Nuprl Lemma : geo-equilateral-exists
∀e:BasicGeometry-. ∀a,b,c:Point. (a # bc ⇒ (∃x:Point. ((ax ≅ ac ∧ out(a bx)) ∧ a # xc)))
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
basic-geometry-: BasicGeometry-,
geo-lsep: a # bc,
geo-congruent: ab ≅ cd,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
geo-lsep: a # bc,
or: P ∨ Q,
member: t ∈ T,
basic-geometry-: BasicGeometry-,
guard: {T},
and: P ∧ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
euclidean-plane: EuclideanPlane,
exists: ∃x:A. B[x],
prop: ℙ,
geo-strict-between: a-b-c,
iff: 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,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
select: L[n],
cons: [a / b],
subtract: n - m
Lemmas referenced :
left-implies-sep,
euclidean-plane-axioms,
geo-proper-extend-exists,
euclidean-plane-subtype-basic,
basic-geometry--subtype,
subtype_rel_transitivity,
basic-geometry-_wf,
euclidean-plane_wf,
basic-geometry_wf,
geo-O_wf,
geo-X_wf,
geo-sep-O-X,
geo-lsep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf,
geo-congruent_wf,
geo-out_wf,
geo-out-iff-between1,
geo-between-symmetry,
geo-strict-between-implies-between,
geo-out_inversion,
colinear-lsep,
lsep-symmetry,
lsep-all-sym,
geo-colinear-permute,
geo-colinear-is-colinear-set,
geo-out-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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalHypSubstitution,
unionElimination,
thin,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
productElimination,
independent_pairFormation,
because_Cache,
applyEquality,
instantiate,
isectElimination,
independent_isectElimination,
sqequalRule,
setElimination,
rename,
universeIsType,
inhabitedIsType,
dependent_pairFormation_alt,
productIsType,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
approximateComputation,
lambdaEquality_alt
Latex:
\mforall{}e:BasicGeometry-. \mforall{}a,b,c:Point. (a \# bc {}\mRightarrow{} (\mexists{}x:Point. ((ax \mcong{} ac \mwedge{} out(a bx)) \mwedge{} a \# xc)))
Date html generated:
2019_10_16-PM-01_25_33
Last ObjectModification:
2019_01_04-AM-11_51_35
Theory : euclidean!plane!geometry
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