Nuprl Lemma : geo-add-length-between
∀[e:BasicGeometry]. ∀[a,b,c:Point]. |ac| = |ab| + |bc| ∈ Length supposing B(abc)
Proof
Definitions occuring in Statement :
geo-add-length: p + q,
geo-length: |s|,
geo-length-type: Length,
geo-mk-seg: ab,
basic-geometry: BasicGeometry,
geo-between: B(abc),
geo-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
squash: ↓T,
sq_stable: SqStable(P),
uiff: uiff(P;Q),
basic-geometry-: BasicGeometry-,
iff: P ⇐⇒ Q,
stable: Stable{P},
geo-eq: a ≡ b,
false: False,
not: ¬A,
or: P ∨ Q,
cand: A c∧ B,
so_apply: x[s],
so_lambda: λ2x.t[x],
and: P ∧ Q,
top: Top,
geo-length: |s|,
geo-add-length: p + q,
implies: P ⇒ Q,
so_apply: x[s1;s2],
guard: {T},
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-length-type: Length,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
geo-eq_transitivity,
geo-between-same,
sq_stable__geo-congruent,
geo-congruence-identity-sym,
geo-congruent-iff-length,
geo-three-segment,
geo-congruent_functionality,
stable__geo-congruent,
geo-between-exchange4,
geo-between-exchange3,
geo-between-inner-trans,
geo-between-symmetry,
basic-geometry-_wf,
subtype_rel_self,
geo-construction-unicity,
minimal-not-not-excluded-middle,
geo-sep_functionality,
geo-eq_weakening,
geo-between_functionality,
minimal-double-negation-hyp-elim,
not_wf,
false_wf,
geo-Op-sep,
subtype_rel_sets_simple,
stable__not,
geo-congruent_wf,
geo-extend_wf,
geo-sep_wf,
geo-extend-property,
geo_seg2_mk_seg_lemma,
istype-void,
geo_seg1_mk_seg_lemma,
geo-sep-O-X,
geo-add-length_wf1,
geo-mk-seg_wf,
geo-length_wf1,
geo-length-equiv,
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-eq_wf,
geo-X_wf,
geo-O_wf,
geo-between_wf,
geo-point_wf,
quotient-member-eq
Rules used in proof :
imageElimination,
baseClosed,
imageMemberEquality,
independent_pairFormation,
promote_hyp,
unionElimination,
unionIsType,
functionIsType,
functionEquality,
unionEquality,
productEquality,
productElimination,
isectIsTypeImplies,
axiomEquality,
equalitySymmetry,
equalityTransitivity,
equalityIstype,
productIsType,
lambdaFormation_alt,
dependent_set_memberEquality_alt,
voidElimination,
isect_memberEquality_alt,
independent_functionElimination,
independent_isectElimination,
instantiate,
universeIsType,
setIsType,
inhabitedIsType,
lambdaEquality_alt,
rename,
setElimination,
dependent_functionElimination,
hypothesis,
because_Cache,
applyEquality,
hypothesisEquality,
setEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalRule,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[e:BasicGeometry]. \mforall{}[a,b,c:Point]. |ac| = |ab| + |bc| supposing B(abc)
Date html generated:
2019_10_29-AM-09_13_58
Last ObjectModification:
2019_10_18-PM-03_16_48
Theory : euclidean!plane!geometry
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