Nuprl Lemma : eu-inner-pasch-property
∀e:EuclideanPlane
∀[a,b:Point]. ∀[c:{c:Point| ¬Colinear(a;b;c)} ]. ∀[p:{p:Point| a-p-c} ]. ∀[q:{q:Point| b_q_c} ].
(p-eu-inner-pasch(e;a;b;c;p;q)-b ∧ q-eu-inner-pasch(e;a;b;c;p;q)-a)
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,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
and: P ∧ Q,
set: {x:A| B[x]}
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,
let: let,
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
euclidean-plane_wf,
eu-colinear_wf,
not_wf,
eu-between_wf,
eu-between-eq_wf,
eu-point_wf,
set_wf,
eu-inner-pasch_wf,
sq_stable__eu-between
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{}[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\} ].
(p-eu-inner-pasch(e;a;b;c;p;q)-b \mwedge{} q-eu-inner-pasch(e;a;b;c;p;q)-a)
Date html generated:
2016_05_18-AM-06_33_41
Last ObjectModification:
2016_01_16-PM-10_31_48
Theory : euclidean!geometry
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