-inner-pasch(e;a;b;c;p;q),
eu-between-eq: a_b_c,
eu-colinear: Colinear(a;b;c),
eu-between: a-b-c,
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,
prop: ℙ,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B
Lemmas referenced :
set_wf,
eu-point_wf,
eu-between-eq_wf,
eu-between_wf,
not_wf,
eu-colinear_wf,
euclidean-plane_wf,
eu-inner-pasch_wf,
eu-inner-pasch-property,
and_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
dependent_pairFormation,
dependent_functionElimination,
productElimination,
independent_pairFormation
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b:Point. \mforall{}c:\{c:Point| \mneg{}Colinear(a;b;c)\} . \mforall{}p:\{p:Point| a-p-c\} . \mforall{}q:\{q:Point|
b\_q\_c\} .
\mexists{}x:Point. (p-x-b \mwedge{} q-x-a \mwedge{} (x = eu-inner-pasch(e;a;b;c;p;q)))
Date html generated:
2016_05_18-AM-06_45_29
Last ObjectModification:
2015_12_28-AM-09_21_57
Theory : euclidean!geometry
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