Nuprl Lemma : geo-colinear-sep-cases
∀e:BasicGeometry. ∀a,b,c:Point. (a ≠ b ⇒ Colinear(a;b;c) ⇒ ((¬((¬a_c_b) ∧ (¬a_b_c))) ∨ (¬((¬b_c_a) ∧ (¬b_a_c)))))
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-colinear: Colinear(a;b;c),
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
or: P ∨ Q,
exists: ∃x:A. B[x],
geo-midpoint: a=m=b,
and: P ∧ Q,
uimplies: b supposing a,
cand: A c∧ B,
guard: {T},
not: ¬A,
false: False,
basic-geometry-: BasicGeometry-,
stable: Stable{P},
uiff: uiff(P;Q),
geo-eq: a ≡ 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),
less_than: a < b,
squash: ↓T,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
decidable: Dec(P)
Lemmas referenced :
geo-sep-or,
geo-sep_wf,
geo-colinear_wf,
geo-point_wf,
basic-geometry_wf,
symmetric-point-construction,
extend-using-SC,
geo-sep-sym,
geo-congruent-symmetry,
geo-congruent-sep,
stable__geo-between,
false_wf,
or_wf,
geo-between_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
basic-geometry-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
not_wf,
geo-between-same-side2,
geo-between-symmetry,
geo-between-inner-trans,
geo-between-exchange4,
subtype_rel_self,
basic-geometry-_wf,
geo-between-outer-trans,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
geo-between-sep,
geo-simple-colinear-cases,
stable__false,
geo-construction-unicity,
geo-congruent-iff-length,
geo-colinear-is-colinear-set,
length_of_cons_lemma,
length_of_nil_lemma,
lelt_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
dependent_set_memberEquality,
isectElimination,
applyEquality,
because_Cache,
sqequalRule,
unionElimination,
independent_functionElimination,
productElimination,
independent_isectElimination,
independent_pairFormation,
instantiate,
functionEquality,
voidElimination,
inlFormation,
productEquality,
inrFormation,
equalitySymmetry,
isect_memberEquality,
voidEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
promote_hyp
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c:Point.
(a \mneq{} b {}\mRightarrow{} Colinear(a;b;c) {}\mRightarrow{} ((\mneg{}((\mneg{}a\_c\_b) \mwedge{} (\mneg{}a\_b\_c))) \mvee{} (\mneg{}((\mneg{}b\_c\_a) \mwedge{} (\mneg{}b\_a\_c)))))
Date html generated:
2018_05_22-AM-11_56_35
Last ObjectModification:
2018_04_09-PM-05_51_51
Theory : euclidean!plane!geometry
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