Nuprl Lemma : lsep-iff-all-sep
∀g:EuclideanPlane. ∀a,b,c:Point. (a # bc ⇐⇒ (∀x:Point. (Colinear(x;b;c) ⇒ a ≠ x)) ∧ b ≠ c)
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],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
rev_implies: P ⇐ Q,
oriented-plane: OrientedPlane,
exists: ∃x:A. B[x],
sq_exists: ∃x:A [B[x]],
euclidean-plane: EuclideanPlane,
sq_stable: SqStable(P),
squash: ↓T,
basic-geometry: BasicGeometry,
geo-equilateral: EQΔ(a;b;c),
geo-midpoint: a=m=b,
uiff: uiff(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,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
or: P ∨ Q,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3]
Lemmas referenced :
geo-colinear_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
lsep-implies-sep,
geo-lsep_wf,
geo-sep_wf,
geo-point_wf,
lsep-colinear-sep,
colinear-equidistant-points-exist,
Euclid-midpoint,
sq_stable__midpoint,
midpoint-sep,
Euclid-Prop1,
upper-dimension-axiom,
geo-congruent-iff-length,
geo-length-flip,
colinear-lsep-cycle,
lsep-all-sym,
geo-sep-sym,
geo-colinear-is-colinear-set,
geo-between-implies-colinear,
length_of_cons_lemma,
istype-void,
length_of_nil_lemma,
istype-false,
istype-le,
istype-less_than,
colinear-lsep,
oriented-colinear-append,
cons_wf,
nil_wf,
cons_member,
l_member_wf,
list_ind_cons_lemma,
list_ind_nil_lemma,
geo-sep-or,
colinear-lsep'
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
sqequalRule,
inhabitedIsType,
dependent_functionElimination,
independent_functionElimination,
productElimination,
because_Cache,
productIsType,
functionIsType,
dependent_set_memberEquality_alt,
setElimination,
rename,
imageMemberEquality,
baseClosed,
imageElimination,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality_alt,
voidElimination,
natural_numberEquality,
dependent_pairFormation_alt,
inrFormation_alt,
inlFormation_alt,
equalityIsType1,
unionElimination
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c:Point. (a \# bc \mLeftarrow{}{}\mRightarrow{} (\mforall{}x:Point. (Colinear(x;b;c) {}\mRightarrow{} a \mneq{} x)) \mwedge{} b \mneq{} c)
Date html generated:
2019_10_16-PM-01_42_53
Last ObjectModification:
2018_11_08-PM-02_12_29
Theory : euclidean!plane!geometry
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