Nuprl Lemma : geo-parallel-points-symmetry2
∀e:EuclideanPlane. ∀a,b,c,d:Point.
(geo-parallel-points(e;a;b;c;d)
⇒ (geo-parallel-points(e;b;a;c;d) ∧ geo-parallel-points(e;a;b;d;c) ∧ geo-parallel-points(e;b;a;d;c)))
Proof
Definitions occuring in Statement :
geo-parallel-points: geo-parallel-points(e;a;b;c;d),
euclidean-plane: EuclideanPlane,
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,
cand: A c∧ B,
geo-parallel-points: geo-parallel-points(e;a;b;c;d),
member: t ∈ T,
not: ¬A,
exists: ∃x:A. B[x],
false: False,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
select: L[n],
cons: [a / b],
subtract: n - m
Lemmas referenced :
geo-sep-sym,
geo-colinear_wf,
geo-left_wf,
geo-parallel-points_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf,
subtype_rel_sets_simple,
geo-colinear-is-colinear-set,
length_of_cons_lemma,
istype-void,
length_of_nil_lemma,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-le,
istype-less_than
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
voidElimination,
sqequalRule,
productIsType,
setIsType,
because_Cache,
universeIsType,
isectElimination,
applyEquality,
setElimination,
rename,
instantiate,
independent_isectElimination,
inhabitedIsType,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
dependent_set_memberEquality_alt,
natural_numberEquality,
unionElimination,
approximateComputation
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,d:Point.
(geo-parallel-points(e;a;b;c;d)
{}\mRightarrow{} (geo-parallel-points(e;b;a;c;d)
\mwedge{} geo-parallel-points(e;a;b;d;c)
\mwedge{} geo-parallel-points(e;b;a;d;c)))
Date html generated:
2019_10_16-PM-01_47_14
Last ObjectModification:
2019_08_23-PM-10_05_42
Theory : euclidean!plane!geometry
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