Nuprl Lemma : eu-colinear-same-side2
∀e:EuclideanPlane
∀[A,B,C,D:Point]. (Colinear(A;C;D)) supposing ((¬(A = C ∈ Point)) and (¬(A = B ∈ Point)) and A_B_C and A_B_D)
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
eu-between-eq: a_b_c,
eu-colinear: Colinear(a;b;c),
eu-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
equal: s = t ∈ T
Definitions unfolded in proof :
prop: ℙ,
euclidean-plane: EuclideanPlane,
false: False,
implies: P ⇒ Q,
not: ¬A,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
cand: A c∧ B
Lemmas referenced :
eu-between-eq-def,
eu-colinear-def,
eu-between_wf,
eu-between-eq-same-side,
eu-point_wf,
not_wf,
equal_wf,
eu-between-eq_wf,
euclidean-plane_wf
Rules used in proof :
rename,
setElimination,
equalityEquality,
voidElimination,
lambdaEquality,
sqequalRule,
introduction,
independent_isectElimination,
isectElimination,
isect_memberFormation,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
hypothesis,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
lemma_by_obid,
cut,
because_Cache,
productEquality,
independent_pairFormation,
independent_functionElimination,
productElimination,
equalitySymmetry
Latex:
\mforall{}e:EuclideanPlane
\mforall{}[A,B,C,D:Point]. (Colinear(A;C;D)) supposing ((\mneg{}(A = C)) and (\mneg{}(A = B)) and A\_B\_C and A\_B\_D)
Date html generated:
2016_05_18-AM-06_39_56
Last ObjectModification:
2016_01_01-PM-03_17_21
Theory : euclidean!geometry
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