Nuprl Lemma : lsep-implies-sep-or-not-colinear
∀e:EuclideanPlane. ∀a,b,c,x:Point. (a # bc ⇒ (c ≠ x ∨ (¬Colinear(a;b;x))))
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],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
oriented-plane: OrientedPlane,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
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,
cand: A c∧ B,
euclidean-plane: EuclideanPlane,
or: P ∨ Q,
basic-geometry: BasicGeometry,
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
basic-geometry-: BasicGeometry-,
geo-eq: a ≡ b,
geo-strict-between: a-b-c,
uiff: uiff(P;Q),
stable: Stable{P},
geo-gt: cd > ab
Lemmas referenced :
geo-lsep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf,
lsep-colinear-sep1,
geo-colinear-is-colinear-set,
length_of_cons_lemma,
istype-void,
length_of_nil_lemma,
istype-false,
istype-le,
istype-less_than,
geo-colinear_wf,
lsep-all-sym,
euclidean-plane-axioms,
geo-sep-or,
lsep-implies-sep,
geo-sep_wf,
geo-sep-sym,
geo-strict-between-sep3,
geo-proper-extend-exists,
geo-strict-between-implies-colinear,
oriented-colinear-append,
cons_wf,
nil_wf,
cons_member,
l_member_wf,
list_ind_cons_lemma,
list_ind_nil_lemma,
geo-colinear-cases,
false_wf,
stable__false,
geo-eq_wf,
geo-strict-between_wf,
geo-construction-unicity,
geo-strict-between-implies-between,
geo-congruent-iff-length,
not_wf,
minimal-double-negation-hyp-elim,
geo-sep_functionality,
geo-eq_weakening,
geo-congruent_functionality,
geo-strict-between_functionality,
minimal-not-not-excluded-middle,
geo-between-same-side-or,
geo-between-exchange3,
geo-between-exchange4,
geo-between-symmetry,
geo-between-outer-trans,
geo-between-inner-trans,
geo-congruent-refl,
geo-between_wf,
geo-congruent_wf,
geo-le-from-be,
geo-lt_wf,
geo-length_wf,
geo-mk-seg_wf,
geo-gt-implies-lt,
not-lt-and-symm-le,
geo-between-congruent,
geo-between_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
sqequalRule,
inhabitedIsType,
because_Cache,
dependent_functionElimination,
dependent_set_memberEquality_alt,
independent_functionElimination,
isect_memberEquality_alt,
voidElimination,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
productIsType,
productElimination,
setElimination,
rename,
unionElimination,
inlFormation_alt,
functionIsType,
dependent_pairFormation_alt,
inrFormation_alt,
equalityIstype,
equalitySymmetry,
unionEquality,
functionEquality,
unionIsType,
hyp_replacement,
equalityTransitivity,
applyLambdaEquality
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,x:Point. (a \# bc {}\mRightarrow{} (c \mneq{} x \mvee{} (\mneg{}Colinear(a;b;x))))
Date html generated:
2019_10_16-PM-02_56_01
Last ObjectModification:
2018_12_05-PM-10_06_04
Theory : euclidean!plane!geometry
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