Nuprl Lemma : Euclid-Prop28_1
∀e:EuclideanPlane. ∀a,b,c,d,x,y,p:Point.
(((Colinear(x;a;b) ∧ Colinear(y;c;d)) ∧ (a leftof yx ∧ a-x-b) ∧ (c leftof xy ∧ c-y-d) ∧ p-x-y ∧ bxp ≅a cyx)
⇒ geo-parallel-points(e;a;b;c;d))
Proof
Definitions occuring in Statement :
geo-parallel-points: geo-parallel-points(e;a;b;c;d),
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-colinear: Colinear(a;b;c),
geo-strict-between: a-b-c,
geo-left: a leftof bc,
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,
member: t ∈ T,
heyting-geometry: HeytingGeometry,
basic-geometry-: BasicGeometry-,
geo-triangle: a # bc,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
basic-geometry: BasicGeometry,
cand: A c∧ B,
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,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
select: L[n],
cons: [a / b],
subtract: n - m
Lemmas referenced :
vert-angles-congruent,
geo-strict-between-sym,
geo-colinear_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-left_wf,
geo-strict-between_wf,
geo-cong-angle_wf,
geo-point_wf,
colinear-lsep-cycle,
lsep-all-sym2,
geo-strict-between-sep2,
geo-colinear-is-colinear-set,
geo-strict-between-implies-colinear,
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,
lsep-all-sym,
colinear-lsep,
geo-sep-sym,
geo-strict-between-sep3,
Euclid-Prop27,
euclidean-plane-axioms,
left-implies-sep,
geo-cong-angle-symm2,
geo-cong-angle-transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
sqequalRule,
hypothesisEquality,
independent_functionElimination,
because_Cache,
hypothesis,
productIsType,
universeIsType,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
inhabitedIsType,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,d,x,y,p:Point.
(((Colinear(x;a;b) \mwedge{} Colinear(y;c;d))
\mwedge{} (a leftof yx \mwedge{} a-x-b)
\mwedge{} (c leftof xy \mwedge{} c-y-d)
\mwedge{} p-x-y
\mwedge{} bxp \mcong{}\msuba{} cyx)
{}\mRightarrow{} geo-parallel-points(e;a;b;c;d))
Date html generated:
2019_10_16-PM-02_38_00
Last ObjectModification:
2019_08_24-PM-06_48_55
Theory : euclidean!plane!geometry
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