Nuprl Lemma : geo-colinear-incident
∀[e:EuclideanPlane]. ∀[l:LINE]. ∀[a,b,v:Point]. (a ≠ b ⇒ Colinear(a;b;v) ⇒ a I l ⇒ b I l ⇒ v I l)
Proof
Definitions occuring in Statement :
geo-incident: p I L,
geoline: LINE,
euclidean-plane: EuclideanPlane,
geo-colinear: Colinear(a;b;c),
geo-sep: a ≠ b,
geo-point: Point,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
geo-incident: p I L,
all: ∀x:A. B[x],
geo-colinear: Colinear(a;b;c),
not: ¬A,
false: False,
and: P ∧ Q,
geo-line: Line,
pi1: fst(t),
pi2: snd(t),
uiff: uiff(P;Q),
oriented-plane: OrientedPlane,
exists: ∃x:A. B[x],
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
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],
geo-colinear-set: geo-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 :
geo-incident_wf,
geo-colinear_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-sep_wf,
not_wf,
geo-between_wf,
equal_wf,
geoline_wf,
geoline-subtype1,
geo-line_wf,
geo-point_wf,
and_wf,
geo-incident-line,
oriented-colinear-append,
cons_wf,
nil_wf,
cons_member,
l_member_wf,
exists_wf,
geo-colinear-is-colinear-set,
list_ind_cons_lemma,
list_ind_nil_lemma,
length_of_cons_lemma,
length_of_nil_lemma,
false_wf,
lelt_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
lambdaEquality,
dependent_functionElimination,
productEquality,
productElimination,
isect_memberEquality,
voidElimination,
rename,
addLevel,
hyp_replacement,
equalitySymmetry,
dependent_set_memberEquality,
independent_pairFormation,
equalityTransitivity,
applyLambdaEquality,
setElimination,
levelHypothesis,
independent_functionElimination,
dependent_pairFormation,
inlFormation,
inrFormation,
voidEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[e:EuclideanPlane]. \mforall{}[l:LINE]. \mforall{}[a,b,v:Point].
(a \mneq{} b {}\mRightarrow{} Colinear(a;b;v) {}\mRightarrow{} a I l {}\mRightarrow{} b I l {}\mRightarrow{} v I l)
Date html generated:
2018_05_22-PM-01_04_39
Last ObjectModification:
2018_05_11-PM-11_08_25
Theory : euclidean!plane!geometry
Home
Index