Nuprl Lemma : geo-Aparallel_weakening2
∀g:EuclideanParPlane. ∀l,m:LINE.  ((l = m ∈ (l,m:Line//l || m)) 
⇒ l || m)
Proof
Definitions occuring in Statement : 
euclidean-parallel-plane: EuclideanParPlane
, 
geo-Aparallel: l || m
, 
geoline: LINE
, 
geo-line: Line
, 
quotient: x,y:A//B[x; y]
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
geo-Aparallel: l || m
, 
not: ¬A
, 
member: t ∈ T
, 
prop: ℙ
, 
uall: ∀[x:A]. B[x]
, 
euclidean-parallel-plane: EuclideanParPlane
, 
subtype_rel: A ⊆r B
, 
guard: {T}
, 
uimplies: b supposing a
, 
so_lambda: λ2x y.t[x; y]
, 
so_apply: x[s1;s2]
, 
quotient: x,y:A//B[x; y]
, 
false: False
, 
and: P ∧ Q
, 
cand: A c∧ B
Lemmas referenced : 
geo-intersect_wf, 
equal_wf, 
quotient_wf, 
geo-line_wf, 
euclidean-plane-structure-subtype, 
euclidean-plane-subtype, 
euclidean-planes-subtype, 
subtype_rel_transitivity, 
euclidean-parallel-plane_wf, 
euclidean-plane_wf, 
euclidean-plane-structure_wf, 
geo-primitives_wf, 
geo-Aparallel_wf, 
geoline-subtype1, 
geo-Aparallel-equiv, 
geoline-subtype2, 
geoline_wf, 
quotient_subtype_quotient, 
geo-Aparallel-equiv2, 
equal_functionality_wrt_subtype_rel2, 
member_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
setElimination, 
rename, 
because_Cache, 
hypothesis, 
hypothesisEquality, 
dependent_functionElimination, 
applyEquality, 
instantiate, 
independent_isectElimination, 
sqequalRule, 
lambdaEquality, 
equalityTransitivity, 
equalitySymmetry, 
independent_functionElimination, 
pertypeElimination, 
productElimination, 
voidElimination, 
productEquality
Latex:
\mforall{}g:EuclideanParPlane.  \mforall{}l,m:LINE.    ((l  =  m)  {}\mRightarrow{}  l  ||  m)
Date html generated:
2018_05_22-PM-01_11_38
Last ObjectModification:
2018_05_14-AM-11_56_37
Theory : euclidean!plane!geometry
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