Nuprl Lemma : geo-line-sep-or-lemma
∀g:EuclideanParPlane. ∀l,m,p:Line.
(((∀L,M,N:Line. (L \/ M ⇒ (L \/ N ∨ M \/ N))) ∧ l \/ m) ⇒ (geo-line-sep(g;p;l) ∨ geo-line-sep(g;p;m)))
Proof
Definitions occuring in Statement :
euclidean-parallel-plane: EuclideanParPlane,
geo-intersect: L \/ M,
geo-line-sep: geo-line-sep(g;l;m),
geo-line: Line,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q
Definitions unfolded in proof :
iff: P ⇐⇒ Q,
so_apply: x[s],
euclidean-parallel-plane: EuclideanParPlane,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
or: P ∨ Q,
member: t ∈ T,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
geo-line-sep_wf,
geo-intersect-symmetry,
geo-intersect-lines-iff,
or_wf,
geoline-subtype1,
geo-intersect_wf,
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,
all_wf
Rules used in proof :
inrFormation,
inlFormation,
rename,
setElimination,
functionEquality,
because_Cache,
lambdaEquality,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
isectElimination,
extract_by_obid,
introduction,
productEquality,
unionElimination,
independent_functionElimination,
hypothesisEquality,
dependent_functionElimination,
hypothesis,
cut,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}g:EuclideanParPlane. \mforall{}l,m,p:Line.
(((\mforall{}L,M,N:Line. (L \mbackslash{}/ M {}\mRightarrow{} (L \mbackslash{}/ N \mvee{} M \mbackslash{}/ N))) \mwedge{} l \mbackslash{}/ m)
{}\mRightarrow{} (geo-line-sep(g;p;l) \mvee{} geo-line-sep(g;p;m)))
Date html generated:
2018_05_23-PM-06_09_34
Last ObjectModification:
2018_05_23-PM-04_36_06
Theory : euclidean!plane!geometry
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