Nuprl Lemma : unique-angles-in-half-plane
∀e:EuclideanPlane. ∀a,b,c,x,y,z:Point.
(a # bc
⇒ x # yz
⇒ ((∃f:Point. (acb ≅a fzy ∧ (x leftof yz ⇐⇒ f leftof yz) ∧ (x leftof zy ⇐⇒ f leftof zy)))
∧ (∀f1,f2:Point.
((acb ≅a f1zy ∧ acb ≅a f2zy)
⇒ (((x leftof yz ⇐⇒ f1 leftof yz) ∧ (x leftof zy ⇐⇒ f1 leftof zy))
∧ (x leftof yz ⇐⇒ f2 leftof yz)
∧ (x leftof zy ⇐⇒ f2 leftof zy))
⇒ Colinear(z;f1;f2)))))
Proof
Definitions occuring in Statement :
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-colinear: Colinear(a;b;c),
geo-lsep: a # bc,
geo-left: a leftof bc,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
append: as @ bs,
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
ge: i ≥ j ,
nat: ℕ,
l_member: (x ∈ l),
basic-geometry-: BasicGeometry-,
geo-cong-tri: Cong3(abc,a'b'c'),
geo-cong-angle: abc ≅a xyz,
uiff: uiff(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),
less_than: a < b,
squash: ↓T,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
geo-lsep: a # bc,
or: P ∨ Q,
geo-out: out(p ab),
not: ¬A,
false: False,
exists: ∃x:A. B[x],
basic-geometry: BasicGeometry,
rev_implies: P ⇐ Q,
prop: ℙ,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
cand: A c∧ B,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
list_ind_nil_lemma,
list_ind_cons_lemma,
l_member_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
istype-int,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
select_wf,
length_wf,
nil_wf,
cons_wf,
geo-colinear-append,
left-convex2,
geo-eq_inversion,
Euclid-Prop7,
geo-sep_wf,
left-convex,
geo-between_functionality,
geo-eq_weakening,
geo-congruent_functionality,
geo-inner-five-segment,
geo-construction-unicity,
geo-congruent-refl,
geo-between-symmetry,
geo-inner-three-segment,
geo-cong-angle-transitivity,
geo-cong-angle-symm2,
euclidean-plane-axioms,
geo-between-trivial,
geo-length-flip,
geo-congruent_wf,
cong3-in-half-plane,
geo-congruent-iff-length,
geo-add-length-between,
geo-add-length_wf,
squash_wf,
true_wf,
geo-length-type_wf,
basic-geometry_wf,
geo-add-length-comm,
not-left-and-right,
lsep-all-sym2,
colinear-lsep-cycle,
lsep-all-sym,
geo-colinear-is-colinear-set,
geo-between-implies-colinear,
length_of_cons_lemma,
length_of_nil_lemma,
istype-false,
istype-le,
istype-less_than,
geo-left-out,
geo-between-sep,
istype-void,
geo-between_wf,
geo-extend-exists,
geo-sep-sym,
lsep-implies-sep,
geo-point_wf,
geo-lsep_wf,
geo-cong-angle_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-left_wf
Rules used in proof :
int_eqEquality,
approximateComputation,
equalityIstype,
setElimination,
inlFormation_alt,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation_alt,
promote_hyp,
isect_memberEquality_alt,
dependent_set_memberEquality_alt,
natural_numberEquality,
imageMemberEquality,
baseClosed,
unionElimination,
voidElimination,
dependent_functionElimination,
independent_functionElimination,
rename,
inhabitedIsType,
because_Cache,
independent_isectElimination,
instantiate,
applyEquality,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
universeIsType,
functionIsType,
productIsType,
sqequalRule,
thin,
productElimination,
sqequalHypSubstitution,
hypothesis,
independent_pairFormation,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,x,y,z:Point.
(a \# bc
{}\mRightarrow{} x \# yz
{}\mRightarrow{} ((\mexists{}f:Point. (acb \mcong{}\msuba{} fzy \mwedge{} (x leftof yz \mLeftarrow{}{}\mRightarrow{} f leftof yz) \mwedge{} (x leftof zy \mLeftarrow{}{}\mRightarrow{} f leftof zy)))
\mwedge{} (\mforall{}f1,f2:Point.
((acb \mcong{}\msuba{} f1zy \mwedge{} acb \mcong{}\msuba{} f2zy)
{}\mRightarrow{} (((x leftof yz \mLeftarrow{}{}\mRightarrow{} f1 leftof yz) \mwedge{} (x leftof zy \mLeftarrow{}{}\mRightarrow{} f1 leftof zy))
\mwedge{} (x leftof yz \mLeftarrow{}{}\mRightarrow{} f2 leftof yz)
\mwedge{} (x leftof zy \mLeftarrow{}{}\mRightarrow{} f2 leftof zy))
{}\mRightarrow{} Colinear(z;f1;f2)))))
Date html generated:
2019_10_29-AM-09_18_40
Last ObjectModification:
2019_10_18-PM-04_51_29
Theory : euclidean!plane!geometry
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