Nuprl Lemma : geo-add-length-comm
∀[e:BasicGeometry]. ∀[x,y:Length]. (x + y = y + x ∈ Length)
Proof
Definitions occuring in Statement :
geo-add-length: p + q,
geo-length-type: Length,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uiff: uiff(P;Q),
stable: Stable{P},
false: False,
not: ¬A,
or: P ∨ Q,
iff: P ⇐⇒ Q,
basic-geometry-: BasicGeometry-,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
prop: ℙ,
so_apply: x[s],
uimplies: b supposing a,
guard: {T},
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
euclidean-plane: EuclideanPlane,
basic-geometry: BasicGeometry,
geo-add-length: p + q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
geo-eq: a ≡ b,
and: P ∧ Q,
quotient: x,y:A//B[x; y],
geo-length-type: Length,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
geo-congruence-identity-sym,
geo-three-segment,
geo-length-flip,
geo-congruent-iff-length,
geo-congruent-symmetry,
geo-sep-O-X,
geo-construction-unicity,
minimal-not-not-excluded-middle,
minimal-double-negation-hyp-elim,
istype-void,
not_wf,
false_wf,
stable__not,
geo-congruent_functionality,
geo-eq_weakening,
geo-between_functionality,
geo-congruent_wf,
geo-between-exchange4,
basic-geometry-_wf,
subtype_rel_self,
geo-between-exchange3,
geo-between-inner-trans,
geo-between-symmetry,
geo-length-equiv,
geo-eq_wf,
quotient-member-eq,
geo-extend_wf,
geo-Op-sep,
geo-sep_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-X_wf,
geo-between_wf,
geo-point_wf,
subtype_rel_sets_simple,
geo-O_wf,
geo-extend-property
Rules used in proof :
voidElimination,
unionElimination,
unionIsType,
functionIsType,
functionEquality,
unionEquality,
isectIsTypeImplies,
axiomEquality,
isect_memberEquality_alt,
applyLambdaEquality,
sqequalBase,
equalityIstype,
productIsType,
independent_functionElimination,
setIsType,
setEquality,
dependent_set_memberEquality_alt,
independent_isectElimination,
instantiate,
universeIsType,
lambdaEquality_alt,
isectElimination,
applyEquality,
setElimination,
hypothesisEquality,
dependent_functionElimination,
extract_by_obid,
lambdaFormation_alt,
inhabitedIsType,
equalitySymmetry,
equalityTransitivity,
rename,
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:Length]. (x + y = y + x)
Date html generated:
2019_10_29-AM-09_13_43
Last ObjectModification:
2019_10_18-PM-03_16_55
Theory : euclidean!plane!geometry
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