Nuprl Lemma : geo-le-null-segment
∀e:BasicGeometry. ∀[p:Length]. ∀[a:Point].  uiff(p ≤ |aa|;p = |aa| ∈ Length)
Proof
Definitions occuring in Statement : 
geo-le: p ≤ q
, 
geo-length: |s|
, 
geo-length-type: Length
, 
geo-mk-seg: ab
, 
basic-geometry: BasicGeometry
, 
geo-point: Point
, 
uiff: uiff(P;Q)
, 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
subtype_rel: A ⊆r B
, 
guard: {T}
, 
uimplies: b supposing a
, 
basic-geometry: BasicGeometry
, 
euclidean-plane: EuclideanPlane
, 
uiff: uiff(P;Q)
, 
and: P ∧ Q
, 
prop: ℙ
, 
squash: ↓T
, 
true: True
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
implies: P 
⇒ Q
, 
sq_stable: SqStable(P)
, 
geo-length-type: Length
, 
quotient: x,y:A//B[x; y]
, 
so_lambda: λ2x y.t[x; y]
, 
so_apply: x[s1;s2]
Lemmas referenced : 
subtype-geo-length-type, 
geo-point_wf, 
euclidean-plane-structure-subtype, 
euclidean-plane-subtype, 
basic-geometry-subtype, 
subtype_rel_transitivity, 
basic-geometry_wf, 
euclidean-plane_wf, 
euclidean-plane-structure_wf, 
geo-primitives_wf, 
geo-length-type_wf, 
geo-le_wf, 
geo-length_wf, 
geo-mk-seg_wf, 
equal_wf, 
geo-between-trivial, 
geo-O_wf, 
geo-X_wf, 
geo-between_wf, 
squash_wf, 
true_wf, 
geo-length-null-segment, 
iff_weakening_equal, 
sq_stable__geo-le, 
sq_stable__uiff, 
sq_stable__equal, 
geo-le_weakening, 
equal-wf-base, 
geo-eq_wf, 
geo-le_imp, 
quotient-member-eq, 
geo-length-equiv, 
geo-between-same, 
geo-eq_inversion
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation, 
isect_memberFormation, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
hypothesis, 
applyEquality, 
instantiate, 
independent_isectElimination, 
sqequalRule, 
setElimination, 
rename, 
because_Cache, 
independent_pairFormation, 
dependent_functionElimination, 
dependent_set_memberEquality, 
lambdaEquality, 
imageElimination, 
equalityTransitivity, 
equalitySymmetry, 
universeEquality, 
natural_numberEquality, 
imageMemberEquality, 
baseClosed, 
productElimination, 
independent_functionElimination, 
pointwiseFunctionalityForEquality, 
pertypeElimination, 
productEquality, 
setEquality, 
promote_hyp
Latex:
\mforall{}e:BasicGeometry.  \mforall{}[p:Length].  \mforall{}[a:Point].    uiff(p  \mleq{}  |aa|;p  =  |aa|)
Date html generated:
2017_10_02-PM-04_53_53
Last ObjectModification:
2017_08_17-PM-01_28_24
Theory : euclidean!plane!geometry
Home
Index