Nuprl Lemma : geo-Aparallel-equiv
∀g:EuclideanParPlane. EquivRel(Line;l,m.l || m)
Proof
Definitions occuring in Statement :
euclidean-parallel-plane: EuclideanParPlane,
geo-Aparallel: l || m,
geo-line: Line,
equiv_rel: EquivRel(T;x,y.E[x; y]),
all: ∀x:A. B[x]
Definitions unfolded in proof :
trans: Trans(T;x,y.E[x; y]),
prop: ℙ,
sym: Sym(T;x,y.E[x; y]),
cand: A c∧ B,
uimplies: b supposing a,
guard: {T},
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
euclidean-parallel-plane: EuclideanParPlane,
member: t ∈ T,
refl: Refl(T;x,y.E[x; y]),
and: P ∧ Q,
equiv_rel: EquivRel(T;x,y.E[x; y]),
all: ∀x:A. B[x]
Lemmas referenced :
geo-Aparallel_transitivity,
geo-Aparallel_wf,
geo-Aparallel_inversion,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
euclidean-parallel-plane_wf,
subtype_rel_transitivity,
euclidean-planes-subtype,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-line_wf,
geoline-subtype1,
geo-Aparallel_weakening
Rules used in proof :
independent_isectElimination,
instantiate,
independent_functionElimination,
because_Cache,
sqequalRule,
isectElimination,
applyEquality,
hypothesis,
hypothesisEquality,
rename,
setElimination,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}g:EuclideanParPlane. EquivRel(Line;l,m.l || m)
Date html generated:
2018_05_22-PM-01_11_05
Last ObjectModification:
2018_05_21-AM-01_33_28
Theory : euclidean!plane!geometry
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