Nuprl Lemma : lsep-colinear-sep
∀g:OrientedPlane. ∀a,b,c:Point. (a # bc ⇒ (∀y:Point. (Colinear(y;b;c) ⇒ a ≠ y)))
Proof
Definitions occuring in Statement :
oriented-plane: OrientedPlane,
geo-lsep: a # bc,
geo-colinear: Colinear(a;b;c),
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
oriented-plane: Error :oriented-plane,
or: P ∨ Q,
prop: ℙ,
and: P ∧ Q,
uimplies: b supposing a,
guard: {T},
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
cand: A c∧ B,
false: False,
not: ¬A,
geo-colinear: Colinear(a;b;c)
Lemmas referenced :
geo-lsep_wf,
Error :basic-geo-primitives_wf,
geo-point_wf,
geo-colinear_wf,
geo-sep_wf,
lsep-implies-sep,
Error :oriented-plane-subtype,
geo-sep-sym,
Error :basic-geo-structure_wf,
Error :o-geo-structure_wf,
Error :oriented-plane_wf,
subtype_rel_transitivity,
Error :oriented-plane-subtype1,
Error :o-geo-structure-subtype,
geo-sep-or,
lsep-iff,
lsep-all-sym,
not_wf,
geo-between_wf,
geo-between-symmetry
Rules used in proof :
rename,
setElimination,
unionElimination,
dependent_set_memberEquality,
productElimination,
because_Cache,
independent_functionElimination,
sqequalRule,
independent_isectElimination,
isectElimination,
instantiate,
hypothesis,
applyEquality,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
productEquality,
independent_pairFormation,
voidElimination
Latex:
\mforall{}g:OrientedPlane. \mforall{}a,b,c:Point. (a \# bc {}\mRightarrow{} (\mforall{}y:Point. (Colinear(y;b;c) {}\mRightarrow{} a \mneq{} y)))
Date html generated:
2017_10_02-PM-04_47_27
Last ObjectModification:
2017_08_05-AM-10_20_13
Theory : euclidean!plane!geometry
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