Nuprl Lemma : geo-length-equiv
∀e:BasicGeometry. EquivRel({p:Point| O_X_p} ;x,y.x ≡ y)
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-X: X,
geo-O: O,
geo-eq: a ≡ b,
geo-between: a_b_c,
geo-point: Point,
equiv_rel: EquivRel(T;x,y.E[x; y]),
all: ∀x:A. B[x],
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
equiv_rel: EquivRel(T;x,y.E[x; y]),
and: P ∧ Q,
refl: Refl(T;x,y.E[x; y]),
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
prop: ℙ,
sym: Sym(T;x,y.E[x; y]),
implies: P ⇒ Q,
trans: Trans(T;x,y.E[x; y]),
basic-geometry-: BasicGeometry-
Lemmas referenced :
geo-point_wf,
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-between_wf,
geo-O_wf,
geo-X_wf,
geo-eq_inversion,
geo-eq_wf,
geo-eq_transitivity,
subtype_rel_self,
basic-geometry-_wf,
geo-eq_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
setIsType,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
dependent_functionElimination,
setElimination,
rename,
inhabitedIsType
Latex:
\mforall{}e:BasicGeometry. EquivRel(\{p:Point| O\_X\_p\} ;x,y.x \mequiv{} y)
Date html generated:
2019_10_16-PM-01_14_58
Last ObjectModification:
2018_11_12-PM-03_30_14
Theory : euclidean!plane!geometry
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