Nuprl Lemma : not-lt-zero-angle
∀e:EuclideanPlane. ∀a,b,c,x,y,z:Point. (abc < xyz ⇒ (y_z_x ∨ y_x_z) ⇒ False)
Proof
Definitions occuring in Statement :
geo-lt-angle: abc < xyz,
euclidean-plane: EuclideanPlane,
geo-between: a_b_c,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
false: False
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
false: False,
geo-lt-angle: abc < xyz,
and: P ∧ Q,
exists: ∃x:A. B[x],
or: P ∨ Q,
member: t ∈ T,
basic-geometry: BasicGeometry,
geo-out: out(p ab),
not: ¬A,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ
Lemmas referenced :
geo-out_inversion,
geo-between-out,
euclidean-plane-axioms,
geo-sep-sym,
geo-between_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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalHypSubstitution,
productElimination,
thin,
unionElimination,
introduction,
extract_by_obid,
dependent_functionElimination,
sqequalRule,
hypothesisEquality,
independent_functionElimination,
because_Cache,
hypothesis,
voidElimination,
unionIsType,
universeIsType,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
inhabitedIsType
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,x,y,z:Point. (abc < xyz {}\mRightarrow{} (y\_z\_x \mvee{} y\_x\_z) {}\mRightarrow{} False)
Date html generated:
2019_10_16-PM-01_49_25
Last ObjectModification:
2019_09_27-PM-05_58_24
Theory : euclidean!plane!geometry
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