Nuprl Lemma : geo-colinear-append
∀e:EuclideanPlane. ∀L1,L2:Point List.
((∃A,B:Point. (A ≠ B ∧ ((A ∈ L1) ∧ (A ∈ L2)) ∧ (B ∈ L1) ∧ (B ∈ L2)))
⇒ geo-colinear-set(e; L1)
⇒ geo-colinear-set(e; L2)
⇒ geo-colinear-set(e; L1 @ L2))
Proof
Definitions occuring in Statement :
geo-colinear-set: geo-colinear-set(e; L),
euclidean-plane: EuclideanPlane,
geo-sep: a ≠ b,
geo-point: Point,
l_member: (x ∈ l),
append: as @ bs,
list: T List,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
and: P ∧ Q,
geo-colinear-set: geo-colinear-set(e; L),
member: t ∈ T,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
not: ¬A,
false: False,
or: P ∨ Q,
stable: Stable{P},
geo-eq: a ≡ b
Lemmas referenced :
geo-colinear-set_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-sep_wf,
l_member_wf,
list_wf,
l_all_iff,
l_all_wf2,
geo-colinear_wf,
l_all_append,
append_wf,
geo-colinear-transitivity,
geo-colinear-cycle,
stable__colinear,
false_wf,
or_wf,
not_wf,
istype-void,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
geo-colinear-permute,
geo-colinear_functionality,
geo-eq_weakening,
geo-eq_inversion
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
universeIsType,
cut,
introduction,
extract_by_obid,
isectElimination,
setElimination,
rename,
hypothesisEquality,
hypothesis,
sqequalRule,
productIsType,
applyEquality,
instantiate,
independent_isectElimination,
because_Cache,
inhabitedIsType,
dependent_functionElimination,
lambdaEquality_alt,
setIsType,
independent_functionElimination,
independent_pairFormation,
functionEquality,
functionIsType,
unionIsType,
unionElimination,
voidElimination
Latex:
\mforall{}e:EuclideanPlane. \mforall{}L1,L2:Point List.
((\mexists{}A,B:Point. (A \mneq{} B \mwedge{} ((A \mmember{} L1) \mwedge{} (A \mmember{} L2)) \mwedge{} (B \mmember{} L1) \mwedge{} (B \mmember{} L2)))
{}\mRightarrow{} geo-colinear-set(e; L1)
{}\mRightarrow{} geo-colinear-set(e; L2)
{}\mRightarrow{} geo-colinear-set(e; L1 @ L2))
Date html generated:
2019_10_16-PM-01_14_32
Last ObjectModification:
2018_12_11-AM-11_39_39
Theory : euclidean!plane!geometry
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