Nuprl Lemma : geo-add-length-cancel-left
∀[e:BasicGeometry]. ∀[x,y,z:Length]. x = y ∈ Length supposing z + x = z + y ∈ Length
Proof
Definitions occuring in Statement :
geo-add-length: p + q,
geo-length-type: Length,
basic-geometry: BasicGeometry,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
uiff: uiff(P;Q),
so_apply: x[s],
so_lambda: λ2x.t[x],
guard: {T},
basic-geometry-: BasicGeometry-,
implies: P ⇒ Q,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
prop: ℙ,
euclidean-plane: EuclideanPlane,
basic-geometry: BasicGeometry,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
geo-add-length: p + q,
cand: A c∧ B,
and: P ∧ Q,
quotient: x,y:A//B[x; y],
geo-length-type: Length,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
geo-eq_weakening,
geo-congruent_functionality,
geo-congruent-refl,
geo-congruent-iff-length,
geo-sep-O-X,
geo-construction-unicity,
geo-congruent_wf,
geo-extend_wf,
geo-Op-sep,
geo-sep_wf,
subtype_rel_sets_simple,
geo-extend-property,
geo-length-type_wf,
geo-add-length_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-add-length_wf1,
basic-geometry-_wf,
subtype_rel_self,
geo-eq_transitivity,
geo-length-equiv,
geo-eq_wf,
geo-X_wf,
geo-O_wf,
geo-between_wf,
geo-point_wf,
quotient-member-eq
Rules used in proof :
dependent_set_memberEquality_alt,
lambdaFormation_alt,
isectIsTypeImplies,
axiomEquality,
isect_memberEquality_alt,
setIsType,
sqequalBase,
universeIsType,
equalityIstype,
productIsType,
applyLambdaEquality,
instantiate,
independent_functionElimination,
independent_isectElimination,
equalitySymmetry,
equalityTransitivity,
inhabitedIsType,
lambdaEquality_alt,
rename,
setElimination,
dependent_functionElimination,
applyEquality,
hypothesisEquality,
setEquality,
isectElimination,
extract_by_obid,
productElimination,
thin,
promote_hyp,
pertypeElimination,
sqequalRule,
hypothesis,
because_Cache,
pointwiseFunctionalityForEquality,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[e:BasicGeometry]. \mforall{}[x,y,z:Length]. x = y supposing z + x = z + y
Date html generated:
2019_10_29-AM-09_14_12
Last ObjectModification:
2019_10_18-PM-03_16_39
Theory : euclidean!plane!geometry
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