Nuprl Lemma : geo-colinear-symmetry
∀[e:BasicGeometry]. ∀[a,b,c:Point].
{Colinear(b;c;a) ∧ Colinear(c;a;b) ∧ Colinear(c;b;a) ∧ Colinear(a;c;b) ∧ Colinear(b;a;c)} supposing Colinear(a;b;c)
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-colinear: Colinear(a;b;c),
geo-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T},
and: P ∧ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
guard: {T},
and: P ∧ Q,
all: ∀x:A. B[x],
basic-geometry: BasicGeometry,
implies: P ⇒ Q,
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,
prop: ℙ,
less_than: a < b,
squash: ↓T,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
cand: A c∧ B,
geo-colinear: Colinear(a;b;c),
subtype_rel: A ⊆r B
Lemmas referenced :
geo-colinear-is-colinear-set,
length_of_cons_lemma,
length_of_nil_lemma,
false_wf,
lelt_wf,
not_wf,
geo-between_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
basic-geometry-subtype,
subtype_rel_transitivity,
basic-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-colinear_wf,
geo-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
lambdaFormation,
imageMemberEquality,
baseClosed,
isectElimination,
because_Cache,
productElimination,
independent_pairEquality,
lambdaEquality,
productEquality,
applyEquality,
instantiate,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[e:BasicGeometry]. \mforall{}[a,b,c:Point].
\{Colinear(b;c;a) \mwedge{} Colinear(c;a;b) \mwedge{} Colinear(c;b;a) \mwedge{} Colinear(a;c;b) \mwedge{} Colinear(b;a;c)\}
supposing Colinear(a;b;c)
Date html generated:
2018_05_22-AM-11_54_02
Last ObjectModification:
2018_04_20-PM-05_47_44
Theory : euclidean!plane!geometry
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