Nuprl Lemma : geo-lt-angle-in-half-plane-implies-left
∀e:EuclideanPlane. ∀w,x,y,z:Point. (xyz < wyz ⇒ w leftof yz ⇒ x leftof yz ⇒ w leftof yx)
Proof
Definitions occuring in Statement :
geo-lt-angle: abc < xyz,
euclidean-plane: EuclideanPlane,
geo-left: a leftof bc,
geo-point: Point,
all: ∀x:A. B[x],
implies: 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,
and: P ∧ Q,
cand: A c∧ B,
geo-lt-angle: abc < xyz,
exists: ∃x:A. B[x],
geo-lsep: a # bc,
or: P ∨ Q,
geo-out: out(p ab),
basic-geometry: BasicGeometry,
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,
oriented-plane: OrientedPlane
Lemmas referenced :
geo-left_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,
lsep-all-sym2,
geo-lt-angle-symm,
lsep-all-sym,
colinear-lsep,
geo-sep-sym,
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,
colinear-lsep2,
geo-between-implies-colinear,
left-implies-sep,
interior-angles-unique2,
lsep-symmetry,
geo-out_weakening,
geo-eq_weakening,
out-preserves-lsep,
cong-angle-preserves-lsep_strong,
geo-between-sep,
lsep-implies-sep,
geo-between_wf,
geo-cong-angle-symm2,
out-preserves-angle-cong_1,
geo-left-out,
geo-left-out-2,
geo-left-out-3,
geo-out_inversion,
geo-lt-angle-symm2,
geo-cong-angle-symmetry,
geo-lt-angle-left,
lt-angle-irrefl
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,
independent_functionElimination,
productElimination,
inlFormation_alt,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
productIsType,
functionIsType
Latex:
\mforall{}e:EuclideanPlane. \mforall{}w,x,y,z:Point. (xyz < wyz {}\mRightarrow{} w leftof yz {}\mRightarrow{} x leftof yz {}\mRightarrow{} w leftof yx)
Date html generated:
2019_10_16-PM-02_28_56
Last ObjectModification:
2019_09_24-PM-03_28_21
Theory : euclidean!plane!geometry
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