Nuprl Lemma : lsep-colinear
∀g:EuclideanPlane. ∀p,a,b,x,y:Point. (p # ab ⇒ x ≠ y ⇒ Colinear(x;a;b) ⇒ Colinear(y;a;b) ⇒ p # xy)
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
geo-colinear: Colinear(a;b;c),
geo-sep: a ≠ b,
geo-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,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
oriented-plane: OrientedPlane,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
or: P ∨ Q,
prop: ℙ,
guard: {T},
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),
false: False,
not: ¬A,
less_than: a < b,
squash: ↓T,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
uimplies: b supposing a
Lemmas referenced :
lsep-iff-all-sep,
oriented-colinear-append,
cons_wf,
geo-point_wf,
nil_wf,
cons_member,
l_member_wf,
equal_wf,
geo-sep_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,
geo-colinear_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-lsep_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_functionElimination,
sqequalRule,
isectElimination,
applyEquality,
because_Cache,
dependent_pairFormation,
independent_pairFormation,
inlFormation,
inrFormation,
productEquality,
lambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
independent_isectElimination
Latex:
\mforall{}g:EuclideanPlane. \mforall{}p,a,b,x,y:Point.
(p \# ab {}\mRightarrow{} x \mneq{} y {}\mRightarrow{} Colinear(x;a;b) {}\mRightarrow{} Colinear(y;a;b) {}\mRightarrow{} p \# xy)
Date html generated:
2018_05_22-PM-00_08_56
Last ObjectModification:
2018_04_04-PM-05_46_36
Theory : euclidean!plane!geometry
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