Nuprl Lemma : geo-congruent-iff-length
∀e:BasicGeometry. ∀[a,b,c,d:Point]. uiff(ab ≅ cd;|ab| = |cd| ∈ Length)
Proof
Definitions occuring in Statement :
geo-length: |s|,
geo-length-type: Length,
geo-mk-seg: ab,
basic-geometry: BasicGeometry,
geo-congruent: ab ≅ cd,
geo-point: Point,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
guard: {T},
subtype_rel: A ⊆r B,
prop: ℙ,
top: Top,
basic-geometry: BasicGeometry,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
geo-seg-congruent: geo-seg-congruent(e; s1; s2),
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
geo-length-type: Length,
quotient: x,y:A//B[x; y],
squash: ↓T,
sq_stable: SqStable(P)
Lemmas referenced :
geo-point_wf,
geo-length_wf,
geo-length-type_wf,
equal_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-congruent_wf,
geo_seg2_mk_seg_lemma,
geo_seg1_mk_seg_lemma,
geo-mk-seg_wf,
geo-seg-congruent-iff-length,
geo-length_wf1,
geo-length-equiv,
geo-eq_wf,
geo-X_wf,
geo-O_wf,
geo-between_wf,
quotient-member-eq,
member_wf,
sq_stable__geo-eq
Rules used in proof :
axiomEquality,
instantiate,
applyEquality,
independent_isectElimination,
productElimination,
voidEquality,
voidElimination,
isect_memberEquality,
because_Cache,
rename,
setElimination,
isectElimination,
independent_pairFormation,
isect_memberFormation,
sqequalRule,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
hypothesis,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut,
independent_functionElimination,
lambdaEquality,
setEquality,
equalitySymmetry,
equalityTransitivity,
productEquality,
pertypeElimination,
imageElimination,
baseClosed,
imageMemberEquality
Latex:
\mforall{}e:BasicGeometry. \mforall{}[a,b,c,d:Point]. uiff(ab \00D0 cd;|ab| = |cd|)
Date html generated:
2017_10_02-PM-04_52_44
Last ObjectModification:
2017_08_05-AM-09_10_27
Theory : euclidean!plane!geometry
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