Nuprl Lemma : geo-le_transitivity
∀e:BasicGeometry. ∀[p,q,r:Length]. (p ≤ r) supposing (q ≤ r and p ≤ q)
Proof
Definitions occuring in Statement :
geo-le: p ≤ q,
geo-length-type: Length,
basic-geometry: BasicGeometry,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
sq_stable: SqStable(P),
implies: P ⇒ Q,
geo-length-type: Length,
prop: ℙ,
quotient: x,y:A//B[x; y],
and: P ∧ Q,
subtype_rel: A ⊆r B,
guard: {T},
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
basic-geometry-: BasicGeometry-,
squash: ↓T
Lemmas referenced :
sq_stable__geo-le,
subtype-geo-length-type,
geo-le_wf,
geo-le_witness,
geo-le_imp,
geo-between-symmetry,
geo-X_wf,
geo-between-inner-trans,
geo-between-exchange3,
subtype_rel_self,
euclidean-plane-structure_wf,
basic-geo-axioms_wf,
euclidean-plane-structure-subtype,
geo-left-axioms_wf,
geo-between-exchange4,
equal-wf-base,
geo-eq_wf,
euclidean-plane-subtype,
basic-geometry-subtype,
subtype_rel_transitivity,
basic-geometry_wf,
euclidean-plane_wf,
geo-primitives_wf,
geo-length-type_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_functionElimination,
because_Cache,
pointwiseFunctionalityForEquality,
sqequalRule,
pertypeElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
applyEquality,
independent_isectElimination,
dependent_functionElimination,
setElimination,
rename,
lambdaEquality,
instantiate,
setEquality,
productEquality,
cumulativity,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}e:BasicGeometry. \mforall{}[p,q,r:Length]. (p \mleq{} r) supposing (q \mleq{} r and p \mleq{} q)
Date html generated:
2017_10_02-PM-04_52_31
Last ObjectModification:
2017_08_17-PM-01_40_26
Theory : euclidean!plane!geometry
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