Nuprl Lemma : geo-parallel-symmetry
∀e:EuclideanPlane. ∀a,b,c,d:Point. (geo-parallel(e;a;b;c;d) ⇒ geo-parallel(e;c;d;a;b))
Proof
Definitions occuring in Statement :
geo-parallel: geo-parallel(e;a;b;c;d),
euclidean-plane: EuclideanPlane,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
geo-parallel: geo-parallel(e;a;b;c;d),
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
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,
less_than: a < b,
squash: ↓T,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
euclidean-plane: EuclideanPlane,
or: P ∨ Q
Lemmas referenced :
geo-colinear_wf,
geo-parallel_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
lsep-iff-all-sep,
geo-colinear-is-colinear-set,
length_of_cons_lemma,
length_of_nil_lemma,
false_wf,
lelt_wf,
geo-sep-or,
geo-sep_wf,
colinear-lsep,
lsep-all-sym,
geo-sep-sym,
lsep-implies-sep
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
hypothesis,
sqequalHypSubstitution,
productElimination,
thin,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
sqequalRule,
dependent_functionElimination,
instantiate,
independent_isectElimination,
independent_functionElimination,
rename,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
setElimination,
unionElimination
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,d:Point. (geo-parallel(e;a;b;c;d) {}\mRightarrow{} geo-parallel(e;c;d;a;b))
Date html generated:
2018_05_22-PM-00_14_06
Last ObjectModification:
2017_10_12-AM-11_27_33
Theory : euclidean!plane!geometry
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