Nuprl Lemma : use-plane-sep_strict
∀g:EuclideanPlane. ∀a,b,u,v:Point. (u leftof ab ⇒ v leftof ba ⇒ (∃x:Point. (Colinear(a;b;x) ∧ u-x-v)))
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-colinear: Colinear(a;b;c),
geo-strict-between: a-b-c,
geo-left: a leftof bc,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
subtract: n - m,
cons: [a / b],
select: L[n],
true: True,
squash: ↓T,
less_than: a < b,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
top: Top,
l_all: (∀x∈L.P[x]),
geo-colinear-set: geo-colinear-set(e; L),
or: P ∨ Q,
geo-lsep: a # bc,
iff: P ⇐⇒ Q,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
geo-strict-between: a-b-c,
cand: A c∧ B,
and: P ∧ Q,
exists: ∃x:A. B[x],
euclidean-plane: EuclideanPlane,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
geo-sep-sym,
lelt_wf,
false_wf,
length_of_nil_lemma,
length_of_cons_lemma,
geo-colinear-is-colinear-set,
geo-point_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-left_wf,
geo-strict-between_wf,
geo-colinear_wf,
lsep-iff-all-sep,
use-plane-sep
Rules used in proof :
inlFormation,
baseClosed,
imageMemberEquality,
natural_numberEquality,
dependent_set_memberEquality,
voidEquality,
voidElimination,
isect_memberEquality,
inrFormation,
independent_isectElimination,
instantiate,
sqequalRule,
applyEquality,
isectElimination,
productEquality,
because_Cache,
independent_pairFormation,
dependent_pairFormation,
productElimination,
independent_functionElimination,
hypothesis,
hypothesisEquality,
rename,
setElimination,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,u,v:Point.
(u leftof ab {}\mRightarrow{} v leftof ba {}\mRightarrow{} (\mexists{}x:Point. (Colinear(a;b;x) \mwedge{} u-x-v)))
Date html generated:
2018_05_22-PM-00_20_16
Last ObjectModification:
2018_05_21-AM-01_19_22
Theory : euclidean!plane!geometry
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