Nuprl Lemma : test-colinear-sets
∀e:BasicGeometry. ∀A,B,C,X,Y,Z,W,U,V:Point. (A ≠ B ⇒ Colinear(A;B;X) ⇒ A_B_C ⇒ Y_C_A ⇒ Y ≠ C ⇒ Colinear(C;Y;X))
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-colinear: Colinear(a;b;c),
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
basic-geometry: BasicGeometry,
subtract: n - m,
cons: [a / b],
select: L[n],
true: True,
squash: ↓T,
less_than: a < b,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
l_all: (∀x∈L.P[x]),
geo-colinear-set: geo-colinear-set(e; L),
so_apply: x[s1;s2;s3],
top: Top,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
append: as @ bs,
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
or: P ∨ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
cand: A c∧ B,
and: P ∧ Q,
exists: ∃x:A. B[x],
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
geo-colinear_wf,
geo-between_wf,
lelt_wf,
false_wf,
length_of_nil_lemma,
length_of_cons_lemma,
list_ind_nil_lemma,
list_ind_cons_lemma,
geo-between-implies-colinear,
geo-colinear-is-colinear-set,
exists_wf,
geo-sep_wf,
equal_wf,
l_member_wf,
cons_member,
geo-between-sep,
geo-sep-sym,
nil_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-point_wf,
cons_wf,
geo-colinear-append
Rules used in proof :
rename,
setElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
dependent_set_memberEquality,
voidEquality,
voidElimination,
isect_memberEquality,
lambdaEquality,
productEquality,
inlFormation,
inrFormation,
productElimination,
independent_pairFormation,
dependent_pairFormation,
independent_functionElimination,
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
hypothesis,
applyEquality,
isectElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:BasicGeometry. \mforall{}A,B,C,X,Y,Z,W,U,V:Point.
(A \mneq{} B {}\mRightarrow{} Colinear(A;B;X) {}\mRightarrow{} A\_B\_C {}\mRightarrow{} Y\_C\_A {}\mRightarrow{} Y \mneq{} C {}\mRightarrow{} Colinear(C;Y;X))
Date html generated:
2017_10_02-PM-06_20_39
Last ObjectModification:
2017_08_05-PM-04_15_19
Theory : euclidean!plane!geometry
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