Nuprl Lemma : geo-zero-length-iff
∀e:BasicGeometry. ∀a,b:Point.  (|ab| = 0 ∈ Length 
⇐⇒ a ≡ b)
Proof
Definitions occuring in Statement : 
geo-length: |s|
, 
geo-zero-length: 0
, 
geo-length-type: Length
, 
geo-mk-seg: ab
, 
basic-geometry: BasicGeometry
, 
geo-eq: a ≡ b
, 
geo-point: Point
, 
all: ∀x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
uimplies: b supposing a
, 
guard: {T}
, 
subtype_rel: A ⊆r B
, 
rev_implies: P 
⇐ Q
, 
basic-geometry: BasicGeometry
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
member: t ∈ T
, 
implies: P 
⇒ Q
, 
and: P ∧ Q
, 
iff: P 
⇐⇒ Q
, 
all: ∀x:A. B[x]
, 
top: Top
, 
geo-length: |s|
, 
geo-zero-length: 0
, 
cand: A c∧ B
, 
so_apply: x[s]
, 
so_lambda: λ2x.t[x]
, 
quotient: x,y:A//B[x; y]
, 
geo-length-type: Length
, 
squash: ↓T
, 
sq_stable: SqStable(P)
, 
so_apply: x[s1;s2]
, 
so_lambda: λ2x y.t[x; y]
Lemmas referenced : 
geo-point_wf, 
Error :basic-geo-primitives_wf, 
Error :basic-geo-structure_wf, 
basic-geometry_wf, 
subtype_rel_transitivity, 
basic-geometry-subtype, 
geo-eq_wf, 
geo-zero-length_wf, 
geo-mk-seg_wf, 
geo-length_wf, 
geo-length-type_wf, 
equal_wf, 
geo_seg2_mk_seg_lemma, 
geo_seg1_mk_seg_lemma, 
geo-between-trivial, 
geo-congruent_wf, 
subtype_rel_sets, 
geo-sep_wf, 
geo-sep-O-X, 
geo-X_wf, 
geo-O_wf, 
geo-between_wf, 
member_wf, 
sq_stable__geo-eq, 
set_wf, 
geo-Op-sep, 
geo-extend-property, 
geo-congruence-identity, 
geo-congruent_functionality, 
geo-eq_weakening, 
geo-between_functionality, 
geo-length-equiv, 
quotient-member-eq, 
geo-congruence-identity-sym, 
geo-eq_inversion
Rules used in proof : 
because_Cache, 
sqequalRule, 
independent_isectElimination, 
instantiate, 
applyEquality, 
rename, 
setElimination, 
hypothesis, 
hypothesisEquality, 
thin, 
isectElimination, 
sqequalHypSubstitution, 
extract_by_obid, 
introduction, 
cut, 
independent_pairFormation, 
lambdaFormation, 
sqequalReflexivity, 
computationStep, 
sqequalTransitivity, 
sqequalSubstitution, 
voidEquality, 
voidElimination, 
isect_memberEquality, 
dependent_functionElimination, 
equalitySymmetry, 
equalityTransitivity, 
productElimination, 
lambdaEquality, 
dependent_set_memberEquality, 
setEquality, 
productEquality, 
pertypeElimination, 
imageElimination, 
baseClosed, 
imageMemberEquality, 
independent_functionElimination, 
promote_hyp
Latex:
\mforall{}e:BasicGeometry.  \mforall{}a,b:Point.    (|ab|  =  0  \mLeftarrow{}{}\mRightarrow{}  a  \mequiv{}  b)
Date html generated:
2017_10_02-PM-04_53_08
Last ObjectModification:
2017_08_05-AM-09_10_39
Theory : euclidean!plane!geometry
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