Nuprl Lemma : geo-lt_transitivity2
∀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)
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-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 < q {}\mRightarrow{} q \mleq{} r {}\mRightarrow{} p < r)
Date html generated:
2017_10_02-PM-06_19_03
Last ObjectModification:
2017_08_05-PM-04_13_37
Theory : euclidean!plane!geometry
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