Nuprl Lemma : test-colinear-sets
∀e:EuclideanPlane. ∀A,B,C,X,Y,Z,W,U,V:Point.
(Colinear(A;B;X) ⇒ A_B_C ⇒ Y_C_A ⇒ (¬(Y = C ∈ Point)) ⇒ Colinear(C;Y;X))
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
eu-between-eq: a_b_c,
eu-colinear: Colinear(a;b;c),
eu-point: Point,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
false: False,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T},
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
eu-colinear-set: eu-colinear-set(e;L),
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
squash: ↓T,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m
Lemmas referenced :
equal_wf,
eu-point_wf,
eu-colinear-append,
cons_wf,
nil_wf,
eu-between-eq_wf,
eu-between-eq-same,
eu-colinear-def,
cons_member,
l_member_wf,
not_wf,
exists_wf,
eu-colinear-is-colinear-set,
eu-between-eq-implies-colinear,
list_ind_cons_lemma,
list_ind_nil_lemma,
length_of_cons_lemma,
length_of_nil_lemma,
false_wf,
lelt_wf,
eu-colinear_wf,
euclidean-plane_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
thin,
equalitySymmetry,
voidElimination,
introduction,
extract_by_obid,
isectElimination,
setElimination,
rename,
hypothesisEquality,
dependent_functionElimination,
because_Cache,
dependent_pairFormation,
hyp_replacement,
Error :applyLambdaEquality,
sqequalRule,
independent_isectElimination,
productElimination,
independent_pairFormation,
inrFormation,
inlFormation,
productEquality,
lambdaEquality,
isect_memberEquality,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}e:EuclideanPlane. \mforall{}A,B,C,X,Y,Z,W,U,V:Point.
(Colinear(A;B;X) {}\mRightarrow{} A\_B\_C {}\mRightarrow{} Y\_C\_A {}\mRightarrow{} (\mneg{}(Y = C)) {}\mRightarrow{} Colinear(C;Y;X))
Date html generated:
2016_10_26-AM-07_44_10
Last ObjectModification:
2016_07_12-AM-08_11_36
Theory : euclidean!geometry
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