Nuprl Lemma : geo-lt_transitivity
∀e:BasicGeometry. ∀p,q,r:Length.  (p ≤ q 
⇒ q < r 
⇒ p < r)
Proof
Definitions occuring in Statement : 
geo-lt: p < q
, 
geo-le: p ≤ q
, 
geo-length-type: Length
, 
basic-geometry: BasicGeometry
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
Definitions unfolded in proof : 
so_apply: x[s]
, 
so_lambda: λ2x.t[x]
, 
uimplies: b supposing a
, 
guard: {T}
, 
basic-geometry: BasicGeometry
, 
subtype_rel: A ⊆r B
, 
uall: ∀[x:A]. B[x]
, 
cand: A c∧ B
, 
prop: ℙ
, 
and: P ∧ Q
, 
member: t ∈ T
, 
exists: ∃x:A. B[x]
, 
geo-lt: p < q
, 
implies: P 
⇒ Q
, 
all: ∀x:A. B[x]
, 
geo_ge: geo_ge(e; p; q)
, 
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced : 
geo-length-type_wf, 
geo-lt_wf, 
Error :basic-geo-primitives_wf, 
Error :basic-geo-structure_wf, 
basic-geometry_wf, 
subtype_rel_transitivity, 
basic-geometry-subtype, 
geo-point_wf, 
exists_wf, 
geo-mk-seg_wf, 
geo-length_wf, 
geo-add-length_wf, 
geo-le_wf, 
geo-sep_wf, 
geo-le_weakening, 
geo-add-length_functionality_wrt_le, 
geo-le_functionality_wrt_implies
Rules used in proof : 
lambdaEquality, 
independent_isectElimination, 
instantiate, 
rename, 
setElimination, 
sqequalRule, 
because_Cache, 
applyEquality, 
isectElimination, 
extract_by_obid, 
introduction, 
cut, 
productEquality, 
hypothesis, 
promote_hyp, 
independent_pairFormation, 
hypothesisEquality, 
dependent_pairFormation, 
thin, 
productElimination, 
sqequalHypSubstitution, 
lambdaFormation, 
sqequalReflexivity, 
computationStep, 
sqequalTransitivity, 
sqequalSubstitution, 
equalitySymmetry, 
equalityTransitivity
Latex:
\mforall{}e:BasicGeometry.  \mforall{}p,q,r:Length.    (p  \mleq{}  q  {}\mRightarrow{}  q  <  r  {}\mRightarrow{}  p  <  r)
Date html generated:
2017_10_02-PM-06_18_58
Last ObjectModification:
2017_08_05-PM-04_13_33
Theory : euclidean!plane!geometry
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