Nuprl Lemma : not-gt-and-lt
∀g:EuclideanPlane. ∀a,b,c,d:Point. (ab > cd ⇒ |ab| < |cd| ⇒ False)
Proof
Definitions occuring in Statement :
geo-lt: p < q,
geo-length: |s|,
geo-mk-seg: ab,
euclidean-plane: EuclideanPlane,
geo-gt: cd > ab,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
false: False
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
false: False,
member: t ∈ T,
uall: ∀[x:A]. B[x],
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
not: ¬A
Lemmas referenced :
geo-lt-implies-gt-strong,
geo-lt_wf,
geo-length_wf,
geo-mk-seg_wf,
geo-gt_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf,
geo-gt-irrefl,
geo-gt-trans
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
universeIsType,
isectElimination,
sqequalRule,
setElimination,
rename,
because_Cache,
applyEquality,
instantiate,
independent_isectElimination,
inhabitedIsType,
voidElimination
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,d:Point. (ab > cd {}\mRightarrow{} |ab| < |cd| {}\mRightarrow{} False)
Date html generated:
2019_10_16-PM-01_36_49
Last ObjectModification:
2019_08_08-PM-03_24_07
Theory : euclidean!plane!geometry
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