Nuprl Lemma : geo-colinear-preserves-parallel
∀e:EuclideanPlane. ∀a,b,c,d,x:Point.
(geo-parallel-points(e;a;b;c;d) ⇒ Colinear(a;b;x) ⇒ a ≠ x ⇒ geo-parallel-points(e;a;x;c;d))
Proof
Definitions occuring in Statement :
geo-parallel-points: geo-parallel-points(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
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
geo-parallel-points: geo-parallel-points(e;a;b;c;d),
and: P ∧ Q,
not: ¬A,
exists: ∃x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
l_member: (x ∈ l),
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
top: Top,
select: L[n],
cons: [a / b],
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
true: True,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
prop: ℙ,
subtract: n - m,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
guard: {T}
Lemmas referenced :
subtype_rel_sets_simple,
geo-point_wf,
geo-colinear_wf,
geo-colinear-append,
cons_wf,
nil_wf,
istype-void,
istype-le,
length_of_cons_lemma,
length_of_nil_lemma,
istype-less_than,
length_wf,
select_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
geo-sep_wf,
l_member_wf,
geo-colinear-is-colinear-set,
list_ind_cons_lemma,
list_ind_nil_lemma,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
geo-left_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-parallel-points_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
hypothesis,
independent_functionElimination,
dependent_pairFormation_alt,
cut,
hypothesisEquality,
applyEquality,
introduction,
extract_by_obid,
isectElimination,
because_Cache,
sqequalRule,
lambdaEquality_alt,
inhabitedIsType,
independent_isectElimination,
dependent_functionElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
voidElimination,
isect_memberEquality_alt,
imageMemberEquality,
baseClosed,
productIsType,
setElimination,
rename,
equalityIstype,
unionElimination,
approximateComputation,
int_eqEquality,
universeIsType,
equalityTransitivity,
equalitySymmetry,
setIsType,
instantiate
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,d,x:Point.
(geo-parallel-points(e;a;b;c;d) {}\mRightarrow{} Colinear(a;b;x) {}\mRightarrow{} a \mneq{} x {}\mRightarrow{} geo-parallel-points(e;a;x;c;d))
Date html generated:
2019_10_16-PM-01_46_44
Last ObjectModification:
2019_08_23-PM-10_00_50
Theory : euclidean!plane!geometry
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