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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