Nuprl Lemma : geo-colinear-switch2
∀e:BasicGeometry. ∀a,b,c:Point. (Colinear(a;b;c) ⇒ Colinear(b;a;c))
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-colinear: Colinear(a;b;c),
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
basic-geometry: BasicGeometry,
subtract: n - m,
cons: [a / b],
select: L[n],
uall: ∀[x:A]. B[x],
true: True,
squash: ↓T,
less_than: a < b,
prop: ℙ,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
top: Top,
l_all: (∀x∈L.P[x]),
geo-colinear-set: geo-colinear-set(e; L),
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-point_wf,
geo-colinear_wf,
lelt_wf,
false_wf,
length_of_nil_lemma,
length_of_cons_lemma,
geo-colinear-is-colinear-set
Rules used in proof :
independent_isectElimination,
instantiate,
applyEquality,
rename,
setElimination,
because_Cache,
isectElimination,
baseClosed,
imageMemberEquality,
independent_pairFormation,
natural_numberEquality,
dependent_set_memberEquality,
voidEquality,
voidElimination,
isect_memberEquality,
sqequalRule,
hypothesis,
independent_functionElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c:Point. (Colinear(a;b;c) {}\mRightarrow{} Colinear(b;a;c))
Date html generated:
2018_05_22-PM-00_02_09
Last ObjectModification:
2018_04_12-AM-10_33_11
Theory : euclidean!plane!geometry
Home
Index