Nuprl Lemma : eu-col-connect
∀e:EuclideanPlane. ∀a,b,c,d:Point. (Colinear(a;b;c) ⇒ Colinear(c;b;d) ⇒ Colinear(a;b;d))
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
eu-colinear: Colinear(a;b;c),
eu-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
euclidean-plane: EuclideanPlane,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
rev_implies: P ⇐ Q,
guard: {T},
or: P ∨ Q,
prop: ℙ,
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),
false: False,
not: ¬A,
less_than: a < b,
squash: ↓T,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m
Lemmas referenced :
euclidean-plane_wf,
eu-colinear_wf,
lelt_wf,
false_wf,
length_of_nil_lemma,
length_of_cons_lemma,
list_ind_nil_lemma,
list_ind_cons_lemma,
eu-colinear-is-colinear-set,
exists_wf,
not_wf,
equal_wf,
l_member_wf,
cons_member,
nil_wf,
eu-point_wf,
cons_wf,
eu-colinear-append,
eu-colinear-def
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
isectElimination,
hypothesis,
productElimination,
independent_functionElimination,
dependent_pairFormation,
because_Cache,
independent_pairFormation,
sqequalRule,
inrFormation,
inlFormation,
productEquality,
lambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
introduction,
imageMemberEquality,
baseClosed
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,d:Point. (Colinear(a;b;c) {}\mRightarrow{} Colinear(c;b;d) {}\mRightarrow{} Colinear(a;b;d))
Date html generated:
2016_05_18-AM-06_45_47
Last ObjectModification:
2016_01_16-PM-10_29_56
Theory : euclidean!geometry
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