Nuprl Lemma : colinear-preserves-parallel
∀e:EuclideanPlane. ∀a,b,c,d,y:Point. ((geo-parallel(e;a;b;c;d) ∧ Colinear(c;d;y) ∧ c ≠ y) ⇒ geo-parallel(e;a;b;c;y))
Proof
Definitions occuring in Statement :
geo-parallel: geo-parallel(e;a;b;c;d),
euclidean-plane: EuclideanPlane,
geo-colinear: Colinear(a;b;c),
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
geo-parallel: geo-parallel(e;a;b;c;d),
cand: A c∧ B,
member: t ∈ T,
guard: {T},
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
less_than: a < b,
squash: ↓T,
true: True,
uall: ∀[x:A]. B[x],
select: L[n],
cons: [a / b],
subtract: n - m,
subtype_rel: A ⊆r B,
uimplies: b supposing a
Lemmas referenced :
colinear-lsep-cycle,
lsep-all-sym,
geo-sep-sym,
geo-colinear-is-colinear-set,
length_of_cons_lemma,
length_of_nil_lemma,
false_wf,
lelt_wf,
geo-colinear_wf,
geo-parallel_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-sep_wf,
geo-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
cut,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
introduction,
extract_by_obid,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
isectElimination,
because_Cache,
applyEquality,
productEquality,
instantiate,
independent_isectElimination
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,d,y:Point.
((geo-parallel(e;a;b;c;d) \mwedge{} Colinear(c;d;y) \mwedge{} c \mneq{} y) {}\mRightarrow{} geo-parallel(e;a;b;c;y))
Date html generated:
2018_05_22-PM-00_14_14
Last ObjectModification:
2017_10_12-PM-01_38_00
Theory : euclidean!plane!geometry
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