Nuprl Lemma : eu-colinear-append
∀e:EuclideanPlane. ∀L1,L2:Point List.
((∃A,B:Point. ((¬(A = B ∈ Point)) ∧ ((A ∈ L1) ∧ (A ∈ L2)) ∧ (B ∈ L1) ∧ (B ∈ L2)))
⇒ eu-colinear-set(e;L1)
⇒ eu-colinear-set(e;L2)
⇒ eu-colinear-set(e;L1 @ L2))
Proof
Definitions occuring in Statement :
eu-colinear-set: eu-colinear-set(e;L),
euclidean-plane: EuclideanPlane,
eu-point: Point,
l_member: (x ∈ l),
append: as @ bs,
list: T List,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
and: P ∧ Q,
eu-colinear-set: eu-colinear-set(e;L),
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
guard: {T},
stable: Stable{P},
uimplies: b supposing a,
not: ¬A,
or: P ∨ Q,
false: False
Lemmas referenced :
eu-colinear-set_wf,
exists_wf,
eu-point_wf,
not_wf,
equal_wf,
l_member_wf,
list_wf,
euclidean-plane_wf,
l_all_iff,
l_all_wf2,
eu-colinear_wf,
all_wf,
l_all_append,
append_wf,
stable__colinear,
false_wf,
or_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
eu-colinear-transitivity,
eu-colinear-cycle,
eu-colinear-swap,
eu-colinear-def,
member_wf,
eu-between_wf,
eu-colinear-permute
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
because_Cache,
sqequalRule,
lambdaEquality,
productEquality,
addLevel,
dependent_functionElimination,
functionEquality,
setEquality,
independent_functionElimination,
allFunctionality,
levelHypothesis,
promote_hyp,
independent_pairFormation,
impliesFunctionality,
allLevelFunctionality,
impliesLevelFunctionality,
independent_isectElimination,
unionElimination,
voidElimination,
hyp_replacement,
equalitySymmetry,
Error :applyLambdaEquality,
equalityTransitivity
Latex:
\mforall{}e:EuclideanPlane. \mforall{}L1,L2:Point List.
((\mexists{}A,B:Point. ((\mneg{}(A = B)) \mwedge{} ((A \mmember{} L1) \mwedge{} (A \mmember{} L2)) \mwedge{} (B \mmember{} L1) \mwedge{} (B \mmember{} L2)))
{}\mRightarrow{} eu-colinear-set(e;L1)
{}\mRightarrow{} eu-colinear-set(e;L2)
{}\mRightarrow{} eu-colinear-set(e;L1 @ L2))
Date html generated:
2016_10_26-AM-07_44_03
Last ObjectModification:
2016_07_12-AM-08_11_29
Theory : euclidean!geometry
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