Nuprl Lemma : colinear-lsep-general
∀g:OrientedPlane. ∀a,b,c,d:Point. (Colinear(a;b;c) ⇒ Colinear(a;b;d) ⇒ c ≠ d ⇒ (∀y:Point. (y # ab ⇒ y # cd)))
Proof
Definitions occuring in Statement :
oriented-plane: OrientedPlane,
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,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
oriented-plane: OrientedPlane,
euclidean-plane: EuclideanPlane,
or: P ∨ Q,
and: P ∧ Q,
cand: A c∧ B,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
top: Top,
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,
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: 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]),
so_apply: x[s1;s2;s3]
Lemmas referenced :
geo-lsep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
oriented-plane-subtype,
subtype_rel_transitivity,
oriented-plane_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf,
geo-sep_wf,
geo-colinear_wf,
geo-sep-or,
colinear-lsep,
lsep-all-sym,
geo-colinear-is-colinear-set,
length_of_cons_lemma,
length_of_nil_lemma,
false_wf,
lelt_wf,
colinear-lsep',
oriented-colinear-append,
cons_wf,
nil_wf,
lsep-implies-sep,
cons_member,
l_member_wf,
equal_wf,
exists_wf,
list_ind_cons_lemma,
list_ind_nil_lemma,
geo-sep-sym
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
dependent_functionElimination,
setElimination,
rename,
dependent_set_memberEquality,
unionElimination,
independent_functionElimination,
productElimination,
isect_memberEquality,
voidElimination,
voidEquality,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
dependent_pairFormation,
inlFormation,
inrFormation,
productEquality,
lambdaEquality
Latex:
\mforall{}g:OrientedPlane. \mforall{}a,b,c,d:Point.
(Colinear(a;b;c) {}\mRightarrow{} Colinear(a;b;d) {}\mRightarrow{} c \mneq{} d {}\mRightarrow{} (\mforall{}y:Point. (y \# ab {}\mRightarrow{} y \# cd)))
Date html generated:
2018_05_22-AM-11_54_34
Last ObjectModification:
2018_04_20-AM-09_48_40
Theory : euclidean!plane!geometry
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