Nuprl Lemma : geo-ge-cases
∀e:BasicGeometry. ∀a,b,c,d:Point. (¬¬((ab > cd ∨ cd > ab) ∨ ab ≅ cd))
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-gt: cd > ab,
geo-congruent: ab ≅ cd,
geo-point: Point,
all: ∀x:A. B[x],
not: ¬A,
or: P ∨ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
false: False,
member: t ∈ T,
uall: ∀[x:A]. B[x],
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
prop: ℙ,
or: P ∨ Q,
subtype_rel: A ⊆r B,
guard: {T},
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a
Lemmas referenced :
geo-le-cases2,
geo-length_wf,
geo-mk-seg_wf,
double-negation-hyp-elim,
geo-lt_wf,
equal_wf,
geo-length-type_wf,
not_wf,
geo-gt_wf,
geo-congruent_wf,
geo-congruent-iff-length,
istype-void,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
basic-geometry-subtype,
subtype_rel_transitivity,
basic-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf,
geo-lt-implies-gt-strong
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
isectElimination,
setElimination,
rename,
hypothesis,
because_Cache,
sqequalRule,
unionEquality,
applyEquality,
independent_functionElimination,
unionElimination,
inrFormation_alt,
productElimination,
independent_isectElimination,
unionIsType,
universeIsType,
functionIsType,
equalityIstype,
voidElimination,
instantiate,
inhabitedIsType,
inlFormation_alt
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,d:Point. (\mneg{}\mneg{}((ab > cd \mvee{} cd > ab) \mvee{} ab \mcong{} cd))
Date html generated:
2019_10_16-PM-01_37_14
Last ObjectModification:
2019_07_08-PM-00_27_02
Theory : euclidean!plane!geometry
Home
Index