Nuprl Lemma : colinear-lsep'
∀g:OrientedPlane. ∀a,b,c,y:Point. (y # ab ⇒ b ≠ c ⇒ Colinear(a;b;c) ⇒ y # cb)
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 :
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
subtract: n - m,
cons: [a / b],
select: L[n],
l_all: (∀x∈L.P[x]),
geo-colinear-set: geo-colinear-set(e; L),
cand: A c∧ B,
and: P ∧ Q,
guard: {T},
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
oriented-plane: OrientedPlane
Lemmas referenced :
geo-point_wf,
geo-lsep_wf,
geo-sep_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
oriented-plane_wf,
subtype_rel_transitivity,
oriented-plane-subtype,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-colinear_wf,
geo-colinear-is-colinear-set,
geo-sep-sym,
lsep-all-sym,
colinear-lsep
Rules used in proof :
independent_isectElimination,
instantiate,
applyEquality,
isectElimination,
sqequalRule,
productElimination,
hypothesis,
independent_functionElimination,
hypothesisEquality,
because_Cache,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}g:OrientedPlane. \mforall{}a,b,c,y:Point. (y \# ab {}\mRightarrow{} b \mneq{} c {}\mRightarrow{} Colinear(a;b;c) {}\mRightarrow{} y \# cb)
Date html generated:
2017_10_02-PM-04_47_07
Last ObjectModification:
2017_08_08-PM-00_34_39
Theory : euclidean!plane!geometry
Home
Index