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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