Nuprl Lemma : geo-tar-opp-side-mid1
∀e:HeytingGeometry. ∀a,c,r,m:Point.
(geo-tar-opp-side(e;a;c;r;m) ⇒ a=m=c ⇒ (∀b:Point. (r-a-b ⇒ geo-tar-opp-side(e;b;c;r;m))))
Proof
Definitions occuring in Statement :
heyting-geometry: HeytingGeometry,
geo-tar-opp-side: geo-tar-opp-side(e;a;b;p;q),
geo-midpoint: a=m=b,
geo-strict-between: a-b-c,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
geo-tar-opp-side: geo-tar-opp-side(e;a;b;p;q),
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
guard: {T},
subtype_rel: A ⊆r B,
heyting-geometry: HeytingGeometry,
euclidean-plane: EuclideanPlane,
basic-geometry-: BasicGeometry-,
uall: ∀[x:A]. B[x],
prop: ℙ,
basic-geometry: BasicGeometry,
uimplies: b supposing a,
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],
geo-midpoint: a=m=b,
uiff: uiff(P;Q),
geo-strict-between: a-b-c,
geo-triangle: a # bc,
oriented-plane: OrientedPlane,
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]),
so_apply: x[s1;s2;s3]
Lemmas referenced :
colinear-lsep-cycle,
lsep-all-sym,
geo-sep-sym,
geo-strict-between-sep1,
subtype_rel_self,
euclidean-plane-structure_wf,
basic-geo-axioms_wf,
geo-left-axioms_wf,
geo-colinear-is-colinear-set,
geo-strict-between-implies-colinear,
length_of_cons_lemma,
length_of_nil_lemma,
false_wf,
lelt_wf,
geo-strict-between_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
heyting-geometry-subtype,
subtype_rel_transitivity,
heyting-geometry_wf,
euclidean-plane_wf,
geo-primitives_wf,
geo-point_wf,
geo-midpoint_wf,
geo-tar-opp-side_wf,
lsep-implies-sep,
geo-proper-extend-exists,
geo-midpoint-diagonals-between,
geo-between-symmetry,
geo-strict-between-implies-between,
geo-congruent-iff-length,
symmetry-preserves-congruence,
geo-midpoint-symmetry,
colinear-implies-midpoint,
geo-length-flip,
euclidean-plane-axioms,
geo-congruent-symmetry,
geo-congruent-sep,
geo-strict-between-sep3,
colinear-lsep,
geo-inner-pasch-ex,
geo-strict-between-sym,
geo-triangle_wf,
colinear-lsep2,
oriented-colinear-append,
cons_wf,
nil_wf,
cons_member,
l_member_wf,
equal_wf,
geo-sep_wf,
exists_wf,
list_ind_cons_lemma,
list_ind_nil_lemma,
geo-colinear_wf,
geo-between_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
because_Cache,
hypothesisEquality,
independent_functionElimination,
hypothesis,
applyEquality,
sqequalRule,
instantiate,
isectElimination,
setEquality,
productEquality,
cumulativity,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
equalitySymmetry,
equalityTransitivity,
dependent_pairFormation,
inrFormation,
inlFormation,
lambdaEquality
Latex:
\mforall{}e:HeytingGeometry. \mforall{}a,c,r,m:Point.
(geo-tar-opp-side(e;a;c;r;m) {}\mRightarrow{} a=m=c {}\mRightarrow{} (\mforall{}b:Point. (r-a-b {}\mRightarrow{} geo-tar-opp-side(e;b;c;r;m))))
Date html generated:
2017_10_02-PM-07_05_03
Last ObjectModification:
2017_08_10-PM-06_05_48
Theory : euclidean!plane!geometry
Home
Index