Nuprl Lemma : geo-gt-trans
∀e:EuclideanPlane. ∀a,b,c,d,x,y:Point. (ab > cd ⇒ cd > xy ⇒ ab > xy)
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-gt: cd > ab,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
geo-gt: cd > ab,
squash: ↓T,
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
sq_stable: SqStable(P),
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
basic-geometry: BasicGeometry,
uiff: uiff(P;Q),
cand: A c∧ B,
basic-geometry-: BasicGeometry-
Lemmas referenced :
sq_stable__geo-gt,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-gt_wf,
geo-point_wf,
geo-congruent-between-exists2,
geo-between-symmetry,
geo-between-sep,
geo-sep-sym,
geo-congruent-iff-length,
geo-between-inner-trans,
geo-between-exchange3,
geo-between-exchange4,
geo-between_wf,
geo-congruent_wf,
geo-sep_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
imageElimination,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
sqequalRule,
independent_functionElimination,
productElimination,
imageMemberEquality,
baseClosed,
universeIsType,
because_Cache,
inhabitedIsType,
dependent_functionElimination,
equalitySymmetry,
dependent_pairFormation_alt,
independent_pairFormation,
equalityTransitivity,
productIsType
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,d,x,y:Point. (ab > cd {}\mRightarrow{} cd > xy {}\mRightarrow{} ab > xy)
Date html generated:
2019_10_16-PM-01_17_07
Last ObjectModification:
2019_04_19-PM-03_34_21
Theory : euclidean!plane!geometry
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