Nuprl Lemma : geo-lt-angle-triangle-point-exists
∀e:EuclideanPlane. ∀a,b,c,x,y,z:Point. (abc < xyz ⇒ a # bc ⇒ x # yz ⇒ (∃p:Point. (x-p-z ∧ xyp ≅a abc)))
Proof
Definitions occuring in Statement :
geo-lt-angle: abc < xyz,
geo-cong-angle: abc ≅a xyz,
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
geo-strict-between: a-b-c,
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,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
geo-lt-angle: abc < xyz,
and: P ∧ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
oriented-plane: OrientedPlane,
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),
false: False,
select: L[n],
cons: [a / b],
subtract: n - m,
geo-out: out(p ab),
basic-geometry: BasicGeometry,
geo-strict-between: a-b-c
Lemmas referenced :
geo-lsep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-lt-angle_wf,
geo-point_wf,
out-preserves-lsep,
lsep-symmetry,
lsep-all-sym,
colinear-lsep2,
geo-colinear-is-colinear-set,
geo-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,
geo-sep-sym,
geo-out-colinear,
geo-between-sep,
lsep-implies-sep,
geo-between-out,
geo-out_inversion,
geo-out-interior-point-exists,
geo-out_weakening,
geo-eq_weakening,
out-preserves-angle-cong_1,
lsep-cong-angle-implies-sep,
geo-strict-between_wf,
geo-cong-angle_wf,
geo-cong-angle-symm2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
dependent_functionElimination,
inhabitedIsType,
productElimination,
independent_functionElimination,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
productIsType,
rename
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,x,y,z:Point.
(abc < xyz {}\mRightarrow{} a \# bc {}\mRightarrow{} x \# yz {}\mRightarrow{} (\mexists{}p:Point. (x-p-z \mwedge{} xyp \mcong{}\msuba{} abc)))
Date html generated:
2019_10_16-PM-02_00_38
Last ObjectModification:
2019_09_12-AM-11_37_36
Theory : euclidean!plane!geometry
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