Nuprl Lemma : weak-same-side-invariant
∀[e:BasicGeometry]. ∀[A,B,C,D:Point].
(¬¬(((∀P:Point. (P leftof AB ⇐⇒ P leftof CD)) ∧ (∀P:Point. (P leftof BA ⇐⇒ P leftof DC)))
∨ ((∀P:Point. (P leftof AB ⇐⇒ P leftof DC)) ∧ (∀P:Point. (P leftof BA ⇐⇒ P leftof CD))))) supposing
(A ≠ B and
C ≠ D and
Colinear(C;A;B) and
Colinear(D;A;B))
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-colinear: Colinear(a;b;c),
geo-left: a leftof bc,
geo-sep: a ≠ b,
geo-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
or: P ∨ Q,
and: P ∧ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
basic-geometry-: BasicGeometry-,
all: ∀x:A. B[x],
prop: ℙ,
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T},
stable: Stable{P},
or: P ∨ Q,
geo-eq: a ≡ b,
cand: A c∧ B,
geo-out: out(p ab),
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),
less_than: a < b,
squash: ↓T,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
exists: ∃x:A. B[x],
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3]
Lemmas referenced :
basic-geometry_wf,
geo-simple-colinear-cases,
subtype_rel_self,
basic-geometry-_wf,
false_wf,
stable__false,
geo-between_wf,
not_wf,
or_wf,
all_wf,
geo-point_wf,
iff_wf,
geo-left_wf,
geo-sep_wf,
geo-colinear_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
basic-geometry-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
euclidean-plane-subtype-oriented,
oriented-plane_wf,
minimal-double-negation-hyp-elim,
geo-between_functionality,
geo-eq_weakening,
geo-left_functionality,
geo-sep_functionality,
geo-colinear_functionality,
minimal-not-not-excluded-middle,
left-convex2,
left-all-symmetry,
left-convex,
geo-between-out,
geo-out_wf,
geo-sep-sym,
geo-left-out-better-1,
geo-between-symmetry,
left-between-implies-right1,
geo-out_inversion,
geo-left-out-better,
left-between-implies-right2,
geo-sep-or,
geo-colinear-is-colinear-set,
length_of_cons_lemma,
length_of_nil_lemma,
lelt_wf,
oriented-colinear-append,
cons_wf,
nil_wf,
cons_member,
l_member_wf,
equal_wf,
exists_wf,
list_ind_cons_lemma,
list_ind_nil_lemma,
double-negation-hyp-elim
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
hypothesis,
lambdaFormation,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
sqequalRule,
instantiate,
dependent_functionElimination,
independent_functionElimination,
because_Cache,
productEquality,
lambdaEquality,
independent_isectElimination,
functionEquality,
unionElimination,
voidElimination,
productElimination,
addLevel,
impliesFunctionality,
orFunctionality,
independent_pairFormation,
allFunctionality,
andLevelFunctionality,
allLevelFunctionality,
impliesLevelFunctionality,
orLevelFunctionality,
levelHypothesis,
promote_hyp,
inlFormation,
inrFormation,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
dependent_set_memberEquality,
voidEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
dependent_pairFormation
Latex:
\mforall{}[e:BasicGeometry]. \mforall{}[A,B,C,D:Point].
(\mneg{}\mneg{}(((\mforall{}P:Point. (P leftof AB \mLeftarrow{}{}\mRightarrow{} P leftof CD)) \mwedge{} (\mforall{}P:Point. (P leftof BA \mLeftarrow{}{}\mRightarrow{} P leftof DC)))
\mvee{} ((\mforall{}P:Point. (P leftof AB \mLeftarrow{}{}\mRightarrow{} P leftof DC))
\mwedge{} (\mforall{}P:Point. (P leftof BA \mLeftarrow{}{}\mRightarrow{} P leftof CD))))) supposing
(A \mneq{} B and
C \mneq{} D and
Colinear(C;A;B) and
Colinear(D;A;B))
Date html generated:
2018_05_22-PM-00_00_03
Last ObjectModification:
2018_04_23-PM-06_55_04
Theory : euclidean!plane!geometry
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