Nuprl Lemma : geo-add-length-is-zero
∀e:BasicGeometry. ∀x,y:Length.  (x + y = 0 ∈ Length 
⇐⇒ (x = 0 ∈ Length) ∧ (y = 0 ∈ Length))
Proof
Definitions occuring in Statement : 
geo-add-length: p + q
, 
geo-zero-length: 0
, 
geo-length-type: Length
, 
basic-geometry: BasicGeometry
, 
all: ∀x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
implies: P 
⇒ Q
, 
geo-length-type: Length
, 
quotient: x,y:A//B[x; y]
, 
geo-zero-length: 0
, 
geo-add-length: p + q
, 
cand: A c∧ B
, 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
prop: ℙ
, 
so_lambda: λ2x y.t[x; y]
, 
subtype_rel: A ⊆r B
, 
so_apply: x[s1;s2]
, 
uimplies: b supposing a
, 
basic-geometry: BasicGeometry
, 
euclidean-plane: EuclideanPlane
, 
so_lambda: λ2x.t[x]
, 
guard: {T}
, 
so_apply: x[s]
, 
respects-equality: respects-equality(S;T)
, 
squash: ↓T
, 
rev_implies: P 
⇐ Q
, 
true: True
Lemmas referenced : 
quotient-member-eq, 
geo-between_wf, 
geo-point_wf, 
geo-eq_wf, 
geo-length-equiv, 
geo-extend-property, 
geo-O_wf, 
subtype_rel_sets_simple, 
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-X_wf, 
geo-sep_wf, 
geo-Op-sep, 
geo-extend_wf, 
geo-congruent_wf, 
respects-equality-set, 
respects-equality-set-trivial2, 
geo-add-length_wf, 
geo-zero-length_wf, 
geo-length-type_wf, 
geo-between-symmetry, 
geo-between-inner-trans, 
geo-eq_inversion, 
geo-add-length-zero, 
geo-between_functionality, 
geo-eq_weakening, 
geo-congruent_functionality, 
geo-between-same, 
geo-congruence-identity, 
equal_wf, 
squash_wf, 
true_wf, 
istype-universe, 
subtype_rel_self, 
iff_weakening_equal
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation_alt, 
independent_pairFormation, 
cut, 
sqequalHypSubstitution, 
pointwiseFunctionalityForEquality, 
because_Cache, 
hypothesis, 
sqequalRule, 
pertypeElimination, 
promote_hyp, 
thin, 
productElimination, 
introduction, 
extract_by_obid, 
isectElimination, 
setEquality, 
hypothesisEquality, 
lambdaEquality_alt, 
universeIsType, 
applyEquality, 
setElimination, 
rename, 
independent_isectElimination, 
dependent_functionElimination, 
equalityTransitivity, 
equalitySymmetry, 
independent_functionElimination, 
inhabitedIsType, 
instantiate, 
dependent_set_memberEquality_alt, 
productIsType, 
equalityIstype, 
applyLambdaEquality, 
productEquality, 
imageMemberEquality, 
baseClosed, 
imageElimination, 
sqequalBase, 
setIsType, 
natural_numberEquality, 
universeEquality
Latex:
\mforall{}e:BasicGeometry.  \mforall{}x,y:Length.    (x  +  y  =  0  \mLeftarrow{}{}\mRightarrow{}  (x  =  0)  \mwedge{}  (y  =  0))
Date html generated:
2019_10_16-PM-01_16_38
Last ObjectModification:
2018_12_11-AM-11_56_37
Theory : euclidean!plane!geometry
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